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Abstract
We present two approaches to symbolically obtain isoptic curves
in GeoGebra in an automated, interactive process. Both methods
are based on computing implicit locus equations, by using
algebraization of the geometric setup and elimination of the
intermediate variables. These methods can be considered as
automatic discovery.
Our first approach uses pure computer algebra support of
GeoGebra, utilizing symbolic differentiation.
The second approach hides all details in computer algebra from the
user: the input problem is defined by a purely geometric way.
In both approaches the output is dynamically changed when using
a slider bar or the free points are dragged.
Programming the internal GeoGebra computations is an on-going
work with various challenges in optimizing computations and
to avoiding unnecessary extra curves in the output.
Isoptic curves
Let Cbe a plane curve. For a given angle θsuch that
0≤θ≤180◦, a θ-isoptic curve (or simply a θ-isoptic) of Cis the
geometric locus of points Mthrough which passes a pair of
tangents with an angle of θbetween them.
If θ= 90◦, i.e. if the tangents are perpendicular, then the isoptic
curve is called an orthoptic curve.
Isoptic curves may either exist or not, depending on the given
curve and on the angle.
Orthoptics of conics
Hyperbola
The existence of an
orthoptic curve for a
hyperbola depends on the
eccentricity c/a, where
c2=a2−b2.
If it exists, the orthoptic
curve of the hyperbola
with canonical equation
x2
a2−y2
b2= 1 (i.e. the focal
axis is the x=axis) is the
circle whose equation is
x2+y2=a2−b2, also
called the director circle.https://www.geogebra.org/m/tZcGGrCm
Previous and related work
IDana-Picard, Th., Mann, G. and Zehavi, N.: From conic
intersections to toric intersections: the case of the isoptic curves of
an ellipse, The Montana Mathematical Enthusiast 9 (1), pp. 59–76.
2011.
IDana-Picard, Th.: An automated study of isoptic curves of an
astroid, Preprint, JCT, 2018.
IDana-Picard, Th. and Naiman, A.: Isoptics of Fermat curves,
Preprint, JCT, 2018.
IMiernowski, A. and Mosgawa, W.: Isoptics of Pairs of Nested
Closed Strictly Convex Curves and Crofton-Type Formulas, Beitr¨age
zur Algebra und Geometrie Contributions to Algebra and Geometry
42 (1), pp. 281–288. 2001.
ISza lkowski, D.: Isoptics of open rosettes, Annales Universitatis
Mariae Curie-Sk lodowska, Lublin – Polonia LIX, Section A,
pp. 119–128, 2005.
ICsima, G.: Isoptic curves and surfaces . PhD thesis, BUTE,
Math. Institute, Department of Geometry, Budapest, 2017.
Two novel approaches in GeoGebra
An overview
IBoth
Ican be considered as automatic discovery,
Ideliver an algebraic output: a polynomial (with its graphical
representation) via Gr¨obner bases and elimination.
IThe first approach
Iuses pure computer algebra support of GeoGebra:
symbolic differentiation of the input formula,
Iallows the output to be changed dynamically with a slider bar
(dynamic study),
Ican do observations up to quartic curves
(due to computational challenges).
IThe second approach
Ihides all details in computer algebra from the user:
the input problem is given in a a purely geometric way,
Iis a handy method for a new kind of man and machine
communication,
Iworks only for certain conics.
Two novel approaches in GeoGebra
An overview
IBoth
Ican be considered as automatic discovery,
Ideliver an algebraic output: a polynomial (with its graphical
representation) via Gr¨obner bases and elimination.
IThe first approach
Iuses pure computer algebra support of GeoGebra:
symbolic differentiation of the input formula,
Iallows the output to be changed dynamically with a slider bar
(dynamic study),
Ican do observations up to quartic curves
(due to computational challenges).
IThe second approach
Ihides all details in computer algebra from the user:
the input problem is given in a a purely geometric way,
Iis a handy method for a new kind of man and machine
communication,
Iworks only for certain conics.
Two novel approaches in GeoGebra
An overview
IBoth
Ican be considered as automatic discovery,
Ideliver an algebraic output: a polynomial (with its graphical
representation) via Gr¨obner bases and elimination.
IThe first approach
Iuses pure computer algebra support of GeoGebra:
symbolic differentiation of the input formula,
Iallows the output to be changed dynamically with a slider bar
(dynamic study),
Ican do observations up to quartic curves
(due to computational challenges).
IThe second approach
Ihides all details in computer algebra from the user:
the input problem is given in a a purely geometric way,
Iis a handy method for a new kind of man and machine
communication,
Iworks only for certain conics.
The first approach
Let Cbe an algebraic curve given by an implicit equation
F(x,y) = 0.
1. Compute the derivatives dx=F0
xand dy=F0
y.
2. Consider points A(xA,yA) and B(xB,yB) that are assumed to
be points of the curve, that is,
F(xA,yA) = 0 (1)
and
F(xB,yB) = 0 (2)
hold.
3. Compute the partial derivatives px,A=F0
x(xA,yA),
px,B=F0
x(xB,yB), py,A=F0
y(xA,yA) and py,B=F0
y(xB,yB).
The first approach
Let Cbe an algebraic curve given by an implicit equation
F(x,y) = 0.
1. Compute the derivatives dx=F0
xand dy=F0
y.
2. Consider points A(xA,yA) and B(xB,yB) that are assumed to
be points of the curve, that is,
F(xA,yA) = 0 (1)
and
F(xB,yB) = 0 (2)
hold.
3. Compute the partial derivatives px,A=F0
x(xA,yA),
px,B=F0
x(xB,yB), py,A=F0
y(xA,yA) and py,B=F0
y(xB,yB).
The first approach
Let Cbe an algebraic curve given by an implicit equation
F(x,y) = 0.
1. Compute the derivatives dx=F0
xand dy=F0
y.
2. Consider points A(xA,yA) and B(xB,yB) that are assumed to
be points of the curve, that is,
F(xA,yA) = 0 (1)
and
F(xB,yB) = 0 (2)
hold.
3. Compute the partial derivatives px,A=F0
x(xA,yA),
px,B=F0
x(xB,yB), py,A=F0
y(xA,yA) and py,B=F0
y(xB,yB).
The first approach
Let Cbe an algebraic curve given by an implicit equation
F(x,y) = 0.
1. Compute the derivatives dx=F0
xand dy=F0
y.
2. Consider points A(xA,yA) and B(xB,yB) that are assumed to
be points of the curve,
that is,
F(xA,yA) = 0 (1)
and
F(xB,yB) = 0 (2)
hold.
3. Compute the partial derivatives px,A=F0
x(xA,yA),
px,B=F0
x(xB,yB), py,A=F0
y(xA,yA) and py,B=F0
y(xB,yB).
The first approach
Let Cbe an algebraic curve given by an implicit equation
F(x,y) = 0.
1. Compute the derivatives dx=F0
xand dy=F0
y.
2. Consider points A(xA,yA) and B(xB,yB) that are assumed to
be points of the curve, that is,
F(xA,yA) = 0 (1)
and
F(xB,yB) = 0 (2)
hold.
3. Compute the partial derivatives px,A=F0
x(xA,yA),
px,B=F0
x(xB,yB), py,A=F0
y(xA,yA) and py,B=F0
y(xB,yB).
The first approach
Let Cbe an algebraic curve given by an implicit equation
F(x,y) = 0.
1. Compute the derivatives dx=F0
xand dy=F0
y.
2. Consider points A(xA,yA) and B(xB,yB) that are assumed to
be points of the curve, that is,
F(xA,yA) = 0 (1)
and
F(xB,yB) = 0 (2)
hold.
3. Compute the partial derivatives px,A=F0
x(xA,yA),
px,B=F0
x(xB,yB), py,A=F0
y(xA,yA) and py,B=F0
y(xB,yB).
The first approach
(cont’’d)
5. When defining a point P(x,y) that is an element of both
tangents t1and t2to c, the points
A,A0= (xA+py,A,yA−px,A) and Pmust be collinear;
for the same reason, also
B,B0= (xB+py,B,yB−px,B) and Pare collinear.
So the following equations hold:
xAyA1
xA+py,AyA−px,A1
x y 1
= 0,(4)
xByB1
xB+py,ByB−px,B1
x y 1
= 0.(5)
The first approach
(cont’’d)
5. When defining a point P(x,y) that is an element of both
tangents t1and t2to c, the points
A,A0= (xA+py,A,yA−px,A) and Pmust be collinear;
for the same reason, also
B,B0= (xB+py,B,yB−px,B) and Pare collinear.
So the following equations hold:
xAyA1
xA+py,AyA−px,A1
x y 1
= 0,(4)
xByB1
xB+py,ByB−px,B1
x y 1
= 0.(5)
The first approach
(cont’’’d)
6. Now we have 5 equations.
By eliminating all variables
but xand ywe obtain an implicit equation whose graphical
representation is, at least partly, the θ-isoptic curve.
This technique (“elimination theory”, “automated geometry
theorem proving”, “automated discovery”) is discussed
in detail in:
ICox, D., Little, J. and O’Shea, D.: Ideals, varieties and
algorithms. Third edition. Springer, 2007.
IChou, S.-C.: Mechanical Geometry Theorem Proving,
Reidel Dordrecht, 1987.
IAb´anades, M. A., Botana, F., Kov´acs, Z., Recio, T. and
S´olyom-Gecse, C.: Development of automatic reasoning tools
in GeoGebra. Software Demonstration at the ISSAC 2016
Conf. ACM Comm. in Comp. Alg. 50 (3), pp. 85–88. 2016.
Theoretically, the obtained implicit equation is a multiple of
the algebraic closure of the geometrically expected set.
That is, some factors of the obtained implicit equation will
contain the expected curve.
The first approach
(cont’’’d)
6. Now we have 5 equations. By eliminating all variables
but xand ywe obtain an implicit equation whose graphical
representation is, at least partly, the θ-isoptic curve.
This technique (“elimination theory”, “automated geometry
theorem proving”, “automated discovery”) is discussed
in detail in:
ICox, D., Little, J. and O’Shea, D.: Ideals, varieties and
algorithms. Third edition. Springer, 2007.
IChou, S.-C.: Mechanical Geometry Theorem Proving,
Reidel Dordrecht, 1987.
IAb´anades, M. A., Botana, F., Kov´acs, Z., Recio, T. and
S´olyom-Gecse, C.: Development of automatic reasoning tools
in GeoGebra. Software Demonstration at the ISSAC 2016
Conf. ACM Comm. in Comp. Alg. 50 (3), pp. 85–88. 2016.
Theoretically, the obtained implicit equation is a multiple of
the algebraic closure of the geometrically expected set.
That is, some factors of the obtained implicit equation will
contain the expected curve.
The first approach
(cont’’’d)
6. Now we have 5 equations. By eliminating all variables
but xand ywe obtain an implicit equation whose graphical
representation is, at least partly, the θ-isoptic curve.
This technique (“elimination theory”, “automated geometry
theorem proving”, “automated discovery”) is discussed
in detail in:
ICox, D., Little, J. and O’Shea, D.: Ideals, varieties and
algorithms. Third edition. Springer, 2007.
IChou, S.-C.: Mechanical Geometry Theorem Proving,
Reidel Dordrecht, 1987.
IAb´anades, M. A., Botana, F., Kov´acs, Z., Recio, T. and
S´olyom-Gecse, C.: Development of automatic reasoning tools
in GeoGebra. Software Demonstration at the ISSAC 2016
Conf. ACM Comm. in Comp. Alg. 50 (3), pp. 85–88. 2016.
Theoretically, the obtained implicit equation is a multiple of
the algebraic closure of the geometrically expected set.
That is, some factors of the obtained implicit equation will
contain the expected curve.
The first approach
(cont’’’d)
6. Now we have 5 equations. By eliminating all variables
but xand ywe obtain an implicit equation whose graphical
representation is, at least partly, the θ-isoptic curve.
This technique (“elimination theory”, “automated geometry
theorem proving”, “automated discovery”) is discussed
in detail in:
ICox, D., Little, J. and O’Shea, D.: Ideals, varieties and
algorithms. Third edition. Springer, 2007.
IChou, S.-C.: Mechanical Geometry Theorem Proving,
Reidel Dordrecht, 1987.
IAb´anades, M. A., Botana, F., Kov´acs, Z., Recio, T. and
S´olyom-Gecse, C.: Development of automatic reasoning tools
in GeoGebra. Software Demonstration at the ISSAC 2016
Conf. ACM Comm. in Comp. Alg. 50 (3), pp. 85–88. 2016.
Theoretically, the obtained implicit equation is a multiple of
the algebraic closure of the geometrically expected set.
That is, some factors of the obtained implicit equation will
contain the expected curve.
Examples
The orthoptic of y=x4(cont’d)
After eliminating all variables but xand yfrom this system by
using a CAS, we obtain the equation
65536x6+ 196608x4y2+ 196608x2y4
−41472x2y+ 65536y6+ 13824y3+ 729·
(16777216x6y3+ 50331648x4y5+ 5308416x4y2+ 50331648x2y7+
5308416x2y4+ 559872x2y+ 16777216y9
−1769472y6
−186624y3+ 19683) = 0.
This can be written as f1·f2= 0. Both f1and f2are reducible
over C. After numerical and visual experiments, it turns out that f1
has no real geometrical meaning, but f2has. Also, f2has a divisor
f=x2y+y3−3/8y23
√2−9y3
√4
64 +27
256.
According to GeoGebra’s numerical precision the cubic f= 0 is
indeed the orthoptic of y=x4.
Examples
The orthoptic of y=x4(cont’d)
After eliminating all variables but xand yfrom this system by
using a CAS, we obtain the equation
65536x6+ 196608x4y2+ 196608x2y4
−41472x2y+ 65536y6+ 13824y3+ 729·
(16777216x6y3+ 50331648x4y5+ 5308416x4y2+ 50331648x2y7+
5308416x2y4+ 559872x2y+ 16777216y9
−1769472y6
−186624y3+ 19683) = 0.
This can be written as f1·f2= 0.
Both f1and f2are reducible
over C. After numerical and visual experiments, it turns out that f1
has no real geometrical meaning, but f2has. Also, f2has a divisor
f=x2y+y3−3/8y23
√2−9y3
√4
64 +27
256.
According to GeoGebra’s numerical precision the cubic f= 0 is
indeed the orthoptic of y=x4.
Examples
The orthoptic of y=x4(cont’d)
After eliminating all variables but xand yfrom this system by
using a CAS, we obtain the equation
65536x6+ 196608x4y2+ 196608x2y4
−41472x2y+ 65536y6+ 13824y3+ 729·
(16777216x6y3+ 50331648x4y5+ 5308416x4y2+ 50331648x2y7+
5308416x2y4+ 559872x2y+ 16777216y9
−1769472y6
−186624y3+ 19683) = 0.
This can be written as f1·f2= 0. Both f1and f2are reducible
over C.
After numerical and visual experiments, it turns out that f1
has no real geometrical meaning, but f2has. Also, f2has a divisor
f=x2y+y3−3/8y23
√2−9y3
√4
64 +27
256.
According to GeoGebra’s numerical precision the cubic f= 0 is
indeed the orthoptic of y=x4.
Examples
The orthoptic of y=x4(cont’d)
After eliminating all variables but xand yfrom this system by
using a CAS, we obtain the equation
65536x6+ 196608x4y2+ 196608x2y4
−41472x2y+ 65536y6+ 13824y3+ 729·
(16777216x6y3+ 50331648x4y5+ 5308416x4y2+ 50331648x2y7+
5308416x2y4+ 559872x2y+ 16777216y9
−1769472y6
−186624y3+ 19683) = 0.
This can be written as f1·f2= 0. Both f1and f2are reducible
over C. After numerical and visual experiments, it turns out that f1
has no real geometrical meaning, but f2has.
Also, f2has a divisor
f=x2y+y3−3/8y23
√2−9y3
√4
64 +27
256.
According to GeoGebra’s numerical precision the cubic f= 0 is
indeed the orthoptic of y=x4.
Examples
The orthoptic of y=x4(cont’d)
After eliminating all variables but xand yfrom this system by
using a CAS, we obtain the equation
65536x6+ 196608x4y2+ 196608x2y4
−41472x2y+ 65536y6+ 13824y3+ 729·
(16777216x6y3+ 50331648x4y5+ 5308416x4y2+ 50331648x2y7+
5308416x2y4+ 559872x2y+ 16777216y9
−1769472y6
−186624y3+ 19683) = 0.
This can be written as f1·f2= 0. Both f1and f2are reducible
over C. After numerical and visual experiments, it turns out that f1
has no real geometrical meaning, but f2has. Also, f2has a divisor
f=x2y+y3−3/8y23
√2−9y3
√4
64 +27
256.
According to GeoGebra’s numerical precision the cubic f= 0 is
indeed the orthoptic of y=x4.
Examples
The orthoptic of y=x4(cont’d)
After eliminating all variables but xand yfrom this system by
using a CAS, we obtain the equation
65536x6+ 196608x4y2+ 196608x2y4
−41472x2y+ 65536y6+ 13824y3+ 729·
(16777216x6y3+ 50331648x4y5+ 5308416x4y2+ 50331648x2y7+
5308416x2y4+ 559872x2y+ 16777216y9
−1769472y6
−186624y3+ 19683) = 0.
This can be written as f1·f2= 0. Both f1and f2are reducible
over C. After numerical and visual experiments, it turns out that f1
has no real geometrical meaning, but f2has. Also, f2has a divisor
f=x2y+y3−3/8y23
√2−9y3
√4
64 +27
256.
According to GeoGebra’s numerical precision the cubic f= 0 is
indeed the orthoptic of y=x4.
Examples
35◦-isoptic of a hyperbola (cont’d)
Algebraically, after elimination, GeoGebra obtains
2x14 −2y14 −c2x12 −c2y12 −10x2y12 −18x4y10 −10x6y8+ 10x8y6+ 18x10y4
+ 10x12y2−6c2x2y10 −15c2x4y8−20c2x6y6−15c2x8y4−6c2x10 y2−23x12
−23y12 + 12c2x10 −12c2y10 −58x2y10 −25x4y8+ 20x6y6−25x8y4−58x10y2
−36c2x2y8−24c2x4y6+ 24c2x6y4+ 36c2x8y2+ 112x10 −112y10 −60c2x8
−60c2y8−80x2y8+ 32x4y6−32x6y4+ 80x8y2−48c2x2y6+ 24c2x4y4−48c2x6y2
−300x8−300y8+ 160c2x6−160c2y6+ 144x2y6−136x4y4+ 144x6y2+ 96c2x2y4
−96c2x4y2+ 480x6−480y6−240c2x4−240c2y4+ 544x2y4−544x4y2+ 288c2x2y2
−464x4−464y4+ 192c2x2−192c2y2+ 608x2y2−64c2+ 256x2−256y2−64 = 0,
where c= cos27
36 π.
After factorization this can be simplified to
cx4+ 2cx2y2+cy 4−x4−2x2y2−4cx 2−y4+ 4cy 2+ 4c= 0,
that is, the isoptic curve is a quartic (containing also the set of
points for the 145◦-isoptic).
Examples
35◦-isoptic of a hyperbola (cont’d)
Algebraically, after elimination, GeoGebra obtains
2x14 −2y14 −c2x12 −c2y12 −10x2y12 −18x4y10 −10x6y8+ 10x8y6+ 18x10y4
+ 10x12y2−6c2x2y10 −15c2x4y8−20c2x6y6−15c2x8y4−6c2x10 y2−23x12
−23y12 + 12c2x10 −12c2y10 −58x2y10 −25x4y8+ 20x6y6−25x8y4−58x10y2
−36c2x2y8−24c2x4y6+ 24c2x6y4+ 36c2x8y2+ 112x10 −112y10 −60c2x8
−60c2y8−80x2y8+ 32x4y6−32x6y4+ 80x8y2−48c2x2y6+ 24c2x4y4−48c2x6y2
−300x8−300y8+ 160c2x6−160c2y6+ 144x2y6−136x4y4+ 144x6y2+ 96c2x2y4
−96c2x4y2+ 480x6−480y6−240c2x4−240c2y4+ 544x2y4−544x4y2+ 288c2x2y2
−464x4−464y4+ 192c2x2−192c2y2+ 608x2y2−64c2+ 256x2−256y2−64 = 0,
where c= cos27
36 π. After factorization this can be simplified to
cx4+ 2cx2y2+cy 4−x4−2x2y2−4cx 2−y4+ 4cy 2+ 4c= 0,
that is, the isoptic curve is a quartic (containing also the set of
points for the 145◦-isoptic).
Computational features of the first approach
IFast computations for conics (dragging of θis possible)
IFeasible (but slow) computations for certain quartics
IInfeasible computations for most quartics
and other higher degree polynomials
IGeoGebra’s CAS View is involved
IIn most cases, the output contains additional factors that
have no geometrical meaning (“extended output”)
IGeoGebra’s Graphics View correctly plots the extended output
IFactorization of the extended output may be incomplete
in GeoGebra (Maple or Singular can be used for absolute
factorization): the minimal algebraic form of the curve is
difficult to determine
Some features of the second approach
IGeoGebra’s CAS View is not involved
IEach type of input (circle, parabola, . . .) must be separately
implemented (=programmed) internally in GeoGebra
IComputations are feasible for orthoptics of circle and parabola
(moderately slow dragging of θis possible)
ITo obtain isoptics, the AreCongruent command must be used
IComputations are slow for isoptics of circle and parabola
IIsoptic curves may contain extra linear components
due to algebraic issues
IOther curves (ellipse, hyperbola and non-conics)
are not yet implemented
IThe output may contain additional factors that have no
geometrical meaning (“extended output”)
IFinding the “best” equation system describing the geometric
setup can be tricky
Conclusion
INo longer a researchers-only topic? Students can be involved!
IAnother application of Gr¨obner bases and elimination
(for polynomial input)
IExperiments exploiting (computer) algebraic and
(dynamic geometric) graphical representations
IFurther studies may involve more efficient computations
and further tricks
