conics & engg_curves
TRANSCRIPT
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Line types
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Line types
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Conics and Engineering Curves
Conics & Engineering Curves
Indian Institute of Technology Kharagpur
Conics & Engineering Curves 1
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Conics and Engineering Curves
Divide a line in n equal segments.
Conics & Engineering 2
C i d E i i C
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Conics and Engineering Curves
Divide a line in n equal segments.
Conics & Engineering Curves 3
C i d E i i C
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Conics and Engineering Curves
Divide a line in n equal segments.
Conics & Engineering 4
Conics and Engineering Curves
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Conics and Engineering Curves
Divide a line in n equal segments.
Conics & Engineering 5
Conics and Engineering Curves
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Conics and Engineering Curves
Divide a line in n equal segments.
Conics & Engineering 6
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Drawing a perpendicularline from a point in a line
A BCE F
D
AB is the line, and C is thepoint on it
With center C and anyradius, describe equal arcsto cut AB at E and F
From E and F describe
equal arcs to intersect at D Join C and D to give the
required perpendicular
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Drawing a line parallel to a given line at agiven distance from it
AB is the given line,and c is the givendistance
From any two pointswell apart of AB, drawtwo arcs of radiusequal to c
Draw a line tangentialto the two arcs to givethe required line
c
A B
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Bisecting an angle
ABC is the given angle
From B describe an arcto cut AB and BC at Eand D respectively
With centers E and D,draw equal arcs tointersect at F
Join BF, the requiredbisector of the angle
A
B CD
EF
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conic sections
Conics and Engineering Curves
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Conic Sections
Eccentricity = distance of the point from the focusdistance of the point from directrix
e< 1Ellipse, e = 1 Parabola and e > 1Hyperbola.
Conics & Engineering 7
Conics and Engineering Curves
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Conic Sections
1
An ellipse
Section planeAA, inclined to the axis cuts
all the generators of the
cone.
2A parabola Section plane
BB, parallel to one of the
generators cuts the cone.
A hyperbola Section
plane CC, inclined to the
axis cuts the cone on oneside of the axis.
3
4A rectangular hyperbola
Section plane DD, parallel
to the axis cuts the cone.
Conics & Engineering 8
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 9
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 10
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 11
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 12
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 13
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 14
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 15
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 16
Conics and Engineering Curves
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Ellipse by Directrix and Focus
e = 2/3 and focus distance from directrix is known.
1Divide CF in five equal
segments. Draw VE CF
and VF = VE.
Extend CE to cover the
horizontal span of the
ellipse.
2
3 Draw a vertical at 1.
Use the compass to measure4
11
and mark P1 and P
1
from F.
Any point, P2 or Pt , is C2 distance apart from the directrix and FP22or FPt apart from the focus.2
Conics & Engineering 17
Conics and Engineering Curves
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Tangent and Normals to Conics
1When a tangent at any point
on the curve is produced to
meet the directrix, the linejoining the focus with this
meeting point will be at
right angles to the line
joining the focus with the
point of contact.
Conics & Engineering 18
Conics and Engineering Curves
T d N l C i
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Tangent and Normals to Conics
1When a tangent at any point
on the curve is produced to
meet the directrix, the linejoining the focus with this
meeting point will be at
right angles to the line
joining the focus with the
point of contact.
Conics & Engineering 19
Conics and Engineering Curves
T t d N l t C i
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Tangent and Normals to Conics
1When a tangent at any point
on the curve is produced to
meet the directrix, the linejoining the focus with this
meeting point will be at
right angles to the line
joining the focus with the
point of contact.
Conics & Engineering 20
Conics and Engineering Curves
T t d N l t C i
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Tangent and Normals to Conics
1When a tangent at any point
on the curve is produced to
meet the directrix, the linejoining the focus with this
meeting point will be at
right angles to the line
joining the focus with the
point of contact.
Conics & Engineering 21
Conics and Engineering Curves
Elli f M j d Mi A i
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Ellipse from Major and Minor Axis
1Draw major and minor axes
and two circles.
2Divide it for sufficient points
to draw the ellipse.
3Draw a vertical at 1.
Draw horizontal at 1
Repeat the procedure
described in the last two
steps
4
5
Conics & Engineering 22
Conics and Engineering Curves
Ellipse from Major and Minor Axis
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Ellipse from Major and Minor Axis
1Draw major and minor axes
and two circles.
2Divide it for sufficient points
to draw the ellipse.
3Draw a vertical at 1.
Draw horizontal at 1
Repeat the procedure
described in the last two
steps
4
5
Conics & Engineering 23
Conics and Engineering Curves
Ellipse from Major and Minor Axis
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Ellipse from Major and Minor Axis
1Draw major and minor axes
and two circles.
2Divide it for sufficient points
to draw the ellipse.
3Draw a vertical at 1.
Draw horizontal at 1
Repeat the procedure
described in the last two
steps
4
5
Conics & Engineering 24
Conics and Engineering Curves
Ellipse from Major and Minor Axis
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Ellipse from Major and Minor Axis
1Draw major and minor axes
and two circles.
2Divide it for sufficient points
to draw the ellipse.
3Draw a vertical at 1.
Draw horizontal at 1
Repeat the procedure
described in the last two
steps
4
5
Conics & Engineering 25
Conics and Engineering Curves
Ellipse from Major and Minor Axis
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Ellipse from Major and Minor Axis
1Draw major and minor axes
and two circles.
2Divide it for sufficient points
to draw the ellipse.
3Draw a vertical at 1.
Draw horizontal at 1
Repeat the procedure
described in the last two
steps
4
5
Conics & Engineering 26
Conics and Engineering Curves
Ellipse from Major and Minor Axis
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Ellipse from Major and Minor Axis
1Draw major and minor axes
and two circles.
2Divide it for sufficient points
to draw the ellipse.
3Draw a vertical at 1.
Draw horizontal at 1
Repeat the procedure
described in the last two
steps
4
5
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Conics and Engineering Curves
Ellipse by Oblong method
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Ellipse by Oblong method
1Divide AO and AE in equal
segments (4)
Join 1, 2 and 3 to C
Join 1, 2 and 3 with D and
extend
2
3
4 On vertical P2P11 cut
P2x = xP11
On horizontal P2P5 cut
P2y = yP5
Find other points similarly
5
6
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Conics and Engineering Curves
Ellipse by Oblong method
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Ellipse by Oblong method
1Divide AO and AE in equal
segments (4)
Join 1, 2 and 3 to C
Join 1, 2 and 3 with D and
extend
2
3
4 On vertical P2P11 cut
P2x = xP11
On horizontal P2P5 cut
P2y = yP5
Find other points similarly
5
6
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Conics and Engineering Curves
Ellipse by Oblong method
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Ellipse by Oblong method
1Divide AO and AE in equal
segments (4)
Join 1, 2 and 3 to C
Join 1, 2 and 3 with D and
extend
2
3
4 On vertical P2P11 cut
P2x = xP11
On horizontal P2P5 cut
P2y = yP5
Find other points similarly
5
6
Conics & Engineering 31 Conics and Engineering Curves
Ellipse by Oblong method
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Ellipse by Oblong method
1Divide AO and AE in equal
segments (4)
Join 1, 2 and 3 to C
Join 1, 2 and 3 with D and
extend
2
3
4 On vertical P2P11 cut
P2x = xP11
On horizontal P2P5 cut
P2y = yP5
Find other points similarly
5
6
Conics & Engineering 32 Conics and Engineering Curves
Ellipse by Oblong method
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Ellipse by Oblong method
1Divide AO and AE in equal
segments (4)
Join 1, 2 and 3 to C
Join 1, 2 and 3 with D and
extend
2
3
4 On vertical P2P11 cut
P2x = xP11
On horizontal P2P5 cut
P2y = yP5
Find other points similarly
5
6
Conics & Engineering 33 Conics and Engineering Curves
Ellipse by Oblong method
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p y g
1Divide AO and AE in equal
segments (4)
Join 1, 2 and 3 to C
Join 1, 2 and 3 with D and
extend
2
3
4 On vertical P2P11 cut
P2x = xP11
On horizontal P2P5 cut
P2y = yP5
Find other points similarly
5
6
Conics & Engineering 34 Conics and Engineering Curves
Ellipse by Oblong method
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p y g
1Divide AO and AE in equal
segments (4)
Join 1, 2 and 3 to C
Join 1, 2 and 3 with D and
extend
2
3
4 On vertical P2P11 cut
P2x = xP11
On horizontal P2P5 cut
P2y = yP5
Find other points similarly
5
6
Conics & Engineering 35 Conics and Engineering Curves
Parabola for a given focus distance
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g
1Bisect CF for the vertex V
with proper process.
Draw a vertical at 1 and cut2
FP and FP1 = C1.1
3 Repeat for more points.
Conics & Engineering 36 Conics and Engineering Curves
Parabola for a given focus distance
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g
1Bisect CF for the vertex V
with proper process.
Draw a vertical at 1 and cut2
FP and FP1 = C1.1
3 Repeat for more points.
Conics & Engineering 37 Conics and Engineering Curves
Parabola for a given focus distance
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1Bisect CF for the vertex V
with proper process.
Draw a vertical at 1 and cut2
FP and FP1 = C1.1
3 Repeat for more points.
Conics & Engineering 38 Conics and Engineering Curves
Parabola for a given focus distance
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1Bisect CF for the vertex V
with proper process.
Draw a vertical at 1 and cut2
FP and FP1 = C1.1
3 Repeat for more points.
Conics & Engineering 39 Conics and Engineering Curves
Parabola for a given base and axis
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1Draw the base AB and axis
EF from mid point E.
2Construct rectangle ABCD
and divide AB and CD in
equal parts.
3Draw lines F1, F2, F3 and
perpendiculars 1P1, 2P2
and 3P3.
4 Cut points like P ....1
Conics & Engineering 40 Conics and Engineering Curves
Parabola for a given base and axis
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1Draw the base AB and axis
EF from mid point E.
2Construct rectangle ABCD
and divide AB and CD in
equal parts.
3Draw lines F1, F2, F3 and
perpendiculars 1P1, 2P2
and 3P3.
4 Cut points like P ....1
Conics & Engineering 41 Conics and Engineering Curves
Parabola for a given base and axis
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1Draw the base AB and axis
EF from mid point E.
2Construct rectangle ABCD
and divide AB and CD in
equal parts.
3Draw lines F1, F2, F3 and
perpendiculars 1P1, 2P2
and 3P3.
4 Cut points like P ....1
Conics & Engineering 42 Conics and Engineering Curves
Parabola for a given base and axis
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1Draw the base AB and axis
EF from mid point E.
2Construct rectangle ABCD
and divide AB and CD in
equal parts.
3Draw lines F1, F2, F3 and
perpendiculars 1P1, 2P2
and 3P3.
4 Cut points like P ....1
Conics & Engineering 43 Conics and Engineering Curves
Parabola for a given base and axis
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1Draw the base AB and axis
EF from mid point E.
2Construct rectangle ABCD
and divide AB and CD in
equal parts.
3Draw lines F1, F2, F3 and
perpendiculars 1P1, 2P2
and 3P3.
4 Cut points like P ....1
Conics & Engineering 44 Conics and Engineering Curves
Parabola in Parrallelogram
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Follow similar process.
Conics & Engineering 45 Conics and Engineering Curves
Hyperbola for a given eccentricity
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Follow similar process.
Conics & Engineering 46 Conics and Engineering Curves
Rectangular Hyperbola through a given point
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( e =
2 and equation may be assumed as xy = 1 not x2
y
2
= 1)
1Draw the axes OA, OB and
mark point P.
2 Draw CD I OA, EF I OB .
Mark points 1, 2, 3 ... (may
not be equidistant)
3
4Join O1 and 1P1 I OB and
1P1 I OA to find P1
Find the other points
similarly.
5
Conics & Engineering 47 Conics and Engineering Curves
Rectangular Hyperbola through a given point
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( e =
2 and equation may be assumed as xy = 1 not x2
y2
= 1)
1Draw the axes OA, OB and
mark point P.
2 Draw CD I OA, EF I OB .
Mark points 1, 2, 3 ... (may
not be equidistant)
3
4Join O1 and 1P1 I OB and
1P1 I OA to find P1
Find the other points
similarly.
5
Conics & Engineering 48 Conics and Engineering Curves
Rectangular Hyperbola through a given point
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( e =
2 and equation may be assumed as xy = 1 not x2
y
2
= 1)
1Draw the axes OA, OB and
mark point P.
2 Draw CD I OA, EF I OB .
Mark points 1, 2, 3 ... (may
not be equidistant)
3
4Join O1 and 1P1 I OB and
1P1 I OA to find P1
Find the other points
similarly.
5
Conics & Engineering 49 Conics and Engineering Curves
Rectangular Hyperbola through a given point
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( e =
2 and equation may be assumed as xy = 1 not x2
y
2
= 1)
1Draw the axes OA, OB and
mark point P.
2 Draw CD I OA, EF I OB .
Mark points 1, 2, 3 ... (may
not be equidistant)
3
4Join O1 and 1P1 I OB and
1P1 I OA to find P1
Find the other points
similarly.
5
Conics & Engineering 50 Conics and Engineering Curves
Rectangular Hyperbola through a given point
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( e =
2 and equation may be assumed as xy = 1 not x2
y
2
= 1)
1Draw the axes OA, OB and
mark point P.
2 Draw CD I OA, EF I OB .
Mark points 1, 2, 3 ... (may
not be equidistant)
3
4Join O1 and 1P1 I OB and
1P1 I OA to find P1
Find the other points
similarly.
5
Conics & Engineering51 Conics and Engineering Curves
Rectangular Hyperbola through a given point
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( e =
2 and equation may be assumed as xy = 1 not x
2y
2
= 1)
1Draw the axes OA, OB and
mark point P.
2 Draw CD I OA, EF I OB .
Mark points 1, 2, 3 ... (may
not be equidistant)
3
4Join O1 and 1P1 I OB and
1P1 I OA to find P1
Find the other points
similarly.
5
Conics & Engineering52 Conics and Engineering Curves
Length of an Arc
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Length of an arc subtending an angle less than 600 and length of
circumference.
Conics & Engineering53 Conics and Engineering Curves
Cycloidal Curves
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The curve generated by a fixed point on the circumference of a circle, which rolls
without slipping along a fixed straight line or a circle. Circle = generating circle. The
line or circle = directing line or circle. Used in tooth profile of gears of a dial gauge.1
4
Let P is the generating point and PA is the circumference of the circle.
Divide PA and the circle in equal parts say 12.
Draw CBI PA and CB = PA and mark 1, 2, 3 .. and C1, C2, .. .
Draw lines through 1, 2, ... I PA.
With centerC1 and radius R mark P1 on line through 1.
2
3
5
Conics & Engineering54 Conics and Engineering Curves
Cycloidal Curves
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The curve generated by a fixed point on the circumference of a circle, which rolls
without slipping along a fixed straight line or a circle. Circle = generating circle. The
line or circle = directing line or circle. Used in tooth profile of gears of a dial gauge.1
4
Let P is the generating point and PA is the circumference of the circle.
Divide PA and the circle in equal parts say 12.
Draw CBI PA and CB = PA and mark 1, 2, 3 .. and C1, C2, .. .
Draw lines through 1, 2, ... I PA.
With centerC1 and radius R mark P1 on line through 1.
2
3
5
Conics & Engineering55 Conics and Engineering Curves
Cycloidal Curves
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The curve generated by a fixed point on the circumference of a circle, which rolls
without slipping along a fixed straight line or a circle. Circle = generating circle. The
line or circle = directing line or circle. Used in tooth profile of gears of a dial gauge.1
4
Let P is the generating point and PA is the circumference of the circle.
Divide PA and the circle in equal parts say 12.
Draw CBI PA and CB = PA and mark 1, 2, 3 .. and C1, C2, .. .
Draw lines through 1, 2, ... I PA.
With centerC1 and radius R mark P1 on line through 1.
2
3
5
Conics & Engineering56 Conics and Engineering Curves
Cycloidal Curves
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The curve generated by a fixed point on the circumference of a circle, which rolls
without slipping along a fixed straight line or a circle. Circle = generating circle. The
line or circle = directing line or circle. Used in tooth profile of gears of a dial gauge.1
4
Let P is the generating point and PA is the circumference of the circle.
Divide PA and the circle in equal parts say 12.
Draw CBI PA and CB = PA and mark 1, 2, 3 .. and C1, C2, .. .
Draw lines through 1, 2, ... I PA.
With centerC1 and radius R mark P1 on line through 1.
2
3
5
Conics & Engineering57 Conics and Engineering Curves
Tangent to a Cycloid Curve
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The normal at any point on a cycloidal curve will pass through the correspondingpoint of contact between the generating circle and the directing line or circle.
Conics & Engineering58 Conics and Engineering Curves
Trochoid Curves
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curve generated by a point fixed to a circle, within or outside its circumference, as the
circle rolls along a straight line. Point within the circle = inferior trochoid, Point
outside the circle = superior trochoid,1
Process is similar to the previous. We need to extend the lines C1P1, C2P2, ..
and cut the appropriate lengths like radius R1 or R2 from the lines.
Conics & Engineering59 Conics and Engineering Curves
Involute
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The curve traced out by an end of a piece of thread unwound from a circle or a
plygon, the thread being kept tight.
1Divide PQ (= perimeter)
and circle in equal parts
(say 12).
2
Draw tangents at 1 and cutP1 on the tangent.
3 Repeat for further points.
To draw tangent and normal
at any point N draw CN,
With diameter CN describe
a semicircle to find tangent
point M.
4
5
Conics & Engineering60 Conics and Engineering Curves
Involute
-
8/12/2019 Conics & Engg_curves
68/71
The curve traced out by an end of a piece of thread unwound from a circle or a
plygon, the thread being kept tight.
1Divide PQ (= perimeter)
and circle in equal parts
(say 12).
2
Draw tangents at 1 and cutP1 on the tangent.
3 Repeat for further points.
To draw tangent and normal
at any point N draw CN,
With diameter CN describe
a semicircle to find tangent
point M.
4
5
Conics & Engineering61 Conics and Engineering Curves
Involute
-
8/12/2019 Conics & Engg_curves
69/71
The curve traced out by an end of a piece of thread unwound from a circle or a
plygon, the thread being kept tight.
1Divide PQ (= perimeter)
and circle in equal parts
(say 12).
2
Draw tangents at 1 and cutP1 on the tangent.
3 Repeat for further points.
To draw tangent and normal
at any point N draw CN,
With diameter CN describe
a semicircle to find tangent
point M.
4
5
Conics & Engineering62 Conics and Engineering Curves
Involute
-
8/12/2019 Conics & Engg_curves
70/71
The curve traced out by an end of a piece of thread unwound from a circle or a
plygon, the thread being kept tight.
1Divide PQ (= perimeter)
and circle in equal parts
(say 12).
2
Draw tangents at 1 and cutP1 on the tangent.
3 Repeat for further points.
To draw tangent and normal
at any point N draw CN,
With diameter CN describe
a semicircle to find tangent
point M.
4
5
Conics & Engineering63 Conics and Engineering Curves
Involute
-
8/12/2019 Conics & Engg_curves
71/71
The curve traced out by an end of a piece of thread unwound from a circle or a
plygon, the thread being kept tight.1
Divide PQ (= perimeter)
and circle in equal parts
(say 12).
2
Draw tangents at 1 and cutP1 on the tangent.
3 Repeat for further points.
To draw tangent and normal
at any point N draw CN,
With diameter CN describe
a semicircle to find tangent
point M.
4
5
Conics & Engineering64