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CADGME
2012
Jakub Jareš
Exploring loci of points by DGS and CASCADGME 2012
Jakub Jareš
University of South Bohemia, Czech Republic
22. - 24.06.2012
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CADGME
2012
Jakub Jareš
About me
Jakub Jareš
1st year PhD. student Theory of education inmathematics, University of South Bohemia
Interested in searching for Loci of points with computer
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CADGME
2012
Jakub Jareš
Introduction
What are loci of points?
![Page 4: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/4.jpg)
CADGME
2012
Jakub Jareš
Introduction
What are loci of points?
Loci of points belong to difficult topics of school curriculaat all levels of mathematics education
![Page 5: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/5.jpg)
CADGME
2012
Jakub Jareš
Introduction
What are loci of points?
Loci of points belong to difficult topics of school curriculaat all levels of mathematics education
There are many interesting loci of points which studentscan recognize (students can also do experiments)
![Page 6: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/6.jpg)
CADGME
2012
Jakub Jareš
Introduction
What are loci of points?
Loci of points belong to difficult topics of school curriculaat all levels of mathematics education
There are many interesting loci of points which studentscan recognize (students can also do experiments)
What technologies can we use?
![Page 7: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/7.jpg)
CADGME
2012
Jakub Jareš
Introduction
What are loci of points?
Loci of points belong to difficult topics of school curriculaat all levels of mathematics education
There are many interesting loci of points which studentscan recognize (students can also do experiments)
What technologies can we use?
Suitable to use DGS (dynamic geometry system)and CAS (computer algebra system)
![Page 8: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/8.jpg)
CADGME
2012
Jakub Jareš
DGS benefits to Loci of points
The development of computers and mathematicalsoftware allows such activities that previously were notpossible:
![Page 9: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/9.jpg)
CADGME
2012
Jakub Jareš
DGS benefits to Loci of points
The development of computers and mathematicalsoftware allows such activities that previously were notpossible:
Dynamic pictures (construction) and possibility to movewith objects (points) - we can create the hypothesis usingDGS (using trace of point)
![Page 10: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/10.jpg)
CADGME
2012
Jakub Jareš
DGS benefits to Loci of points
The development of computers and mathematicalsoftware allows such activities that previously were notpossible:
Dynamic pictures (construction) and possibility to movewith objects (points) - we can create the hypothesis usingDGS (using trace of point)
Verification of hypothesis in DGS (button locus)
![Page 11: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/11.jpg)
CADGME
2012
Jakub Jareš
DGS benefits to Loci of points
The development of computers and mathematicalsoftware allows such activities that previously were notpossible:
Dynamic pictures (construction) and possibility to movewith objects (points) - we can create the hypothesis usingDGS (using trace of point)
Verification of hypothesis in DGS (button locus)
But!!! DGS are based on numerical calculations, the resultcan not be considered absolutely true
![Page 12: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/12.jpg)
CADGME
2012
Jakub Jareš
Using DGS in exploring loci of points
There exist many DGS systems with some minordifferences which behave in a similar way whenobtaining loci
![Page 13: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/13.jpg)
CADGME
2012
Jakub Jareš
Using DGS in exploring loci of points
There exist many DGS systems with some minordifferences which behave in a similar way whenobtaining loci
In general, two objects must be selected:
![Page 14: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/14.jpg)
CADGME
2012
Jakub Jareš
Using DGS in exploring loci of points
There exist many DGS systems with some minordifferences which behave in a similar way whenobtaining loci
In general, two objects must be selected:
driving point or mover (as name says it is bound to apath)
![Page 15: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/15.jpg)
CADGME
2012
Jakub Jareš
Using DGS in exploring loci of points
There exist many DGS systems with some minordifferences which behave in a similar way whenobtaining loci
In general, two objects must be selected:
driving point or mover (as name says it is bound to apath)
locus point (must depend somehow on the first one)
![Page 16: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/16.jpg)
CADGME
2012
Jakub Jareš
Using DGS in exploring loci of points
There exist many DGS systems with some minordifferences which behave in a similar way whenobtaining loci
In general, two objects must be selected:
driving point or mover (as name says it is bound to apath)
locus point (must depend somehow on the first one)
Since the element dependency is preserved the drivingpoint traverses its path, the locus is a trajectory of thelocus point.
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CADGME
2012
Jakub Jareš
Exploring loci by DGS
To formulate a problem to students
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CADGME
2012
Jakub Jareš
Exploring loci by DGS
To formulate a problem to students
we construct the problem with DGS
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CADGME
2012
Jakub Jareš
Exploring loci by DGS
To formulate a problem to students
we construct the problem with DGS
we can guess, what it can be
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CADGME
2012
Jakub Jareš
Exploring loci by DGS
To describe and demonstrate the problem by DGS and
![Page 21: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/21.jpg)
CADGME
2012
Jakub Jareš
Exploring loci by DGS
To describe and demonstrate the problem by DGS and
move with the driving point
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CADGME
2012
Jakub Jareš
Exploring loci by DGS
To describe and demonstrate the problem by DGS and
move with the driving point
watch, what locus is generated with locus point
![Page 23: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/23.jpg)
CADGME
2012
Jakub Jareš
Exploring loci by DGS
To describe and demonstrate the problem by DGS and
move with the driving point
watch, what locus is generated with locus point
discuss with students about, what the locus can be
![Page 24: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/24.jpg)
CADGME
2012
Jakub Jareš
Exploring loci by DGS
To verify the locus, we can use a button LOCUS andwe obtain the curve, but only in plane / 2D
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CADGME
2012
Jakub Jareš
Exploring loci by CAS
To identify locus we need locus equations or itscharacteristic property
Translation of a geometry problem into equations orinequations
![Page 26: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/26.jpg)
CADGME
2012
Jakub Jareš
Exploring loci by CAS
To identify locus we need locus equations or itscharacteristic property
Translation of a geometry problem into equations orinequations
The use of CAS to obtain locus equation from the systemof equations or inequations
![Page 27: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/27.jpg)
CADGME
2012
Jakub Jareš
Exploring loci by CAS
To identify locus we need locus equations or itscharacteristic property
Translation of a geometry problem into equations orinequations
The use of CAS to obtain locus equation from the systemof equations or inequations
Elimination of variables is necessary
![Page 28: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/28.jpg)
CADGME
2012
Jakub Jareš
Exploring loci by CAS
To identify locus we need locus equations or itscharacteristic property
Translation of a geometry problem into equations orinequations
The use of CAS to obtain locus equation from the systemof equations or inequations
Elimination of variables is necessary
Gröbner basis of ideals
![Page 29: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/29.jpg)
CADGME
2012
Jakub Jareš
Exploring loci by CAS
To identify locus we need locus equations or itscharacteristic property
Translation of a geometry problem into equations orinequations
The use of CAS to obtain locus equation from the systemof equations or inequations
Elimination of variables is necessary
Gröbner basis of idealsCharacteristic sets
![Page 30: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/30.jpg)
CADGME
2012
Jakub Jareš
Elimination
Elimination of variables is a basic procedure in exploring loci.
Realize that we eliminate variables in the system ofnon-linear algebraic equations.
By elimination we used the program CoCoA which is freelydistributed athttp://cocoa.dima.unige.it.It is based on Gröbner basis of ideals.
Another elimination program is Geother which is freelydistributed athttp://www-calfor.lip6.fr/∼wang/epsilon/.It is based on Wu–Ritt characteristic sets.
![Page 31: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/31.jpg)
CADGME
2012
Jakub Jareš
Asteroidintroduction
Problem:
Let k be a circle centered at O and two perpendicular lines x ,y through point O. Denote A, B the feet of perpendicular linesdropped to x , y from an arbitrary C ∈ k. Let M be anintersection of a segment AB and perpendicular line from C toAB .
Find the locus M when C moves along circle k.
First we construct with GeoGebra this problem. Using thebutton LOCUS we construct the locus of M when C movesalong k.
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CADGME
2012
Jakub Jareš
Asteroidintroduction
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CADGME
2012
Jakub Jareš
Asteroidintroduction
”It looks like curve of asteroid”
We have no equation
Is it true ?
What is the solution?
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CADGME
2012
Jakub Jareš
Asteroidintroduction
Cooperation between
DGS and CAS
is needed!
![Page 35: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/35.jpg)
CADGME
2012
Jakub Jareš
Asteroidlocus equations
Place a coordinate system so that A = [p, 0], B = [0, q],C = [p, q], M = [m, n] and let k be a circle with the equationk : x2 − y2 − a2 = 0.
We translate the geometry situation into the set of polynomialequations.
M ∈ AB ⇒ H1 : qm + pn − pq = 0,
M ∈ a⇒ H2 : pm − qn − p2 + q2 = 0.
Further
C ∈ k ⇒ H3 : p2 − q2 − a2 = 0
![Page 36: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/36.jpg)
CADGME
2012
Jakub Jareš
Asteroidlocus equations
We get the system of three equations H1 = 0, H2 = 0, H3 = 0in variables p, q,m, n, a.
To find the locus of M = [m, n] we eliminate variables p, q inthe ideal I = (H1,H2,H3) to get a relation in m, n whichdepends on a. We enter in CoCoA
UseR ::= Q[p, q, m, n, a];I := Ideal(qm+ pn− pq, pm− qn− p2 + q2, p2 + q2 − a2);Elim(p..q, I);
![Page 37: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/37.jpg)
CADGME
2012
Jakub Jareš
Asteroidlocus equations
and get
Ideal(−2/9m6a2 − 2/3m4n2a2 − 2/3m2n4a2 − 2/9n6a2++2/3m4a4 − 14/3m2n2a4 + 2/3n4a4 − 2/3m2a6−−2/3n2a6 + 2/9a8).
Solve equation
−2/9m6a2 − 2/3m4n2a2 − 2/3m2n4a2 − 2/9n6a2 + 2/3m4a4 −14/3m2n2a4 + 2/3n4a4 − 2/3m2a6 − 2/3n2a6 + 2/9a8 = 0
get
n2 − (a2/3 −m2/3)3 = 0
![Page 38: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/38.jpg)
CADGME
2012
Jakub Jareš
Asteroidlocus equations
The equation can be expressed as a function of two variablesm, n
n = ±√
(a2/3 −m2/3)3
Or we can display the function as an implicitplot, when a = 1
Ast := −2/9m6− 2/3m4n2 − 2/3m2n4− 2/9n6+ 2/3m4−14/3m2n2 + 2/3n4 − 2/3m2 − 2/3n2 + 2/9 = 0
The locus above was found by algebraic and computer tools.
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CADGME
2012
Jakub Jareš
Asteroid - resultsfunction of two variables
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CADGME
2012
Jakub Jareš
Asteroid - resultsimplicitplot of: −2/9m6 − 2/3m4n2 − 2/3m2n4 − 2/9n6 + 2/3m4 −14/3m2n2 + 2/3n4 − 2/3m2 − 2/3n2 + 2/9 = 0
-0,8
0
m
n
0
0,4
-0,8 0,8
0,8
0,4-0,4
-0,4
![Page 41: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/41.jpg)
CADGME
2012
Jakub Jareš
Strophoidintroduction
Problem:
Let ABC be a triangle with the given side AB and the vertex Con a circle k centered at A and radius |AB |.
Find the locus of the orthocenter M of ABC when C moves onk.
First we construct in GeoGebra the triangle ABC with the pointC on the circle k . Using the button LOCUS we construct thelocus of the orthocenter M when C moves along k.
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CADGME
2012
Jakub Jareš
Strophoidintroduction
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CADGME
2012
Jakub Jareš
Strophoidlocus equations
Derivation of the locus is as follows:
Suppose that A = [0, 0], B = [a, 0], C = [p, q] and M = [m, n].
Then:
M ∈ hAB ⇒ H1 : m − p = 0,
M ∈ hBC ⇒ H2 : (p − a)m + qn = 0,
C ∈ k ⇒ H3 : p2 + q2 − a2 = 0.
![Page 44: Exploring loci of points by DGS and CAS - CADGME 2012 · Exploring loci of points by DGS and CAS CADGME 2012 ... Using DGS in exploring loci of points There exist many DGS systems](https://reader033.vdocuments.us/reader033/viewer/2022041721/5e4e92d4a4632f62db5b445c/html5/thumbnails/44.jpg)
CADGME
2012
Jakub Jareš
Strophoidlocus equations
Elimination of p, q in the system H1 = 0, H2 = 0, H3 = 0 givesin the program Epsilon
with(epsilon);U := [m − p, (p − a) ∗m + q ∗ n, p2 + q2 − a2] :X := [m, n, p, q] :CharSet(U,X );
the equationan2 −m2a +m3 +mn2 = 0
which is the equation of a cubic curve called strophoid.
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CADGME
2012
Jakub Jareš
Strophoidlocus equations
Solve this equation
solve(a ∗ n2 − m2 ∗ a+ m3 + m ∗ n2 = 0, n2);
and get
n2(a +m)−m2(a −m) = 0
The equation can be expressed as a function of two variablesm, n
n = ±m ·
√
a −m
a +m
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CADGME
2012
Jakub Jareš
Strophoid - resultsfunction of two variables
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CADGME
2012
Jakub Jareš
Strophoid - resultsimplicitplot a ∗ n2 −m2 ∗ a +m3 +m ∗ n
2= 0
n
1
0,5
-0,5
-0,40
0
m
0,4
-1
0,8
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CADGME
2012
Jakub Jareš
Strophoid spaceintroduction
We try to do this problem in space:
Problem:
Let ABC be a triangle with the given side AB and thevertex C on a ball κ centered at A and radius |AB|.
Find the locus of the orthocenter M of ABC when Cmoves on κ.
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CADGME
2012
Jakub Jareš
Strophoid spaceintroduction
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CADGME
2012
Jakub Jareš
Strophoid spaceintroduction
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CADGME
2012
Jakub Jareš
Strophoid spacelocus equations
Place a coordinate system so that A = [0, 0, 0], B = [a, 0, 0],C = [p, q, r ], M = [m, n, o] and let κ be a ball in the center Awith the equation κ : x2 + y2 + z2 − a2 = 0.
We translate the geometry situation into the set of polynomialequations.
M ∈ ρ1 ⇒ H1 : m − p = 0,
M ∈ ρ2 ⇒ H2 : pm + qn + ro − pa = 0,
M ∈ ABC ⇒ H3 : −arn+ aqo = 0,
C ∈ k ⇒ H4 : p2 + q2 + r2 − a2 = 0.
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CADGME
2012
Jakub Jareš
Strophoid spacelocus equations
We get the system of four equations H1 = 0, H2 = 0, H3 = 0and H4 = 0 in variables p, q, r ,m, n, o, a.
To find the locus of M = [m, n, o] we eliminate variables p, q, rin the ideal I = (H1,H2,H3,H4) to get a relation in m, n, owhich depends on a. We enter in CoCoA
UseR ::= Q[p, q, r, m, n, o, a];I := Ideal(m− p, pm+ qn+ ro− pa,−arn+ aqo,p2 + q2 + r2 − a2);Elim(p..r, I);
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CADGME
2012
Jakub Jareš
Strophoid spacelocus equations
and get
Ideal(m4a+ m2n2a+ m2o2a− 2m3a2 + m2a3 − n2a3 − o2a3)
Factor this equation in Maple and get
a(−m+ a)(am2 − an2 − ao2 −m3 −mn2 −mo2) = 0.
Equation
am2 − an2 − ao2 −m3 − mn2 −mo2 = 0
is equation of our searching locus.
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CADGME
2012
Jakub Jareš
Strophoid space - resultsimplicitplot am2 − an2 − ao2 −m3 −mn2 −mo2 = 0
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CADGME
2012
Jakub Jareš
Strophoid space - resultsimplicitplot am2 − an2 − ao2 −m3 −mn2 −mo2 = 0 with sphere
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CADGME
2012
Jakub Jareš
The end
Future vision
New technologies shows newpossibilities for exploringLOCI, not only in plane, butalso in space. . .
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CADGME
2012
Jakub Jareš
The end
Thank you