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Analytic Geometry Formulas

Don Peterson [email protected]

Version: 6 Feb 2011

This document is a collection of formulas from analytic geometry. ince !ectors are used a lot for

conciseness" there are also a fe# formulas from !ector analysis.$f you find an error or ob!ious omission" %lease notify me at the abo!e email address and it #ill getfi&ed in a subse'uent re!ision.

Document maintainer: ma(e the )hec( %aragra%h style !isible to see document maintenanceinstructions. *ritten in +%en +ffice ,.2.1.

Table of Contents

)o%yright.....................................................................................................................................-otation.......................................................................................................................................-Plane geometry...........................................................................................................................-

/rea of a triangle.....................................................................................................................-'uation of a line....................................................................................................................

Polar coordinates................................................................................................................6 /ngle bet#een t#o lines.........................................................................................................6Distance from a line to a %oint ...............................................................................................3isectors of angles bet#een t#o lines....................................................................................'uation of circle....................................................................................................................

Polar e'uation of circle.......................................................................................................'uation of %arabola...............................................................................................................4'uation of elli%se...................................................................................................................5'uation of hy%erbola...........................................................................................................11

)onic sections.......................................................................................................................12)onic sections #ith matri& notation......................................................................................1,)ircle through three %oints ...................................................................................................1Tangent to a circle................................................................................................................1

/ngle of intersection of t#o circles.......................................................................................16)enter of triangles inscribed circle.......................................................................................1uclidean Transformations...................................................................................................1

Translations......................................................................................................................17otations...........................................................................................................................1

+bli'ue transformations........................................................................................................1+bli'ue coordinates..........................................................................................................14

/rchimedean s%iral...............................................................................................................14Products....................................................................................................................................14

Pseudo!ectors and %seudoscalars.......................................................................................14calar 8dot9 %roduct..............................................................................................................14Vector 8cross9 %roduct..........................................................................................................15calar tri%le %roduct..............................................................................................................20

Three nonco%lanar !ectors as a base 8reci%rocal system9..............................................21Vector tri%le %roduct..............................................................................................................22

Direction cosines.......................................................................................................................22

mailto:[email protected]

mailto:[email protected]



'uation of a line......................................................................................................................2,Forms....................................................................................................................................2,

Vector ...............................................................................................................................2,nd%oints of t#o !ectors..................................................................................................2,Parametric........................................................................................................................2,ymmetric.........................................................................................................................2,

T#o%oint form..................................................................................................................2,'uation from t#o %oints on line...........................................................................................2-From t#o %lanes...................................................................................................................2-

'uation of a %lane...................................................................................................................2-;eneral form.........................................................................................................................2-

<essian normal form........................................................................................................2-Dual form..........................................................................................................................2

From a&ial interce%ts.............................................................................................................26Through origin and %arallel to t#o !ectors............................................................................26From three noncollinear %oints.............................................................................................26Through a %oint and %er%endicular to a !ector .....................................................................26From t#o simultaneous e'uations: %ro=ecting %lanes.........................................................2

To find the %ro=ecting %lane of a gi!en line.......................................................................2Points........................................................................................................................................2

)ollinear %oints.....................................................................................................................2)o%lanar %oints.....................................................................................................................24

>ines and Planes.......................................................................................................................24Point di!iding a line segment into a gi!en ratio....................................................................24>ines in s%ace.......................................................................................................................24>ine in a %lane.......................................................................................................................24>ine through a %oint %arallel to a !ector ...............................................................................25$ntersection of t#o lines........................................................................................................25$ntersection of t#o %lanes.....................................................................................................25$ntersection of three %lanes..................................................................................................,0$ntersection of four %lanes....................................................................................................,0$ntersection of a line and %lane.............................................................................................,0

/ngle bet#een t#o lines in s%ace........................................................................................,1 /ngle bet#een a line and a %lane........................................................................................,1 /ngle bet#een t#o %lanes....................................................................................................,1Parallel %lanes......................................................................................................................,1Per%endicular %lanes............................................................................................................,1Plane through a %oint %arallel to t#o !ectors........................................................................,1Plane through a %oint %arallel to another %lane....................................................................,2Plane through a line %arallel to another line.........................................................................,2Plane through a gi!en %oint and line....................................................................................,2Plane through a %oint and normal to a !ector ......................................................................,2>ine %er%endicular to a %lane containing three %oints..........................................................,2>ine %er%endicular to t#o lines.............................................................................................,,Distance of a %lane to the origin...........................................................................................,,Distance of a %oint to a line..................................................................................................,,Distance of a %oint from a %lane...........................................................................................,,Distance bet#een t#o nonintersecting lines.......................................................................,,

)ircle in s%ace...........................................................................................................................,-



Vectors......................................................................................................................................,-First degree !ector e'uation.................................................................................................,-$n!ariance.............................................................................................................................,

%heres.....................................................................................................................................,?iscellaneous............................................................................................................................,6

)enter of mass......................................................................................................................,6

Pro=ections............................................................................................................................,6$sometric...........................................................................................................................,6Pro=ection angles..............................................................................................................,ine stic(s.........................................................................................................................,4

7otations...............................................................................................................................,57otating about a gi!en a&is..............................................................................................-0Direction cosine matri& to uler angles............................................................................-0Direction cosine matri& to uler a&is and angle...............................................................-1

7otating lines in s%ace..........................................................................................................-1<eli&......................................................................................................................................-2

/reas and Volumes...................................................................................................................-2Volume of a tetrahedron.......................................................................................................-2

>ayout........................................................................................................................................-,)utting %i%e ends for #elding...............................................................................................-,Di!iding a line e'ually...........................................................................................................-->ine segment %er%endicular bisector ....................................................................................-Per%endicular to a %oint interior to a line segment...............................................................-Per%endicular at end of line segment...................................................................................-63isecting an angle.................................................................................................................-6>aying out right angles..........................................................................................................-6)o%ying an e&isting angle.....................................................................................................-Di!iding an angle..................................................................................................................->aying out angles..................................................................................................................-4

sing tangents..................................................................................................................-460A....................................................................................................................................0-A....................................................................................................................................0,0A" 1A.............................................................................................................................0+ther s%ecial angles.........................................................................................................0

Triangles...............................................................................................................................1$nscribed circle..................................................................................................................1)ircumscribed circle.........................................................................................................1

lli%se....................................................................................................................................1Piece of string and t#o foci...............................................................................................1)alculated %oints..............................................................................................................2sing a mar(ed stic(........................................................................................................2

/%%ro&imate lli%se #ith circles.......................................................................................,7egular %olygons..................................................................................................................

/rcs.......................................................................................................................................6Ta%ered bo&es......................................................................................................................6

7eferences................................................................................................................................



Copyright

)o%yright B 2010 Don Peterson

Permission to use" co%y" modify" distribute and sell this document for any %ur%ose is herebygranted #ithout fee" %ro!ided that the abo!e co%yright notice" this %aragra%h" and the follo#ing%aragra%h a%%ear unchanged in all co%ies.

The author is identified by an </26 hash of the follo#ing information: the string CDonPetersonC" a number familiar to the author" and a %ass#ord used by the author are concatentatedand %ut into a $style file. The resulting hash is2,ae--4fb1f,a-a-fdbd2a-e5,a1d0--225ae1-0c5a-,411e626620fc.

Notation

/ !ector is gi!en in bold: AE the magnitude of A is A. /ll scalars are real unless other#ise stated.

r is the %osition !ector of a general %oint in s%ace 8&" y" G9" r 1 8&1" y1" G19" etc.

/ caret denotes a unit !ector and is gotten by di!iding a !ector by its magnitude:

v = v

v

The usual )artesian unit !ectors are i , j , k .

n denotes a normal !ectorE #hat it is normal to #ill be clear from the conte&t.

$n %arametric e'uations" s" t" and u are %arameters and s ,t ,u∈ℝ .

= , , #here " " and are the three direction cosines of a direction in s%ace.

The >e!i)i!ita symbol is

ijk =1 if 8i"="(9 is 81"2",9" 8,"1"29" or 82","19 8e!en %ermutation9−1 if 8i"="(9 is 81","29" 8,"2"19" or 82"1",9 8odd %ermutation9

0 other#iseor

i = 1 i = 2 i = ,

∣0 0 0

0 0 10 −1 0∣ ∣0 0 1

0 0 01 0 0∣ ∣0 1 0

−1 0 00 0 0∣

*hen the >e!i)i!ita symbol is used" the usual instein summation con!ention is assumed o!erthe three s%acial indices 1" 2" and ,.

ijk klm =il jm−i m jl

HnI is a reference to reference n. Hn:mI is a reference to %age m in reference n. Hn:m:%I uses % to

%oint to the %age number in the PDF for documents that are accessible in PDF format.

Plane geometry

Area of a triangle

$f a" b" and c are the !ectors of the sides of a triangle" then H#ilson:41:104I

abc = 0

and the area / of the triangle is H#ilson:41:104I



2 A =∣a×b∣=∣b×c ∣=∣a×c ∣=

a2sinB sinC

sin A =

b2sin A sinC

sin B =

c 2sin A sinB

sinC

These relationshi%s hold in three dimensions also as long as the !ectors form a closed %olygon.

Equation of a line

;i!en 8&1" y19 and 8&2" y29 on a line" the e'uation of the line is Hschmall:-5:6-Iy −y 1 x − x 1

= y 2−y 1 x 2− x 1

#hich can be #ritten

∣ x y 1

x 1 y 1 1

x 2 y 2 1∣= 0

The symmetrical form in terms of the interce%ts of the a&es is Hschmall:0:6I

x

a

y

b =1

*ith a slo%e m and the y interce%t of b" the e'uation is

y = mx b

The slo%e m is the tangent of the angle bet#een the line and the & a&is. The slo%e of the line

%er%endicular to this line is −1

m.

The %ointslo%e form is

y −y 1 = m x − x 1

The normal form of the e'uation is Hschmall:,:64I

x cosy sin= p

#here p is the distance of the line from the origin and is the angle of the %er%endicular to the line

from the & a&is. The normals angle #ith the & a&is #ill be from J2 to ,J2.



This can be re#ritten as

y =− x

tan

p

sin

#hich sho#s that the y interce%t is b = p

sin.

/ny first degree e'uation in t#o !ariables re%resents a straight line

Ax By C = 0

These are ob!ious sim%lifications of the corres%onding e'uations for three dimensions.

Polar coordinates

3y con!erting the slo%einterce%t form of the e'uation of a line" #e get the %olar form

sin−m cos = b

$n %olar coordinates" the e'uation of a straight line bet#een t#o %oints 81" 19 and 82" 29 isHschmall:66:41I

sin1−2

sin2−

1

sin−1

2

= 0

or

1

cos sin

1

1

cos 1 sin1

1

2

cos 2 sin2

= 0

Angle between two lines

$f y m1& K b1 and y m2& K b2" #e ha!e Hschmall:60:I

tan = m1− m2

1m1m2

&

y

%



Distance from a line to a point

$f the e'uation of the line is x cosy sin = p , the distance from the %oint r 1 is Hschmall:6,:4I

∣ x 1 cosy 1 sin− p∣

$f the e'uation of the line is /& K 3y K ) 0 and the %oint is 8&" y9" the distance is Hschmall:6,:4I

d =

Ax By C

A2B2

Bisectors of angles between two lines

;i!en the lines /& K 3y K ) 0 and /1& K 31y K )1 0. The %oint 8&" y9 on the bisector isHschmall:6-:5I

Ax By C

A2B

2 = ±

A1 x B1 y C 1

A1

2B1

2

The K sign is used if the bisector and the origin lies #ithin the same one of the four %ossible angles.

$f the e'uations of the lines are

x cosy sin= p x cos1y sin1 = p1

then the e'uation of the bisector is

x cos1±cosy sin1±sin= p1± p

Equation of circle

/ circle #ith center at 8&0" y09 and radius r has the e'uation

x − x 02y −y 0

2 = r 2

!ery e'uation of the form &2 K y2 K 2a& K 2by K c 0 re%resents a circle #ith center at 8a" b9 and

radius a2b

2−c Hschmall:4:102I.

The e'uation of the tangent line to a circle at %oint x 1" y 1 is Hschmall:,21:,I

x x 1y y 1 = r 2

Polar equation of circle

$f the circles center is 81" 19 and the radius is r" then the e'uation is Hschmall:11:1,1I

2−2 1 cos−11

2−r 2 = 0

For the circle &2 K y2 K 2;& K 2Fy K ) 0" the %olar form is Hschmall:115:1,-I

22Gcos F sin = 0

#here the %ole is on the circle 8i.e." ) 09.



Equation of parabola

The e'uation is y

2

=-a&. 3ecauseFP = MP

is al#ays true and=1

for the %arabola" the focus isat a distance a from the origin on the & a&is" the distance % a. For any t#o %oints x 1" y 1 and

x 2" y 2 on the %arabola" #e ha!ey 1

2

y 22 =

x 1 x 2

. The tangent at P bisects the angle FP?. >7 is the

latus rectum and its length is -a.

The e'uation of the tangent line to the %arabola at x 1" y 1 is Hschmall:1,:12I

y y 1 = 2a x x 1

;i!en a %oint x 1" y 1 #e ha!e that there is t#o distinct" one" or no lines tangent to the %arabola

corres%onding to #hether the %oint is outside" on" or inside of the %arabola. The slo%e8s9 of theline8s9 are Hschmall:1-,:14I

m = y 1± y 1

2−-a&1

2&1

T#o tangents to the %arabola that are %er%endicular to each other intersect on the directri&Hschmall:1--:15I.

The line y = mx a

m is tangent to the %arabola for all nonGero !alues of m Hschmall:1,4:1,I.

The %olar e'uation for a righto%ening %arabola #ith the origin at the focus is Hschmall:1,:164I

r =2a

1−cos

*ith the origin at the !erte&" the %olar e'uation is Hschmall:1,:164I

r =-acos

sin2

D i r e c t r i &

+

y

&% F8a" 09

P8&" y9?

>

7



Equation of ellipse

/lso see the section Ellipse on %age 1. Hhtt%:JJen.#i(i%edia.orgJ#i(iJlli%se I

F 1 and F 2 are called the foci and ha!e the %ro%erty that if a light ray is emitted at one foci" it #ill bereflected to the other foci if the inside of the elli%se is %erfectly reflecting. $t is also true that the

distance F 1P F 2P is a constant for any %oint P on the elli%se.

The general e'uation of an elli%se is

Ax 2Bxy Cy

2Dx Ey F = 0

%ro!ided F ≠0 and F B2−-/) 0E this elli%se is at an angle to the a&es. $f 3 0 and /) L 0"

then the elli%ses a&es are %arallel to the coordinate a&es.

/nother form is

Ax 2Bxy Cy

2Dx Ey = 1

#ith B2−-/) 0.

3y a translation and rotation" the elli%ses e'uation can be %ut in canonical form 8elli%ses center atorigin and the ma=or diameter lies along the & a&is9:

x

a 2

y

b 2

= 1

The ma=or diameter is 2a and the minor diameter is 2b. This e'uation can also be #ritten

y = ±b 1− x

a 2

=± a2− x

21−

2

The foci are 8a" 09 and 8a" 09 #here is the eccentricity

= 1−b

a 2

/ %oint is outside" on" or inside the elli%se if x /a2 y /b2 is L 1" 1" or M 1" res%ecti!ely.

The e'uation of a tangent line #ith nonGero slo%e m to the elli%se is y = mx ± am2b

2.

The distance to the directri&es from the origin is aJ. FN1 7 is called the latus rectum and is2b

2

a .

Figure 1

/ 3

)

b

+F1

F2

P

aa

7

http://en.wikipedia.org/wiki/Ellipse





The distance from the center to any focus is a2−b

2. The dashed line is the directri& 8theres one

at the other end of the elli%se also9 and is a distancea

from the center.

/n elli%se centered at 8h" (9 and #ith the ma=or a&is %arallel to the & a&is has the e'uation

x −h2

a2

y −k 2

b2 = 1

#hich can also be gi!en in %arametric form 8−≤t ≤9

x = hacos t y = k bsint

/ general %arametric form is

x = x 0acost cos−bsint sin

y = y 0acos t sin−bsint cos

#here 0≤ t ≤ 2

8&0" y09 is the center of the elli%se and is the angle bet#een the & a&is and the ma=or diameter a&isof the elli%se. For an elli%se in canonical %osition 8center at origin" ma=or diameter along the &

a&is9" the e'uation is x = acos t y = b sint

$n %olar coordinates #ith the origin at the center of the elli%se and the %olar angle measured fromthe ma=or diameter a&is" the e'uation is

r = ab

asin2bcos 2 =

b

1−2cos

2

$f instead the origin is translated to a focus" then the other focus is rotated to an angle " the %olare'uation is

r = a1−

1±cos−

The general %olar form is

r = P Q

R

#here

P = r 0 [b2−a2cos0−2a2b

2cos−0 ]Q = 2ab R −2r 0

2sin

2−0

R = b2−a

2cos2−2a

2b

2

and the center of the elli%se is at 8r 0" 09" the ma=or diameter is 2a" the minor diameter is 2b" and thema=or diameter a&is is rotated by relati!e to the %olar a&is.

The circumference of the elli%se is -a #here is the com%lete elli%tic integral of the second(ind. / %o#er series is



C = 2a∑n=0

{−[!m=1

n

2m−1

2m ]2

2n

2n−1 "= 2a[1−1

2 2

2−1#,

2#- 2-

, −1#,#

2#-#6 26

−$]

Equation of hyperbola

The basic %ro%erty of the hy%erbola is that MP = FP . The e'uation of the asym%totes are

y =±b

a x .

The hy%erbola is re%resented by

Ax 2Bxy Cy

2Dx Ey F = 0

#here B2 -/). For the hy%erbolas o%ening in the & direction sho#n in the figure" the e'uation is

x 2−y 2 = 1 . For hy%erbolas o%ening in the y direction" the e'uation is y 2− x 2 = 1 . From here on"

the formulas are for a hy%erbola o%ening in the & direction. For a hy%erbola centered at 8h" (9" thee'uation is

x −h2

a2 −

y −k 2

b2 = 1

The eccentricity is = 1b

a 2

. The foci are at 8h K c" (9 #here c 2 = a

2b2. The directri& is

located at & aJ.

The %olar e'uation is r 2=

a

cos2 #here a is a constant. $f the origin is at the left focus" the

b

a

directri&

F

+

D

?P



e'uation is r = a 2−1

cos −1 Hschmall:216:2,1I.

Parametric e'uations are

x = a

cos t h

y = b tant k

or x =±acosh t hy =bsinh t k

The e'uation of the tangent to the hy%erbola at x 1"y x is Hschmall:202:21I

x x 1

a2 −

y y 1

b2 = 1

The line y = m x ± ma2−b

2 #ith nonGero slo%e m is tangent to the hy%erbola for all !alues of m

Hschmall:20,:214I. The tangent lines to the con=ugate hy%erbola are y = m x ± b2−ma

2.

;i!en the asym%totes A1 x B1 y C 1 = 0 and A2 x B2 y C 2 = 0" the e'uation of the hy%erbola is

A1 x B1 y C 1 A2 x B2 y C 2 = k #here ( is some constant Hschmall:211:226I. The area of the

triangle formed by any tangent and the asym%totes is constant Hschmall:216:2,1I.

Conic sectionshtt%:JJen.#i(i%edia.orgJ#i(iJ)onicNsection

/ conic section is the intersection of a %lane #ith a right circular cone. The conic section isre%resented by the general algebraic e'uation of degree 2. )onic sections ha!e the geometrical%ro%erty that the horiGontal distance DP is e'ual to the distance FP for any %oint P on the conicsection. The %oint F is the focus.

)onics ha!e %olar e'uation r = /1± cos #here > is the semilatus rectum and is theeccentricity.

Name Equationa c F! " F# p EF

circle x 2y 2=a2 0 0 a

D

F

P

+

7

Directri&

;

?

http://en.wikipedia.org/wiki/Conic_section

http://en.wikipedia.org/wiki/Conic_section



Name Equationa c F! " F# p EF

elli%se x /a2y /b2=1 1−b/a2

a2−b

2 b2 /a b

2/ a

2−b

2

%arabola y 2 = -a& 1 a 2a 2a

hy%erbola x /a2−y /b2=1 1b/a2

a2b

2 b2/a b2/ a

2b

2

a

The origin is at the !erte&.ote the relations p = and a! = c .

For the case #here the origin is at a focus" #e ha!e 8 = a2b

2" =a2−b

29

Name $%a& y%a& Polar Cartesian

circle acos t asin t r = a x 2y

2=a2

elli%se acos t b sin t b2/a− cos x − /a

2y / b2 = 1

%arabola a t 2 2at 2a /1−cos y

2 = -a x a

hy%erbola a /cost b tant b2/a− cos x / a2− y /b 2 = 1

8a9 Parametric e'uations.

Fi!e %oints determine a conic if no three are collinearE i.e." there is one uni'ue conic %assingthrough them.

The e'uation Ax 2Bxy Cy

2Dx Ey F = 0 can be rearranged to gi!e

Ax 2Bxy Cy 2 =−Dx Ey F

#hich sho#s that the 'uadratic form " = Ax 2Bxy Cy

2 and the %lane " = −Dx Ey F intersect to yield a conic section. Parabolas and hy%erbolas are gotten by a horiGontal %lane 8D = E =09 #hile elli%ses re'uire the %lane to intersect the &y a&is at an angle. Degenerate conicscorres%ond to degenerate intersections.

Conic sections with matri$ notation

htt%:JJen.#i(i%edia.orgJ#i(iJ?atri&Nre%resentationNofNconicNsections

Directri&

0 M e M 1 elli%se

e 0 circle

e 1 %arabola

e L1 hy%erbola

http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections

http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections



)onic sections in the )artesian %lane are re%resented by the e'uation

ax 2bxy cy

2dx !y # = 0

or in homogeneous coordinates as r TOr 0 #here O is the symmetric matri&

r =

x

y

1

and Q=

a b /2 d / 2

b /2 c ! /2

d / 2 ! /2 #

>et M = a b / 2

b/ 2 c , D det8?9" $ = c ! / 2

!/ 2 # , and det89.

$f D 0" the conic is degenerate:

D 0 : t#o intersecting linesD = 0 : t#o 8%ossibly coincident9 %arallel straight lines

D 0 : em%ty

$f D is not Gero" the conics can be classified by the determinant of the minor O11:

E 0 : hy%erbolaE = 0 : %arabola

E 0 : elli%se 8a circle if Q11=Q22

The center of the conic is

x c = b!−2cd

-D

y c =

db−2ae

-D

The ma=or and minor a&es %ass through the center of the conic and their directions are theeigen!ectors of ?.

Through a %oint %" there are generally t#o tangents. The %oints of tangency are the intersection ofthe conic #ith the line pTOr 0. *hen p is on the conic" the line is tangent there. *hen p is insidean elli%se" the line is the set of all %oints #hose o#n associated line %asses through p.

To #rite the canonical form of the e'uation of the conic" the conic is translated and rotated so that

the conics center is at the origin and its a&es are %arallel to the coordinate a&es. >et %1 and %2 bethe eigen!alues of ?. *e ha!e

%1 x % 2%2 y %

2

D

d!t M = 0

#here the coordinates are



Di!ide through by DJdet8?9 to get the canonical form.

Circle through three points

For each %oint" #rite the e'uation &2 K y2 K 2;& K 2Fy K ) 0. This re%resents three e'uations inthree un(no#ns ;" F" and ). ol!e for them. 'ui!alently" use the e'uation 8& h92 K 8y (92 r 2

and sol!e for the constants.

For the three %oints 8&1" y19" 8&2" y29" 8&," y,9" the e'uation is Hschmall:5:112I

∣ x

2 y

2 x y 1

x 12 y 1

2 x 1 y 1 1

x 22

y 22

x 2 y 2 1 x ,

2 y ,2

x , y , 1

∣= 0

Tangent to a circle

;i!en a circle &2 K y2 r 2" the e'uation of the tangent at 8&1" y19 is Hschmall:45:10-I

xx 1yy 1 = r 2

For the circle &2 K y2 K 2;& K 2Fy K ) 0" the e'uation of the tangent at 8&1" y19 is Hschmall:5-:105I

xx 1yy 1G x x 1F y y 1C = 0

For the circle 8& &092 K 8y y09

2 r 2" the e'uation of the tangent at 8&1" y19 is Hschmall:56:111I

x 1− x 0 x − x 0y 1−y 0y −y 0= r 2

For the line y m& K b intersecting the circle &2 K y2 r 2" the line

y =mx ±r 1m2

is tangent to the circle for all !alues of m Hschmall:54:11,I.

For the line y m& K b intersecting the circle &2 K y2 K 2;& K 2Fy K ) 0" the e'uation of thetangent is Hschmall:56:11,I

y F = m x G± G2F

2−C # 1m

2

$ '

y '

$

y



The is used because there #ill be t#o tangents:

$f #e ha!e a %oint P 8&1" y19 and a circle 8& &092 K 8y y092 r 2" the length > of the tangent from the%oint P to the %oint of tangency is Hschmall:54I

2 = x

1− x

02y

1−y

02−r

2

$f the e'uation #as &2 K y2 K 2;& K 2Fy K ) 0" the tangents length > is Hschmall:55:11-I

2 = x 1

2y 122;&12Fy1C

Angle of intersection of two circles

>et Hschmall:110:12I

AB =

AP = r 1BP = r 2

u%%ose the e'uations of the t#o circles are

x 2y

22;1 x 2F1 y C 1 = 0

x 2y

22;2 x 2F2 y C 2 = 0

Then

8&0"y

09

8&1"y19>

A B

P



cos = 2;1 G22 F 1 F 2−C 1−C 2

2 r 1 r 2

The condition that the t#o circles cut orthogonally is

2;1G22F 1F 2−C 1−C 2 = 0

Center of triangle's inscribe( circle

For a triangle #ith !ertices 8&1" y19" 8&2" y29" 8&," y,9" the center of the inscribed circle isHschmall:122:1,I

ax 1bx 2cx ,abc

,ay 1by 2cy ,

abc #here a" b" c are the lengths of the triangles sides.

Eucli(ean Transformations

The rele!ant grou% is 829" com%rised of rotations and translations. /s usual" translations androtations do not commute" as sho#n by ins%ection of the infinitesimal generators& x ,&y , and x &y −y & x .

Translations

To transform from 8&" y9 to 8" Q9 #ith a translation to 8&0" y09" the transformation e'uations are

& = x x 0' = y y 0

The in!erse transformation is

x = & − x 0y = ' −y 0

Rotations

$f the a&es are rotated by an angle " the transformation e'uations can be used to find the old andne# coordinates of a %oint P 8#hich doesnt rotate9. $f the old coordinates are 8&" y9" use thefollo#ing transformation to get things in terms of the ne# coordinates

x = & cos−' siny = & sin' cos

The in!erse transformation is

& = x cos y sin' = − x siny cos

ote the transformation is orthogonal: the in!erse transformation matri& is the trans%ose of thefor#ard transformation matri&.

!blique transformationsu%%ose the %ositi!e direction of the ne# & a&is 8call it 9 ma(es an angle #ith the old & a&isE

analogously" the ne# y a&is 8call it Q9 ma(es an angle #ith the old & a&is. Then #e ha!e thetransformation Hschmall:126:1-1I

x = & cos' cos y = & sin' sin

ee the section Three nonco%lanar !ectors as a base 8reci%rocal system9 on %age 21 for thee&tension to three dimensions.



Oblique coordinates

$f the & and y a&es are se%arated by an angle ' and P1 8&1" y19 and P2 8&2" y29 are any t#o%oints" then Hcrc:22I

Distance bet#een P1 and P2:

x 1− x

22 y

1−y 2

22 x

1− x 2 y

1−y 2cos '

Point di!iding P1P2 in ratio rJs:

rx 2sx 1r s

,ry 2sy 1

r s ?id%oint of P1P2:

x 1 x 22"

y 1y 22

/rea of triangle P1P2P,:

1

2sin' x 1 y 2 x 2 y , x , y 1−y 1 x 2−y 2 x ,−y , x 1

)ircle: center at 8h" (9" radius r:

x −h2y −k 22 x − h y − k cos' = r 2

Archime(ean spiral

Polar e'uation r = a. /rc length s =(0

r 2 dr

d 2

d = a

2 [ln ] #here = 1

2.

ote ln = sinh−1.

Pro(ucts

Pseu(o)ectors an( pseu(oscalars

Pseudo!ectors and %seudoscalars change signs under %arity transformations 8a reflection #hereone coordinate changes sign9.

The cross %roduct is a %seudo!ector. The dot %roduct of t#o !ectors is a scalarE the dot %roduct ofa !ector and a %seudo!ector is a %seudoscalar. Thus" the scalar tri%le %roduct is a %seudoscalar.

*calar %(ot& pro(uct

a#b ) abcos

#here is the smallest angle bet#een the !ectors. Qou can see 0 ≤∣a#b∣≤ ab and a#b ∈ ℝ . This

form em%hasiGes the in!ariant nature of the scalar %roduct under uclidean grou% coordinatetransformationsR because it de%ends only on the magnitudes of the !ectors and the angle bet#eenthem. Pro%erties:

a#b = b#aa#bc = bc #a = a#ba#c

* a#b =a#*b

a#a =a2

T#o nonGero !ectors a and b are %er%endicular iff a#b = 0. $f a#a = 0 , then a must be the Gero

R ee the comments on in!ariance under the ?iscellaneous section.



!ector.

$n terms of com%onents:

a#b = a x b x ay by a" b"

u%%ose a" b" and c are !ectors that ma(e a closed triangle. Then #e can get the cosine la#:

abc = 0 and

a−b#a−b ) a−b2

= a2

b2

−2ab cosThe geometric %icture of the dot %roduct is that it yields the %ro=ection of one !ector along another:

$f a re%resents a %lane area and if b is a !ector inclined to that %lane" then a#b is the !olume of theslanted cylinder as sho#n in the follo#ing figure H#ilson::4-I

This is because a is %arallel to the !ertical a&is and the altitude h is b cos .

+ector %cross& pro(uct

c = a× b

c =ab sin

#here the direction of c is found by the usual right hand rule and is the angle bet#een a and b. /gain" this is an in!ariant #ith res%ect to uclidean grou% coordinate transformationsR" but c is a%seudo!ector because of its beha!ior #ith res%ect to %arity transformations. $f three !ectors form a

triangle 8closed %olygon9" then the cross %roduct bet#een any t#o of the sides is t#ice the area ofthe triangle.

R ee the comments on in!ariance under the ?iscellaneous section.

a

b

a,b

a

hb

S



*e ha!e 2/ =∣a×b∣=∣b×c ∣=∣a×c ∣.

Pro%erties:

a×b =−b×a ab ×c = a×c b×c

a×a = 0 i × j = k j × k = i k × i = j

d a×b = d a×ba×d b

T#o nonGero !ectors a and b are %arallel iff their cross %roduct !anishes.

$f a and b are the sides of a %arallelogram" a×b is the area of the %arallelogram.

$n terms of com%onents:

a×b x = ay b" −a" by

a×by = a" b x −a x b"

a×b " = a x by −ay b x

#hich can be remembered by symbolically e&%anding the determinant along the first ro#:

a×b =∣ i j k

a x ay a"

b x by b" ∣

The com%onent of a %er%endicular to b is

b+ = a× b×a

a#a

and the com%onent %arallel is b, = a#b .

*calar triple pro(uct

[a b c ] ) a#b×c =∣a x ay a"

b x by b"

c x c y c " ∣

=abc cossin= ijk ai b

j c

k

#here is the angle bet#een b and c and is the angle bet#een a and b×c . This 'uantity isin!ariant under coordinate rotation 8its a %seudoscalar because the sign can change under acoordinate system %arity change9. The absolute !alue is the !olume of the %arallelo%i%ed #hoseedges are a" b" c .

The three nonGero !ectors are co%lanar iff a#b×c = 0.

+ther %ro%erties:

a

bc

/ area



a#b×c a = a×b ×a×c a#b×c = b#c ×a = c #a×b

a#b×c = −a#c ×b

d

dt a#b×c = -a#b× c a#-b× c a#b× -c

$n the scalar tri%le %roduct" the dot and cross can be interchanged and the com%onents %ermuted

cyclicallyE a change of cyclic order changes the sign. This is most easily seen by !irtue of theinter%retation as the !olume of a %arallelo%i%ed.

Four !ectors ha!e a linear relationshi% H#ilson:6:10,I:

[a b c ] d [c d a] b = [b c d ] a[d a b ] c

/nother #ay of saying and #riting this is that any !ector r may be re%resented in terms of threeothers a" b" c by Hcoffin:2,2:261I

r [a b c ]= [r b c ] a[r c a ] b[r a b ] c

/ reduction formula H#ilson:46:11,I

[a b c ] [d e f ] =

∣a#d a#e a#f

b#d b#e b#f

c #d c #e c #f

∣+ther reduction formulas H#ilson:11,:1,5I

a×b×c ×d = [a c d ] b−a#b c ×d = b#d a×c −b#c a×d

[a×b b×c c ×a] = [a b c ]#[a b c ]

Three noncoplanar vectors as a base (reciprocal syste!

The scalar tri%le %roduct is of use in e&%ressing a gi!en !ector in terms of three nonco%lanar!ectors a" b" c H#ilson:41:104I. >et

r = aabbc c

3y multi%lying by #b×c and the other corres%onding terms" #e find

r = [r b c ]

[a b c ]a

[r c a]

[b c a ]b

[r a b]

[c a b ]c

*e can define

A = b×c

[ a b c ], " =

c ×a

[a b c ], # =

a×b

[a b c ]

These !ectors constitute the reci%rocal base. ote a# A = b#" = c ## = 1 and all other dot %roductsbet#een the t#o sets are Gero 8this is easy to recogniGe because of the cross %roducts in thedefinitions9. 8 A" "" # 9 is reci%rocal to 8a" b" c 9 iff these dot %roduct relations holdH#ilson:4:112I.

Then the e&%ression for the arbitrary !ector r is

r = r # A a r #" b r ## c

These re%resent a concise formulation to con!ert bet#een )artesian and obli'ue coordinatesystems in three dimensional s%ace. *e can also e&%ress r in the reci%rocal base as

r = r #a Ar #b"r #c #

The %re!ious t#o e'uations can be used to #rite do#n the transformation and its in!erse bet#een

the t#o coordinate systems. ote i , j , k is its o#n reci%rocal systemE this is the only basis for

#hich this is true 8along #ith the e'ui!alent lefthanded basis9 H#ilson:4:11-I.

These reci%rocal !ectors are used in crystallogra%hy to discuss reci%rocal lattices 8see



htt%:JJen.#i(i%edia.orgJ#i(iJ7eci%rocalNlattice 9. / reci%rocal lattice !ector k has the %ro%erty that

!2 i k #r = 1 for all lattice %oint %ositions r E i.e." k #r is an integer.

$f 8 A" "" # 9 is reci%rocal to 8a" b" c 9" then

[ A " # ][ a b c ] =1

+ector triple pro(uct

The definition is d = a×b×c . The %arentheses are necessary because the cross %roduct isnonassociati!e: a×b×c ≠ a×b×c . *e ha!e H#ilson:6:10,I H#ilson:111:1,IH#ilson:11,:1,5I

a× b×c = b a#c −c a#b

a×b ×c = a#c b−b#c a

a×b×c =−c × a×b

a× a×b = a#b a−a2b

a× b×c b× c ×ac × a×b= 0

a× b× c ×d = [ a c d ] b−a#b c ×d = b#d a×c −b#c a×d

The first e'uation yields the Cbac cabC mnemonic to hel% remember the formula.

The !ector tri%le %roduct is a !ector %er%endicular to a lying in the %lane of b and c .

$n terms of >e!i)i!ita

a×b×c i = ijk a j klmb

l c

m = ijk klm a j b

l c

m

The !ector d is %er%endicular to a and bc E since b×c is %er%endicular to the %lane of b and c" dmust lie in the %lane of b and c and thus ta(es the form

a×b×c = s bt c s , t ∈ℝ

and" from the rule abo!e" s = a#c and t = −a#b .

/lso H#ilson:6:10,I

a×b#c ×d = a#c b#d −a#d b#c =∣a#c a#d

b#c b#d ∣a×b×c ×d = [b c d ] a−[c d a] b[d a b] c −[a b c ] d

/n arbitrary combination of dot and cross %roducts can be bro(en into a sum of terms eachin!ol!ing only one cross %roduct Hsalmon::10-I.

Direction cosines

/ direction in s%ace can be s%ecified using direction cosines. u%%ose a !ector r %oints in thedesired direction. Then the direction cosines are

= r # i r = r x

r = r # j

r = r y

r = r # k

r = r "

r

*e #ill use = , , to indicate the !ector made u% of three direction cosines. ote ∣∣= 1.

Direction numbers are * #here * is nonGero.

u%%ose - = 1 ,1 ,1 and . = 2 ,2 ,2. These t#o directions are %arallel iff -=

.. These

directions are %er%endicular iff -#. = 0 . HdrmathI

Three directions are %arallel to a common %lane iff HdrmathI

http://en.wikipedia.org/wiki/Reciprocal_lattice

http://en.wikipedia.org/wiki/Reciprocal_lattice



∣1 1 1

2 2 2

, , ,∣= 0

$f #e are gi!en t#o lines in s%ace #ith direction cosines - and . #e can calculate the direction

cosines of the line that is %er%endicular to both lines from Hsnyder:2-:--I

12−21

= 12−21

= 12−21

= ±1sin

#here is the angle bet#een the t#o lines Hsnyder:4:24I

cos = -#.

and

sin2 = 1−-

#. = -

×.

2

#here the s'uare of a !ector a is a#a 8this relation can be %ro!ed by #riting

1−-#

. = 1

22

2−-#

. and e&%anding the right hand side in com%onents9.

Equation of a line

Forms

$ector

$f r % is a %oint on the line and a !ector v is %arallel to the line" then all %oints r on the line can befound from HcorralI

r = r /t v t ∈ℝ

&ndpoints of t'o vectors

;i!en t#o !ectors r 1 and r " the e'uation for the line bet#een the ends of the t#o !ectors is

r = t r -1−t r

.

Paraetric

This form is com%osed of the com%onents of the !ector form:

x = x 0at , y = y 0bt , " = " 0ct

#here v 8a" b" c9.

)yetric

3y sol!ing the %arametric e'uations for the %arameter and e'uating in %airs" #e get Hsnyder:20:-1I

x − x 0

a =

y −y 0

b =

" −" 0

c

$f" e.g." a 0" #e cant di!ide by Gero" but the %arametric form gi!es & & 0. This means the line liesin the %lane & &0. ote a" b" and c are direction numbers of the line and can be con!erted to

direction cosines by di!iding by a2b

2c

2.

T'o*point for

sing a second %oint r 1 #ith the symmetric form" #e ha!e Hsnyder:20:-1IHdrmathI



x − x 0 x 1− x 0

= y −y 0y 1−y 0

= " −" 0" 1−" 0

Equation from two points on line

$f r 1 and r are %oints on the line" then all %oints on the line can be found from HcorralI

r -t r

.−r

- t ∈ℝ

The %arametric form is

x = x 1 x 2− x 1 t , y = y 1y 2−y 1 t , " = " 1" 2−" 1t

and the symmetric re%resentation is

x − x 1 x 2− x 1

= y −y 1y 2−y 1

= " −" 1" 2−" 1

From two planes

T#o %lanes that intersect result in a line. This im%lies the line is the locus of %oints that satisfy thet#o simultaneous e'uations HsnyderI

A1 x B1 y C 1 " D1 = 0 A2 x B2 y C 2 " D 2 = 0

ee more details in the section 0ntersection of two planes on %age 25.

Equation of a plane

General form

!ery e'uation of the first degree in &" y" and G re%resents a %lane

Ax By C" D = 0 or r #+ =−D

if + is 8/" 3" )9. H#ilson:44:11I /ny scalar e'uation of the first degree in an un(no#n !ector r maybe reduced to this form and" hence" any scalar e'uation of the first degree in r re%resents a %lane.>et

= A2B

2C

2

$f #e di!ide the general e'uation by " #e get

x y " p = 0

#here " " and are the direction cosines of the normal to the %lane and

p = D

,essian noral forThe %receding allo#s us to %ut the e'uation of the %lane in the <essian normal form:

n#r = p

#here p is the %er%endicular distance from the origin to the %lane and n = , , is a unit normalto the %lane. ote that p can be %ositi!e or negati!e" de%ending on #here the %lane is #ith res%ectto the origin. Thus" in #ords: the position )ector of any point on the plane (otte( with a unitnormal to the plane is a constant1

<eres a %icture that illustrates the notion that any %oint in the %lane dotted #ith a normal to the



%lane is a constant Hcorral:,:-,I

>et r and r % be any t#o %oints in the %lane. Then R r r % . ince R lies in the %lane" #e ha!e n-R n,%r r% & 0. Thus" n-r n-r % . ince r and r % are arbitrary %oints in the %lane" #e ha!e that n-r isthe same constant for any %oint in the %lane.

>et n#r = p . ote #ere no# using the unit normal n . ince #e no# (no# this is a constant" letsloo( at the case #here r is %arallel to n this is the case #hen r is the %er%endicular to the %lanefrom the origin. Thus #e ha!e r %" so #e see % is the distance from the origin to the %lane. inceits a dot %roduct" #e could also ha!e r and n be anti%arallel" so % could be negati!e. Qou couldta(e the con!ention that the unit normal should al#ays %oint to#ards the origin" so then % #ouldal#ays be negati!e.

<eres a geometrical %icture that illustrates this further. Dra# the line +3 %er%endicular to the%lane / from the origin +. The cross section #ill be:

The line +D re%resents any !ector r in the %lane /. This !ector dotted #ith the unit normal in thedirection +3 8i.e." its %ro=ection onto the normal direction9 is the !ector +). ince the %oint D isarbitrary" you can see that n-r is constant for all %oints in the %lane because you can rotate the line

/ about +3 to get to all %ossible %oints in the %lane.

ote that (C =∣ p∣.

.ual for

The %lane can be re%resented by a single !ector H#ilson:104:1,I p. The direction of p #ill be from

n normal !ector to %lane

r 8&" y" G9 r / 8&

0" y

0" G

09

# r 2 r /

3

)

/ D

+

r U%U



the origin %er%endicularly to the %lane and the magnitude #ill be the reci%rocal of the length of that%er%endicular The e'uation of the %lane in <essian form is then

r # p = 1

The com%onents of p are the reci%rocals of the interce%ts of the %lane on the a&es.

From a$ial intercepts

$f a" b" and c are the interce%ts on the &" y" and G a&es" then the e'uation of the %lane is

x

a

y

b

"

c =1

Through origin an( parallel to two )ectors

r = s r -t r .

From three noncollinear points

>et the %oints be r 1" r " and r /. Define

n = r .−r

-× r

.−r

3

$f the three %oints are collinear" this !ector is identically Gero. +ther#ise" its a normal to the %lanecontaining the three %oints.

Hcoffin:2,2:261I:

r −r -# r

-−r

.# r

.−r

3= 0

or # r −r -= 0

#here= r

-×r

.r

.×r

3r

3×r

-

The condition that the %oints lie in the %lane /& K 3y K )G K D 0 is

∣ x y " 1

x 1 y 1 " 1 1

x 2 y 2 " 2 1

x , y , " , 1∣= 0

T#o %arametric e'uation forms are

r = r -s r

.−r

-t r

.−r

3

r = s r -t r .1−s−t r 3

/lternati!e e'uations are Hcrc:,6I

r = s r -t r

.u r

3

or

[ r −r - r -− r . r .−r 3] = 0

or [r r

- r

.] [r r

. r

3] [ r r

3 r

-] − [ r

- r

. r

3] = 0

#here s , t ,u ∈ℝ .

Through a point an( perpen(icular to a )ector

The e'uation of a %lane containing the %oint r % 8&0" y0" G09 and %er%endicular to the nonGero !ectorn 8i.e." n is a normal to the %lane9 is Hcorral:,:-,I



n#r −r /= 0

<ere" r 8&" y" G9 is any %oint in the %lane. This is easily seen in the follo#ing %icture:

)learly" for any %oint in the %lane" n-R 0 and R r r % . This can also be #ritten as Hcrc:,6I

r #n=c

#here c is a constant and is n-r % . The %er%endicular distance from any %oint r to this %lane is

c −n#r

r

From two simultaneous equations4 pro5ecting planes

u%%ose #e ha!e the simultaneous e'uations

x = m " by = n" c

are t#o %lanes #hich intersect in a straight line. They are %er%endicular to the &G and yG %lanes"res%ecti!ely. These t#o %lanes are called pro5ecting planes.

To find the projecting plane of a given line

>et the line be gi!en by the <essian forms

)1 = n-#r −d

-= 0

)2 = n.#r −d . = 0

Then the e'uation

*)1. )2=0

re%resents any %lane %assing through the intersection of the abo!e t#o %lanes. )hoose * and . so

that one of the !ariables #ill !anish" then #ell ha!e an e'uation for one of the %ro=ecting %lanes ofthe gi!en line. $n %ractice" the desired !ariable is sim%ly eliminated from the t#o e'uations.

Points

Collinear points

r 1" r " r / are collinear iff HdrmathI



x 2− x 1 x ,− x 1

= y 2−y 1y ,−y 1

= " 2−" 1" ,−" 2

Coplanar points

r 1" r " r /" r 0 are collinear iff HdrmathI

∣ x 1 y 1 " 1 1

x 2 y 2 " 2 1

x , y , " , 1

x - y - " - 1∣= 0

"ines an( Planes

Point (i)i(ing a line segment into a gi)en ratio

u%%ose #e ha!e t#o %oints P1 8&1" y1" G19 and P2 8&2" y2" G29. *e #ant to find the %oint P on theline bet#een P1 and P2 such that

PP 1PP 2

= m1

m2

The solution is Hsnyder:4:24I

x = m 2 x 1m1 x 2

m1m2

y = m 2 y 1m1 y 2

m1m2

" = m2" 1m1" 2

m1m2

ote if m1 and m2 ha!e o%%osite signs" the %oint P lies outside the segment P1P2.

"ines in space

u%%ose line >1 is gi!en by r 1 K sv 1 and >2 is gi!en by r K tv . Then these lines are either

1. $dentical2. Parallel: v 1 is %arallel to v .,. $ntersect

-. (e#: they dont intersect" but lie in %arallel %lanes.

For the lines to be %er%endicular" #e must ha!e v 1 %er%endicular to v . ote that s(e# lines can be%er%endicular.

"ine in a plane

;i!en a line

x = m" ay = n" b

and a %lane /& K 3y K )G K D 0. For the line to be in the %lane" #e must ha!e Hsalmon:25:I

AmBnC = 0 8line is %arallel to %lane9and

AaBbD = 0 8one of the lines %oints is in the %lane9



"ine through a point parallel to a )ector

*e desire the e'uation of a line through the %oint r % and %arallel to a !ector A. $f r is the !ector toany %oint on the line" then r r % is %arallel to AE thus" the !ector %roduct !anishes and #e ha!eH#ilson:10:1,-I Hcoffin:2,2:261I

A× r −r /= 0

0ntersection of two linesTo determine intersection" use the %arametric re%resentation of the lines" then set the t#o 8&" y" G9tri%les e'ual Hcorral:,-:-2I. This #ill result in a system of three e'uations in t#o un(no#ns 8the t#o%arameters9.

For t#o non%arallel lines 8i.e." r and r 0 are not %arallel9

r = r -s r .r = r 3t r 6

a necessary and sufficient condition that the lines intersect is Hcrc:,I

[ r -−r 3 r . r 6 ]= r -−r 3#r .×r 6 = 0

The e'uation of the %lane holding the t#o intersection lines is found from the normal" #hich is r × 0and the intersecting point

$f the t#o lines are gi!en by the e'uations

x = m" b y = n" c x = m1 " b1 y = n1 " c 1

then the lines #ill intersect if Hschmall:25,:,04I

b1−b

m−m1

= c 1−c

n−n1

0ntersection of two planes

>et the t#o %lanes be gi!en by the e'uations HdrmathI Hosgood:-:02I

P 1 = A1 x B1 y C 1 " D1 = 0

P 2 = A2 x B2 y C 2 " D2 = 0

The line of intersection is

x − x 1a =

y −y 1b =

" −" 1c

#here

a=

∣

B1 C 1

B2 C 2

∣ b=

∣

C 1 A1

C 2 A2

∣ c =

∣

A1 B1

A2 B2

∣* = a2b

2c 2

* x 1 = b∣D1 C 1D2 C 2∣− c ∣D1 B1

D2 B2∣, * y 1 = c ∣D1 A1

D2 A2∣− a∣D1 C 1

D2 C 2∣,

* " 1 = a∣D1

B1

D2 B2∣− b∣D

1 A

1

D2 A2∣



$f a b c 0" then the %lanes are %arallel.

0ntersection of three planes

The e'uations of the three %lanes are

A1 x B

1y C

1" D

1= 0

A2 x B2 y C 2 " D2 = 0

A, x B, y C , " D, = 0

and one must sol!e these simultaneous e'uations for the %oint 8&" y" G9. Hsalmon:21:-6I

0ntersection of four planes

For the four %lanes

A1 x B1 y C 1" D1 = 0

A2 x B2 y C 2" D2 = 0

A, x B, y C ," D, = 0

A- x B- y C - " D- = 0

to meet in a %oint" #e must ha!e Hsalmon:21:-6I

∣ A1 B1 C 1 D1

A2 B2 C 2 D2

A, B, C , D,

A- B- C - D-

∣= 0

0ntersection of a line an( plane

>et r r l K tnl be the e'uation of a line #here r l is a %oint on the line and nl is a !ector of thedirection cosines of the line. >et n p,8r r p9 0 be the e'uation of a %lane #here n p is the !ector ofthe direction cosines of the normal to the %lane and r p is a %oint in the %lane. Then the intersection%oint is r i gi!en by

r i = r l G n l

#here

G =n p#r p−r l

nl #n p

; #ont e&ist if the %lane is %arallel to the line 8i.e." the %lanes normal is %er%endicular to the line9.

/nother formulation H#ilson:10:1,-I: let the e'uation of the %lane be

r #n =*

#here * is a constant and let us ha!e a line that %asses through the %oint r % and is %arallel to the!ector AE the e'uation is

A×r −r /= 0

Then the %oint of intersection is gi!en by

A×r /×n* A

A#n

This is sim%ler if n and A are unit !ectors

A×r /× n* A



Angle between two lines in space

u%%ose line >1 is gi!en by r 1 K sv 1 and >2 is gi!en by r K tv . Then the angle bet#een them 8ifnecessary" they must be translated so as to intersect9 is

cos =v -#v .v 1 v 2

$f the t#o lines are characteriGed by !ectors of their direction cosines d 1 81" 1" 19 andd 82" 2" 29" then #e ha!e

cos = d -#d .

Angle between a line an( a plane

;i!en a %lane /& K 3y K )G K D 0 and the line

x − x 0a

= y −y 0

b =

" −" 0c

The angle bet#een the line and the %lane is the com%lement of the angle bet#een the line andthe normal to the %lane:

sin = aAbBcC

A2B2C 2 a2b2c 2

Angle between two planes

;i!en t#o %lanes #ith normals n1 and n . The angle bet#een these t#o %lanes is gi!en by

cos = n

-#n

.

n1 n2

Parallel planes

$f #e ha!e t#o %lanes

A1 x B1 y C 1 " D1 = 0

A2 x B2 y C 2 " D 2 = 0

the condition that the %lanes are %arallel is

A1

A2

= B 1

B2

= C 1C 2

$f you con!ert to <essian normal form" you can see that %arallel %lanes #ill ha!e the same directioncosines in their e'uations. 'ui!alently" the dot %roduct of their unit normal !ectors is unity

n-# n

.= 1

Perpen(icular planesT#o %lanes are %er%endicular if the dot %roduct of their unit normal !ectors is Gero:

n-# n. = 0

Plane through a point parallel to two )ectors

The %lane %arallel to a and b and through r % is Hcoffin:2,2:261I

[ r −r / a b ]= 0

$f #ere gi!en the direction cosines 1 ,1"1 and 2,2" 2 of the t#o !ectors then the e'uation is



∣ x − x 0 y −y 0 " −" 0

1 1 1

2 2 2∣= 0

Plane through a point parallel to another plane

The %lane through r 1 %arallel to the %lane of r " r / is Hcrc:,6I

r = r - s r

.t r

3

or r −r -#r .×r 3 = 0

or [ r −r

- r

. r

3]= [r r

. r

3]−[r

- r

. r

3] = 0

Plane through a line parallel to another line

>et the lines ha!e direction cosines i 8i" i" i9 and %ass through r i 8&i" yi" Gi9 for i 1" 2. Thee'uations of the %lanes containing one line and %arallel to the other line are Hsalmon:,1:6I

x − x i 1 2−21 y − y i 1 2−21

" −" i 1 2−2 1 = 0 for i 1" 2or" in !ector form"

r −r i #-×

. = 0 for i 1" 2

The %er%endicular distance bet#een these t#o %lanes is

r -−r

.#

-×

.

sin

#here is the angle bet#een the lines

sin = 1−-#.

Plane through a gi)en point an( lineFrom Hschmall:256:,11I. >et the %oint be 8&2" y2" G29 and the %lane be gi!en by

x − x 1 A1

= y −y 1

B1

= " −" 1

C 1

The re'uired %lane is /& K 3y K )G K D 0. Then" since the re'uired %lane must %ass through thegi!en %oint and through the gi!en line" #e ha!e the four e'uations

Ax By C" D = 0 Ax 2By 2C" 2 = D

AA1BB1CC 1 = 0

Ax 1By 1C" 1 = D

liminate /" 3" )" and D from these e'uations to get the e'uation of the re'uired %lane.

Plane through a point an( normal to a )ector

The %lane normal to n and %assing through r % is

n#r −r /= 0

"ine perpen(icular to a plane containing three points

u%%ose #e ha!e three noncollinear %oints r 1" r " and r /. The e'uation of the line %er%endicular to



the %lane containing these %oints is Hcrc:,I

r = r -×r .r .×r 3r 3×r -

"ine perpen(icular to two lines

$f #e ha!e t#o lines #ith direction cosines 1 81" 1" 19 and 82" 2" 29 and #e #ant the

direction cosines 8" " 9 of the line %er%endicular to both" #e ha!e Hsalmon:4:,,I

= -×.

sin =

-×.

1−-#.

#here is the angle bet#een the t#o lines. *ritten out in com%onents" this is

sin = 1

2−

2

1

sin = 12−21

sin = 12−21

Distance of a plane to the origin

u%%ose #e ha!e three noncollinear %oints r 1" r " and r /. The distance of the %lane containingthese three %oints from the origin is Hcrc:,I

d = r

-#r

.× r

3

r .− r

-× r

3− r

-

Distance of a point to a line

;i!en the direction cosines = , , and a line through r 1 in this direction. Then the distance dfrom a %oint r to the line is HdrmathI

d = ×r .−r -#× r .−r -

Distance of a point from a plane

>et O 8&0" y0" G09 and n be a normal to the %lane. The %lane has the e'uation a& K by K cG K d 0

and it does not contain the %oint O. o# r 8& &0" y y0" G G09. Then

D = r ∣cos∣ =∣n#r ∣

∣n∣=

∣ax 0by 0c" 0d ∣

a2b2c

2

$f D is %ositi!e" the %oint and the origin are on o%%osite sides of the %lane.

Distance between two non2intersecting lines

>et the direction cosines of the t#o lines be - = 1"

1"

1 and =

2"

2"

2 and let the lines

%ass through the %oints 8&1" y1" G19 and 8&2" y2" G29" res%ecti!ely. The shortest distance bet#een the



lines is

d = x 1− x

2y 1−y

2" 1−"

2

#here

1 2−1 2

=

1 2−21

=

1 2−12

= ±1

sin

#here = , , are the direction cosines of the shortest line connecting the t#o lines and isthe angle bet#een the t#o lines. *e ha!e

=

-×

.

sin =

-×

.

1−-#.

#here is the angle bet#een the lines. $n com%onents" this is

sin = 12−21

sin = 12−21

sin = 12−21

The distance is HdrmathI

±∣ x 2− x 1 y 2−y 1 " 2−" 1

1 1 1

2 2 2∣

∣1 1

2 2∣2

∣1 1

2 2∣2

∣1 1

2 2∣2

The lines intersect iff the numerator is Gero.

Circle in space

Hhtt%:JJmathforum.orgJlibraryJdrmathJ!ie#J6,.html I

/ %arametric e'uation for a circle in s%ace can be found from:

n = unit normal !ector for the %lane containing the circle# = circle center r = circle radius u = unit !ector from ) to#ard a %oint on the circle v = n× u t = %arameter

Then a %oint P is on the circle if

P = # r u cos t v sin t

+ectors

First (egree )ector equation

H#ilson:50:11I /n e'uation of the first degree in an un(no#n !ector r has each term as a !ector'uantity containing the un(no#n !ector not more than once. /n e&am%le is

a×r b#c r r −F = 0

The e'uation may be sol!ed for r by dotting it successi!ely #ith three nonco%lanar !ectors to get

http://mathforum.org/library/drmath/view/63755.html

http://mathforum.org/library/drmath/view/63755.html



three scalar e'uations. $f the e'uation is of the form

A a#r " b#r # c #r = .

then one can dot multi%ly by A " " " and # #here the %rime denotes the reci%rocal basis !ectors to8 A" "" # 9 8of course" they must be nonco%lanar9. Then one #ill get

r = .# A 1 a 1 .#" 1 b 1 .## 1 c 1

#hich is the solution.+ther terms in the e'uation may be of the form r or & ×r . These can be e&%ressed asH#ilson:51:114I

r = a 1 a#r b 1 b#r c 1 c #r

& × r = & ×a 1 a#r & ×b 1 b#r & ×c 1 c #r

3y such means" one #ill be able to reduce the e'uation to the form

2 a#r 3 b#r 4 c #r = 5

and this can sol!ed as abo!e using the reci%rocal basis 8see %age 219 as

r = 5 #2 1 a 1 5 #3 1 b 1 5 #4 1 c 1

0n)arianceVectors that re%resent %hysical 'uantities need to be in!ariant under common coordinatetransformations. Qou can mentally %icture the !ector as an arro# 8either as a free !ector orattached to a certain %oint" such as in a tor'ue9. Thus" for e&am%le" imagine youre a%%lying ator'ue to a lug nut #ith a soc(et #rench. / second %erson must measure the same force and le!erarm and calculate the same tor'ue youre a%%lying" regardless from #here he !ie#s you.

The #ay to !ie# this is that the tor'ue !ector has an Ce&istenceC inde%endent of the coordinatesystem that has been %ut on the s%ace. Thus" #e #ould only #ant to use mathematical constructsand models that ha!e the same beha!ior.

This in!ariance a%%lies to the usual uclidean transformations of e!eryday e&%erience.

There are other cases #here you can en!ision the !ector being attached to the s%ace. /n e&am%le#ould be ma(ing measurements of an ob=ect by %hotogra%hing it. The %osition !ectors youmeasure #ill be affected by distortions from the imaging system. $f you used a %hotogra%h tomeasure the le!er arm of the tor'ue abo!e" the %ers%ecti!e transformation 8and distortion effects9#ould be inherent in the %hysical measurement. )learly" the calculated tor'ue is not an in!ariantmeasured !alue in this case #e assume it is a %hysical in!ariant" but our measurement methodsinclude some distortion.

*pheres

urface area: A =-r 2 = 12. r

2 =D2

Volume: * = -,r , = -.145r , = D

,

6 = 0.2,6 D,.

The general e'uation of a s%here is HdrmathI

x 2y 2" 22d&2ey2fGm = 0

The center is 8d" e" f9 and the radius is r = d 2!

2#

2−m .

The e'uation of a s%here at the origin #ith radius R is Hcoffin:2,2:261I

r #r = R 2



The e'uation of a s%here #ith center at r 1 and radius R is Hcrc:,I

r −r -#r −r -= R 2

The e'uation of a s%here ha!ing %oints r 1 and r as the end%oints of a diameter is Hcrc:,I

r −r -# r −r

. = 0

Four %oint form HdrmathI

∣ x

2y 2"

2 x y " 1

x 12y 1

2" 1

2 x 1 y 1 " 1 1

x 22y 2

2" 2

2 x 2 y 2 " 2 1

x ,2y ,

2" ,2

x , y , " , 1

x -2y -

2" -2

x - y - " - 1∣= 0

For the %lane r-r 1 s to be tangent to the s%here 8r r 98r r 9 R 2" #e must ha!e Hcrc:,I

s −r -#r

.#s −r

-#r

.=r 1

2R 2

The e'uation of the tangent %lane at r on the surface of s%here 8r r 198r r 19 R 2 is Hcrc:,I

r −r .# r

.−r

- = 0

The area of the surface of a s%here of radius r illuminated by a %oint source a distance a a#ay fromthe surface of the s%here is Hha#(es:-51:210I

2 ar 2

ar

$f a hole of length > is drilled through a s%here" the remaining !olume of material is inde%endent ofthe radius of the s%here Hold %uGGleI.

7iscellaneous

Center of mass

The !ector r to the center of masses mi at the %oints r i is

r =1 m i r i

1 mi

Pro5ections

6soetric

7ef. htt%:JJen.#i(i%edia.orgJ#i(iJ$sometricN%ro=ection

The follo#ing is the transformation for transforming a %oint r in the first octant to a t#o dimensional

%oint 8b&" by9 in the &y %lane:

[b x

by

0 ] = [1 0 0

0 1 0

0 0 0][1 0 00 cos sin0 −sin cos][

cos 0 −sin 0 1 0

sin 0 cos ][ x y

" ]= [1 0 0

0 1 00 0 0] 1

6 [ , 0 − ,1 2 1

2 − 2 2 ][ x

y " ]

http://en.wikipedia.org/wiki/Isometric_projection





#here = sin−1tan

6 2 ,.26+ and =

-. The transformation matri& is thus

[

1

20

−1

21

6

2

,

−1

6

0 0 0

]ote its determinant is Gero. The transformation e'uations are

b x = x − "

2

by = x 2y"

6

Projection angles

$n the figure" the unit !ector r has the s%herical coordinates 81" " 9. $t also has the angles and

that are the %ro=ection angles onto the &G and yG %lanes" res%ecti!ely.*e ha!e

r = x 2y

2"

2

x = r sin cosy = r sin sin" = r cos x = " tany = " tan

G

r

W

&

y

Q



From

x = r sincos= " tan = r cos tany = r sinsin= " tan = r cos tan

#e get

tan tancostan tansin 819

#ith in!erses

tantan

tantan

. tan

. tan

.

829

)ine sticks

/ sim%letoma(e de!ice can be used to measure %ro=ections angles and" thus" measure angles ins%ace.

Ta(e a board and drill t#o holes through it a distance > a%art. Then ri% the board in halflongitudinally to get t#o identical %ieces. Xoin the t#o boards #ith a snugfitting bolt or ri!et throughthe holes so that the t#o boards can %i!ot about the hole center. Tighten the bolt enough to let thestic(s angle be ad=usted but held in %lace #ith normal handling.

To use this de!ice" ad=ust the edges on the t#o boards to coincide #ith an angle. ?easure thedistance and calculate the angle by

= 2sin−1

2>

+n a %ractical note" a 12 inch or larger metal folding rule is !ery con!enient to ma(e such an anglemeasuring de!ice. Xust su%erim%ose the t#o legs" clam%" and drill a small hole through both legs these holes #ill automatically be at the same distance from the %i!ot. Qou can then calibrate thede!ice to a right angle to determine the distance >. $n use" di!iders can be used to set or measure

the distance . The author has seen such a rule from the early 1500s that had such mar(s to hel%measure angles.

ome sim%le reflection #ill sho# you this angle measuring de!ice can be sur%risingly accurate.

For an angle of 60A" if you made a measurement error of 0.02 inches for an > of 10 inches" youllsee that this affects the angle by 0.1,A. Thus" ty%ical sho% #or( can be done to a tenth of a degree#ithout undue effort.

For closer #or(" these sine stic(s can be made from metal such as some 0.2C by 0.C aluminumrectangular bar stoc(. Drill and ream the holes" then %ress in 8or use >octite9 some %ins #ith centermar(s located on a lathe. sing trammels" you should be able to measure the se%aration al%ha to0.00C or less. The abo!e 60A angle #ill be in error by 0.0,,A" #hich is nearly , times better thanthe usual machinists !ernier %rotractor can read 8#hich is ty%ically minutes of arc9. !en moreaccurate measurements could be made by using a sine bar and using the sine stic(s as a transfer

te% 1: drill holes

te% 2: ri% longitudinally along /3

/

3>



de!ice.

#otations

ee C7otation matri&C at htt%:JJen.#i(i%edia.orgJ#i(iJ7otationNre%resentationN8mathematics9 andhtt%:JJen.#i(i%edia.orgJ#i(iJ7otationNmatri& . ?atri& re%resentations of +8n9 ha!e a determinant ofK1 and are orthogonal matrices" meaning their trans%ose is e'ual to the in!erse. For n L 2" thegrou% is non/belian 8i.e." rotation matrices do not commute9.

The follo#ing matrices are +8,9 re%resentations and re%resent rotations about the &" y" and Ga&es in ,s%ace:

R x = [1 0 0

0 cos −sin 0 sin cos ]

R y = [cos 0 −sin 0 1 0

sin 0 cos ]R " = [

cos −sin 0sin cos 0

0 0 1]ote that if J2" the 7& ta(es j into k , 7y ta(es i into k , and 7G ta(es i into k7

+ther rotation matrices can be gotten from these three using matri& multi%lication. Thus"R x R y R " re%resents a rotation #hose ya#" %itch" and roll are " " and " res%ecti!ely.

imilarly" R " R

x R

" re%resents a rotation #hose uler angles are " " and .

$n using uclidean transformations" it is handy to use homogeneous coordinates: the general%osition !ector is 8&" y" G" 19. The transformation matrices are then

a,b ,c = translation = [1 0 0 a0 1 0 b

0 0 1 c 0 0 0 1]

R x = [1 0 0 0

0 cos −sin 00 sin cos 0

0 0 0 1]

R y =

[cos 0 −sin 0

0 1 0 0sin 0 cos 0

0 0 0 1]R " = [

cos −sin 0 0sin cos 0 0

0 0 1 0

0 0 0 1]

http://en.wikipedia.org/wiki/Rotation_representation_(mathematics)

http://en.wikipedia.org/wiki/Rotation_matrix

http://en.wikipedia.org/wiki/Rotation_representation_(mathematics)

http://en.wikipedia.org/wiki/Rotation_matrix



+ne can also #rite the rotation matri& as the direction cosine matri&. u%%ose #e ha!e three unit!ectors u , v , ' as the basis of the rotated system

R = [ u 8

v 8 ' 8

u y v y

' y

u 9 v 9

' 9 ]

#here each element is the cosine of the angle bet#een a rotated unit basis !ector and one of thereference a&es. 7 is a real" orthogonal matri& #ith unity determinant and eigen!alues

1" cos i sin , cos−i sin

The eigen!alue of 1 corres%onds to the rotation a&is" as it is the only !ector unchanged by therotation. 7 has three degrees of freedomE they are constrained by the relations

∣ u ∣=∣ v ∣= 1 u # v = 0 ' = u × v

ote that a rotation matri& %reser!es distances bet#een %oints. >et r be an arbitrary !ectorE thenr Tr is the s'uare of its magnitude. $f 7r is a rotated !ector corres%onding to the rotation matri& 7"#e ha!e 87r 9T87r 9 8r T7T987r 9 r T87T79r r Tr " #hich again is the s'uare of r s magnitude.

Rotating about a given a8is

;i!en a unit !ector u = x , y , " , the matri& for a rotation by an angle about an a&is in thedirection of u is 8htt%:JJen.#i(i%edia.orgJ#i(iJ7otationNmatri&YFindingNtheNrotationNmatri& 9

R = [ x

21− x

2cos x y 1−cos−" sin x " 1−cos y sin

x y 1−cos" sin y 21−y

2cos y " 1−cos− x sin

x " 1−cos−y sin y " 1−cos x sin " 21−"

2cos ]$n a normal right handed )artesian coordinate system" this rotation #ill be countercloc(#ise for anobser!er %laced so that the a&is u goes in the obser!ers direction.

7odrigues formula can be used instead:R = P - −P cos Q sin

#here

P =[ x 2

x y x "

x y y 2

y "

x " y " " 2 ]= u u

T , - =[1 0 00 1 0

0 0 1], Q = [ 0 −" y " 0 − x

−y x 0 ]O is the s(e#symmetric re%resentation of the cross %roduct" P is the %ro=ection onto the a&is ofrotation" and $ P is the %ro=ection onto the %lane orthogonal to the a&is.

.irection cosine atri8 to &uler angles

;i!en a ,&, rotation matri& 7" one can calculate the G&G uler angles " " 3 8rotation around G" &"then G9 as:

=atan. R ,1 ,R ,2

= cos−1

R ,, must be in H 0" 9

3=−atan. R 1, ,R 2,

$f 7,, 0" and 3 should be calculated from 711" 712 instead.

http://en.wikipedia.org/wiki/Rotation_matrix#Finding_the_rotation_matrix

http://en.wikipedia.org/wiki/Rotation_matrix#Finding_the_rotation_matrix



.irection cosine atri8 to &uler a8is and angle

$f the uler angle is not a multi%le of " then the uler a&is u 8&" y" G9 and angle are

cos= trac!R −1

2

x = R ,2−R 2,

2sin

y = R 1,−R ,1

2sin

" = R 21−R 12

2sin

ote that this is the eigen!ector associated #ith the eigen!alue of 1.

#otating lines in space

8?oti!ation: htt%:JJ###.mathforum.orgJlibraryJdrmathJ!ie#J6-62.html .9

u%%ose you ha!e a noteboo( 8see figure belo#9. +%en it u% and dra# a line >1 from a %oint / onthe s%ine to any#here on the right %ageE call the angle bet#een the s%ine and the dashed green

line a. +n the left %age" dra# a line >2 from / to any#hereE call the angle bet#een the s%ine andthe left dotted line b. o#" o%en the noteboo( so that the angle bet#een the t#o %ages is c. Then

the angle bet#een the t#o dra#n lines is

cos = cos a cosbsina sinb cosc

seful subcases are:

c 0A: = a−bc 50A: cos = cosacosbc 140A: = ab

Deri!ation: su%%ose that the dashed and dotted lines both %ass through the origin as sho#n and acustomary righthand )artesian coordinate system is %ut #here the lines intersect" as sho#n. >etu be a unit !ector %ointing in the dashed lines direction and v be a unit !ector %ointing in the dotted

lines direction. Then the cosine of the angle is

ab

c

W

Q

>1

>2

u

)

http://www.mathforum.org/library/drmath/view/56462.html

http://www.mathforum.org/library/drmath/view/56462.html



cos = u #v

ince u lies in the &y %lane" its direction cosines are easy to #rite do#n: sin a , cosa , 0.

To get v s direction cosines" lets start #ith c 0. Then v s direction cosines are sin b ,cosb ,0 . 7otate the coordinate system about the y a&is countercloc(#ise by an angle c. )learly" only the &and G com%onents #ill change. The matri& to rotate about the y a&is is

R y =

[cos 0 −sin

0 1 0

sin 0 cos ]<ere are the direction cosines of the rotated direction cosine !ector of the dotted line >2 u% out ofthe &y %lane by the angle c:

[cos c 0 −sinc

0 1 0

sinc 0 cos c ] sinb

cosb

0 = sin b cosc

cos b

sin b sinc

o# #e calculate the angle :

cos = sina , cosa , 0#sinb cosc , cosb , sin b sinc

or

cos = sinasinbcosc cosacosb

8eli$

7ef. *eisstein" ric *. C<eli&.C From ?ath*orld/ *olfram *eb 7esource.htt%:JJmath#orld.#olfram.comJ<eli&.html

/ heli& is a cur!e in s%ace on a right circular cylinder that ad!ances at a uniform rate as the anglearound the cylinders a&is changes. / scre# thread is the %rototy%ical e&am%le. The %arametrice'uations of a heli& are

x = r cos

y = r sin" = p

r is the radius of the cylinder and 2% is the %itch of the heli&. The tangent line to the heli& ma(es afi&ed angle #ith an arbitrary line. The heli& is the shortest %ath bet#een t#o %oints on a cylinder8cut the cylinder" roll it out flat" and dra# a line bet#een the t#o %oints9.

Formulas:

= r 2 p

2

/rc length= )ur!ature= r /Torsion = p /

Areas an( +olumes

+olume of a tetrahe(ron

$f a tetrahedrons four corners are P i 8&i" yi" Gi9" i 1" 2" ," -" then the !olume is Hsalmon:22:-IH#ilson:11-:1-1I

http://mathworld.wolfram.com/Helix.html

http://mathworld.wolfram.com/Helix.html



* = 1

6 ∣ x 1 y 1 " 1 1

x 2 y 2 " 2 1

x , y , " , 1

x - y - " - 1∣

"ayoutThis section contains !arious methods of geometrical layout using sim%le tools.

Cutting pipe en(s for wel(ing

u%%ose cylindrical %i%e 1 of radius 7 runs along the y a&is and another cylindrical %i%e 2 of radiusr #here r ≤R runs along the & a&is. 7otate %i%e 2 by an angle about the G a&is 8the %ositi!e angleis countercloc(#ise #hen loo(ing at the &y %lane from the KG direction9 and then translate the %i%e

in the KG direction by a distance a. ince the generator for the rotation is x &y − y &

x and the

translation is &" " these transformations commute.

Qou can lay out a %attern on a sheet of %a%er and use it as a tem%late to cut the end of the smaller%i%e. The formula for the y distance to trim off gi!en the angle around the circumference of thesmaller %i%e is

y = s!c R 2− r sina

2r cos sin

The angle =0 is in the %lane containing the %i%es a&es.

To deri!e this" start #ith the %arametric e'uations for one cylinder that lies on the y a&is:

# u , = r cos ,u ,r sin 8,9

tart #ith the other cylinder lying along the & a&is:

/ v , 3 = v , R cos3 , R sin3

se a matri& to rotate this !ector an angle al%ha around the G a&is:

M = cos −sin 0

sin cos 0

0 0 1This lea!es the cylinder in the &y %lane. Then translate it in the KG direction.

'uate the com%onents # and / " then sol!e them for 0 " u" and v in terms of 3 and substitute into 8,9 to get a one%arameter !ector for the cur!e of the intersection. Then use the in!erse of ? to rotatethe cur!e bac( to it around the & a&is and translate bac( do#n to the &y %lane. This #ill yield theabo!e formula.

The site htt%:JJ###.harder#oods.comJ%i%etem%late.%h% %ro!ides an online solution !ia thismethod.

To ma(e a tem%late to trace the hole on the larger %i%e" start #ith the %arametric !ector describingthe cur!e:

x = R

2− r sin a 2r cos sin

cos y = r cos" = r sin −a

>oo(ing do#n the y a&is 8the a&is of the larger %i%e9" youll see

http://www.harderwoods.com/pipetemplate.php

http://www.harderwoods.com/pipetemplate.php



The %oint 3 is on the cur!e of intersection. The angle 3 = tan−1 "

x

. The arc length of the arc /3 is

s = R 3. Then #hen #e %lot the %oints 8R tan−1 "

x " y 9 as goes from 0 to 2 " the cutout #ill

be traced in the %lane.

Di)i(ing a line equally

The 'uic(est method is to lay a rule at an angle to the line" and mar( off %oints along a line at nma=or di!isions. Then dra# %arallel lines from the mar(ed %oints to intersect the original line 8this ismost easily done #ith a drafting machine or t#o triangles9. $t the figure belo#" su%%ose /3 is theline to be di!ided e'ually into - %arts. u%%ose #e set a rule along line 3) that ma(es 3) e&actly- units. Then #e mar( the %oints 1" 2" ," and -. Dra# a line from %oint - 8i.e." )9 to /. Then dra#lines through 1" 2" and , %arallel to /). The intersections di!ide line /3 u% as desired.

3ecause calculators are common" its =ust as easy to calculate /3Jn and set the di!iders to this!alue" then ste% off the intermediate %oints. The first method can be more accurate" as theres nocumulati!e error due to setting the di!iders slightly #rong. <o#e!er" you can eliminate thecumulati!e error if you first ste% off the di!ider along /3 and ensure it comes out the %ro%er lengthEif not" ma(e small ad=ustments and re%eat.

/ 3

-

1

2

,

)

G

&

3

3

/

+

7



"ine segment perpen(icular bisector

;i!en line segment /3" construct the %er%endicular bisector by scribing e'ual arcs from theend%oints of the segment. et di!iders to ,J- or more of the line segment length. The intersectionsof the t#o e'ual circular arcs lie on the %er%endicular bisector.

The method #or(s because the diagonals of a rhombus are %er%endicular bisectors of each other.

Perpen(icular to a point interior to a line segment;i!en line /3 and %oint ) to #hich a %er%endicular to /3 must %ass. sing di!iders" construct arcD.

Then construct e'ual radius arcs from centers D and that %ass through %oint F. >ine )F is

/ 3

)

D

r r

/ 3)D

F



%er%endicular to /3.

Perpen(icular at en( of line segment

;i!en line segment /3. To construct a %er%endicular at 3" set di!iders to a reasonable siGe andscribe arc )D #ith center at 3. >ocate %oint ) on arc )D a%%ro&imately -A abo!e the line /3.cribe the dashed circle 3F #ith radius )3. )onstruct line ) and e&tend it to intersect thedashed circle at F. The line 3F is %er%endicular to /3.

The %rinci%le is based on the fact that if any t#o chords 3 and 3F in a circle meet along adiameter F" then angle 3F is a right angle.

Bisecting an angle

To bisect angle /3)" construct arc D #ith center 3. )onstruct arcs DF and F that ha!e thesame radius. 3F is the angles bisector.

"aying out right anglesT#o triangles are commonly used for right angle layout: the isosceles and the ,:-: triangle.

/ 3

)

D

F

/

)

3

D

F



These are often used in car%entry" yard #or(" and sur!eying. The ad!antage is that all thats

needed is a ta%e measure. The isosceles triangle method can be e&tended to measuring thediagonals of a s'uare #hen they are e'ual" then all four corners are right angles. This methoddoesnt e!en need a ta%e measure.

The ,- triangle is also useful #ith di!iders" as it is rare one doesnt ha!e a rule in the sho%. 3e

mindful that angle isnt any CcommonC angle its the arctangent of -J," or ,.1,A.

Copying an e$isting angle

u%%ose angle /3) is gi!en and it is desired to co%y this angle. The ne# !erte& is 3 and one sideof the angle is to fall on line 3D. +n the /3) angle" dra# an arc 3D and" #ithout changing thedi!ider settings" dra# the arc D from the ne# !erte& 3. +n the /3) angle" set the di!iders to thedistance D and scribe D on the ne# dra#ing. 3 com%letes the co%y of the angle.

Di)i(ing an angle

To di!ide an angle /3) into n e'ual %arts" dra# arc D #ith center at 3" then set the di!iders to anestimate of the arc length D. te% off n %oints on the arc D. $terate until the last %oint fallse&actly on %oint D or . The diagram sho#s the case #hen n ,.

1

1

s'rt829

-

,

/

)

3

D 3D

F

F



Qou cannot dra# a line from D and di!ide that into e'ual segments to e'ually di!ide the angle.<o#e!er" if you ha!e a rule and calculator handy" you can measure the line segment D and r"

then calculate the angle by the cosine la#. Then the di!ider setting is

2rsin 2n

"aying out angles

:sing tangents?ost fol(s are familiar #ith %rotractors" but because they are ty%ically small" they cant be used foraccurate angle layout. Drafting triangles are better" as they are ty%ically made to close tolerances.?y fa!orite method" ho#e!er" is to use trigonometric relationshi%s" ty%ically the tangent. <eres ane&am%le of laying out an angle on the table to% of a table sa# 8this is used e.g. to set a guide #henone #ants to cut a co!e9.

Qou can measure distance /3 on the table. Then to get an angle #ith res%ect to /3" measure u%a distance y = x tan on 3). For accurate #or(" this re'uires that /3) is a right angleE if it isnt"you can measure it and still ma(e the layout" although youll ha!e to use the cosine la#.

<eres an estimate of the angle error:

d =

x 2 −y dx x dy #here =

1

1 y

x 2

/ 3

)

sa# blade

&

y

/

)

3

D

r



For my table sa#" the distance & is 24.2C. For an angle of ,0A" #e ha!e y 16.24C. $f #e assume

dx = dy = 0.1 inches , then d =1.,,,

24.22 −1.6242.42 = 2.00×10

−, radians or 0.11A. 7educing

the d& and dy uncertainties #ill reduce the d uncertainty %ro%ortionately.

/n alternati!e techni'ue is to use a car%enters s'uare. ince these are annoyingly graduated infractional inches" one needs to use a table or a calculator. $n the follo#ing table" the settings are

accurate to #ithin 0.01A..69 $ -:9 square -.9 $ :9 square

Angle; < = > = > 1 2 , - 6 4 5

1011121,1-116114152021

222,2-22622425,0,1,2,,

,-,,6,,4,5-0-1-2-,

1- 1J-21 1J22, J421 1J2

2015

1- 1J-16

20 1J2

11420

15 1J22, 1,J16

1-20 1J2

1420

22 1J22,

21 1J2

1,14 1J-2, J422 1J-20 1J21, 1J-2, 1J2

2,1 ,J-2, 1J2

2-15 1J-

21 1J220

21 1J214 1J-

1610 1J2

1-11 1J21 J41- ,J-

1J-,J-

1 1J-1 1J21 ,J-

21 ,J-2 1J-, 1J-

,, 1J2- 1J-- 1J2

1J16, ,J- J4 1J26 1J2 ,J-4 ,J44 1J-

1J- ,J-10 J410 ,J4

106 ,J-12 1J212 ,J-10 1J-1- 1J4

112 1J2

1- 1J21-

1 J41, ,J-12 1J24 1J211 ,J-

101 J41, ,J-

1J410 ,J-

11 1J1610 ,J-

105 1J2 1J4

410 1J-

4 1J25

106 1J2

11 25J,2

10 1J-5

1011 1J-11 1J210 ,J-

6 1J25 1J4

11 1J1611 1J410 1J-6 J4

11 ,J-11 1J24 J4

11 ,J-12

5 J4

10 ,J-10

10 ,J-5 1J4

4 1J-

,J-

4 1,J16 ,J4

1J4,J4J4,J-J41

J41 1J41 J4

1 1J21 ,J-2 1J41 1J2

2 ,1J,21 J4

2 1J162 ,J-, 1J-, J4- ,J16- 1J4

2 J4, J4 J16 ,J16

, ,J46 1J-6 ,J4 1J4 1J16 1J26 1J-

1J-

1,J166 J46 1J-- 1J- J4

1J166 J4



-- 1- 1J2 1- 1J-

$nterestingly" for the larger s'uare" 1-A is the only angle that re'uires a 16th of an inch.

/nother techni'ue for laying out angles is to use di!iders and a rule for construction" as in thefollo#ing e&am%les.

;%<

The radius of a circle is used to layout an angle of J, 60A:



et the di!iders to +/" the radius of the circle" then scribe the arc through 3 #ith center /. +/ is ata 60A angle to +/.

0=<

3isect a right angle. / 'uic( tem%late can be made from %a%er #ith s'uare edges: fold one cornero!er so that its edge lays along the an o%%osite edge. The fold #ill be at -A.

/%<> 1=<

?a(e a 60A angle and bisect it once or t#ice.

Other special angles

<eres an e&am%le of ma(ing a -0A angle. +ther s%ecial angles can be made throughcombinations of the techni'ues sho#n here. $n general" ho#e!er" the method of using the tangentabo!e #ill %robably be found to be the 'uic(est as long as one has a calculator or table of tangentshandy. ometimes these s%ecial methods can be used #hen no tables or calculator are handy" sotheyre useful to ha!e in your bac( %oc(et.

?a(e a -A angle by bisecting a right angle and ma(e a ,0A angle by bisecting a 60A angle. o#di!ide the 1A angle into three e'ual angles and the needed -0A angle #ill be A less than the -Aangle.

Triangles

6nscribed circle

The bisectors of the internal angles of a triangle intersect at the center of the inscribed circle.

#ircuscribed circle

The %er%endicular bisectors of the sides of a triangle intersect at the center of the circumscribedcircle.

Ellipse

$n the follo#ing section on elli%ses" 2a is the ma=or diameter of the elli%se and 2b is the minordiameter.

/+

3

60A



The area of an elli%se is ab and the %erimeter is -a #here is the com%lete elli%tic integral of

the second (ind #ith k = a2−b

2/a . 7amanu=ans 8151-9 a%%ro&imation for the %erimeter is

p=ab [1 4

10 -−4 ], #here 4=, a−b

ab2

Piece of string and t'o foci

et the di!iders to a and scribe the arc from / to intersect the ma=or diameter at %oints 3 and ).Points 3 and ) are the foci. The foci ha!e the %ro%erty that any beam of light 3D) emitted fromone foci #ill arri!e at the other foci" assuming the interior of the elli%se is %erfectly reflecting. The

abscissas of the foci ha!e an absolute !alue of d = a2−b

2.

The elli%se has the %ro%erty that any %oint D on the elli%se has a constant sum of distances fromthe foci 8i.e." 3D K )D 2a9.

To construct the elli%se" %ush t#o %ins or nails in at the foci. Then tie a string such that the loo%causes the %encil %oint" #hen stretching the loo% taut" to mar( at %oint . Then you should be ableto trace the elli%se by (ee%ing the %encil %oint taut in the string loo%. The loo% length needs to be3D K D) K 3) #hich is e'ual to

2ad = 2a a2−b

2 .

*hile the method is e&act" it can be difficult to (ee% the %encil %er%endicular and the stringstretched a uniform amount. Thus" dont e&%ect %erfection.

#alculated points

ince the e'uation of the elli%se is

x 2

a2 y 2

b2 = 1

the re'uired %oints can be calculated and %lotted. / %olygonal a%%ro&imation to the elli%se can begotten" good enough for many sho% tas(s" es%ecially if a little smoothing is done #ith a file orsander.

This method is relati!ely fast if a tem%late is made and only a 'uarter of the elli%ses form has tobe generated" as the other three 'uarters can be gotten by reflection 8turning the tem%late o!er9and rotation.

a

/

3 )

a

D

b



:sing a arked stick

sing the follo#ing elli%se" #e mar( a stic( or card #ith the dimensions a and b.

+nce the stic( is mar(ed" one can lay out the elli%se by (ee%ing the %oints mar(ed a and b on thehoriGontal and !ertical a&es" res%ecti!ely" and mar(ing %oints on the elli%se at the mar( +.

3y mo!ing the stic(" one can mar( out different %oints on the elli%se at %oint +" as long as the%oints mar(ed a and b are in contact #ith the ma=or and minor diameters.

$f youre doing this on a surface that you can attach a framing s'uare to" then instead of usingmar(s" you can dri!e nails through the stic( at the a and b mar(s 8file their ti%s off a bit so theyrenot so shar% after dri!ing them through the #ood9. Drill a hole at + large enough to hold a %enciland you can dra# the elli%se by (ee%ing the nails against the s'uare 8the s'uares right angle isused to mar( the coordinate a&es9. se doublesided ta%e or clam% the s'uare to the #or(.

Appro8iate &llipse 'ith circles

Hblandford:10I

a

+

b

a b +

a

b

+



$n Figure 2" construct the elli%ses a&es and mar( off the semima=or a&is a 8+9 and semiminor a&isb 8+39. ?ar( arc D/ #ith center at + and radius of a. ?ar( 3) to be a length of 1J, of /3.

$n the follo#ing diagram" #ell only sho# the constructions in one 'uadrant" but the e&tensions tothe other 'uadrants is ob!ious.

Figure

a

b

+

/

3

)

D &

y

3) /3J,



$n Figure ," set the di!iders to distance +) in Figure 2 and mar( the arc F;< #ith center at D.*ithout changing the di!iders" mar( the arc FD< #ith center at ;. ?ar( %oint X from theintersection of the line F; #ith the y a&is.

Figure /

a

b

+

D ;

F

X<

Figure 0

a

b

+

/

X

Z



$n Figure -" dra# arc /Z #ith center at X" blending in #ith the green arc. The dots sho# the desiredelli%se and the blac( and green arcs sho# the a%%ro&imation.

#egular polygons

>et the %olygon ha!e n sides. The geometry is

The blue dashed circle is the circumscribed circle and the green circle is the inscribed circle. Thecentral angle is = 2 / n . The relationshi% bet#een 7 and r is

r = R cos

2 = R cos

n

The easiest #ay to lay out the %olygon is to scribe the circumscribed circle /D3" then ste% offconsecuti!e %oints #ith the di!iders set to distance /3. /ny setting error is cumulati!e" so a trialrun should be done to ensure you #ind u% bac( at the starting %osition. The di!ider setting is

AB = 2R sin2 = 2R sin

n

Qou can a!oid the cumulati!e error by calculating the di!ider setting for each %oint using the cosinela# and al#ays ha!ing one leg of the di!ider at %oint /:

di!ider setting =R 21−cosi #here i =2

i

/ccuracy of this method" ho#e!er" is less #hen is near 140A. $n addition" it is more tedious and"thus" more %rone to setting errors. <o#e!er" if n is e!en" then you can scribe t#o %oints %er settingfrom / and t#o more %oints from the %oint corres%onding to / at the other end of the circumscribedcircles diameter.

/nother method #hich a!oids the cumulati!e error but is also more tedious is to use the )artesiancoordinates of each %oint. These are gotten through the usual %olar con!ersions:

Figure =

+ /

3

)7

r

D



x i = R cos i = R cos 2

i

y i = R sin i = R sin 2

i

#here i 0" 1" 2" ..." n 1 and i = 2

i .

Arcs

u%%ose it is necessary to lay out the circular arc /3) as sho#n in the figure" but the center ofcircle is either inaccessible 8e.g." inside a #all9 or incon!eniently long. se the )artesiancoordinates.

$n the diagram" su%%ose #e (no# h and the radius r of the circle. Then the coordinates 8&" y9 of the%oint on the arc #ith the origin at %oint D has the ordinate y

y = r 2− x

2−r h

$f you ha!e the arc /3) and (no# the %oints / and 3" but dont (no# the radius r of the circle" you

can calculate it from measuring /) and h 3D:

r ==2-h2

4h

$t should be e!ident that if r is large and h is small" small measurement errors result in largeuncertainties in r.

/ construction method to get the radius and the center of the arc is to construct the %er%endicularbisectors to t#o chords of the circle. $f the center is not accessible" then the line 3D that it is oncan be gotten by constructing the %er%endicular bisector of the chord /).

Tapere( bo$es

u%%ose #e #ant to ma(e a symmetrical ta%ered bo& from some thin material. The bo& is asfollo#s:

&

yh

/

3

)D

/)

/

3

)

D



$n the !ertical %lane containing the %oints /" 3" )" and D" #e see the follo#ing !ie#:

$n the de!elo%ed %attern for the bo&" the 'uestion is gi!en " #hat is the angle [

cos=

1

1sin2

This #as deri!ed by Xerry at Dr. ?ath.

#eferences

The boo(s from before 152 or so can be do#nloaded from google boo(s.

HalbertI /lbert" /." )1lid Analytic G!1m!try " ?c;ra#<ill" 15-5.

HbartschI 3artsch" <." 3andb11k 1# Math!matical F1rmulas" 8translation of 5th ;erman ed.9" /cademic Press" 15-.

HblandfordI 3landford" P." h! Mast!r 3andb11k 1# )h!!tm!tal41rk " T/3 3oo(s $nc." 1541

HcoffinI )offin" X." *!ct1r Analysis" *iley" 1511.

HcorralI )orral" ?ichael" *!ct1r Calculus" %ublished by author on #eb" 2004"htt%:JJ###.mecmath.netJ .

HcrcI 3eyer" *." CRC )tandard Math!matical abl!s" 26th ed." )7) Press" 1541.

HdrmathI htt%:JJmathforum.orgJdr.mathJfa'JformulasJfa'.ag,.htmlYthree%lanes sed #ith

O

http://www.mecmath.net/

http://mathforum.org/dr.math/faq/formulas/faq.ag3.html#threeplanes

http://www.mecmath.net/

http://mathforum.org/dr.math/faq/formulas/faq.ag3.html#threeplanes



%ermission. Dr. ?ath is a great #eb resource and $ encourage you to !isit the site youll find all (inds of useful stuff %lus great !olunteers to hel% you #ith math'uestions.

Hha#(esI <a#(es" <." >uby" *." Touton" F." )1lid G!1m!try " ;inn \ )o." 1522.

HosgoodI +sgood" *. and ;raustein" *." Plan! and )1lid Analytic G!1m!try " ?acmillan" 1521.

HsalmonI almon" ;." A tr!atis! 1n th! analytic /!1m!try 1# thr!! dim!nsi1ns" -th ed."<odges" Figgis \ )o." Dublin" 1442.

HschmallI chmall" )." A First C1urs! in Analytical G!1m!try " 2nd ed." !an ostrand" 1521.

HsnyderI nyder" V. and isam" )." Analytic G!1m!try 1# )pac!" <enry <olt and )o." 151-.

H#ilsonI *ilson" ." *!ct1r Analysis" Qale ni!ersity Press" 1522. This is a nice referenceand is based on the lectures of ;ibbs.

analyticgeometry_6feb2011 (1)

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