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Computing real inflection points of cubic algebraic curves

Falai Chen, Wenping Wang
2003 Computer Aided Geometric Design  
In this paper we present an algorithm for computing the real inflection points of a real planar cubic algebraic curve.  ...  Shape modeling using planar cubic algebraic curves calls for computing the real inflection points of these curves since inflection points represents important shape feature.  ...  This article is about computing the real inflection points of an irreducible planar cubic algebraic curve.  ... 
doi:10.1016/s0167-8396(03)00022-0 fatcat:wzxfoplrajdhnoonrq6qlrd2om

Page 3268 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
; Hong Kong) Computing real inflection points of cubic algebraic curves.  ...  In this paper we present an al- gorithm for computing the real inflection points of a real planar cubic algebraic curve.  ... 

Distance Product Cubics

Boris Odehnal
2020 KoG  
All distance product cubics share the real points of inflection which lie on the line at infinity.  ...  The locus of points that determine a constant product of their distances to the sides of a triangle is a cubic curve in the projectively closed Euclidean triangle plane.  ...  The sextic curves (7) have three real and six complex cusps corresponding to the three real and six complex points of inflection on the cubics (2).  ... 
doi:10.31896/k.24.3 fatcat:46akfkh6ijajza6nyxx5glorg4

On algebraically integrable outer billiards [article]

S. Tabachnikov
2007 arXiv   pre-print
We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain non-degenerate sense then the oval is an ellipse.  ...  Kharlamov for providing a proof of Lemma 3 and to Rich Schwartz for interest and criticism. The author was partially supported by an NSF grant DMS-0555803.  ...  Unless C is a conic, this curve has inflection points (not necessarily real). Let d be the degree of C.  ... 
arXiv:0708.0255v1 fatcat:4o2iakyxprelponyt2mbd2nfqi

Characterization of Planar Cubic Alternative curve [article]

Azhar Ahmad, R.Gobithasan, Jamaluddin Md.Ali
2013 arXiv   pre-print
In this paper, we analyze the planar cubic Alternative curve to determine the conditions for convex, loops, cusps and inflection points.  ...  Thus cubic curve is represented by linear combination of three control points and basis function that consist of two shape parameters.  ...  The necessary condition for the occurrence of the inflection points along with numbers of inflection points of cubic Alternative curve can now be stated. where cusp occur at that value.  ... 
arXiv:1304.7848v1 fatcat:uqoh7cfmrvg6jf5jnun6y6wrg4

On algebraically integrable outer billiards

Serge Tabachnikov
2008 Pacific Journal of Mathematics  
We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain nondegenerate sense then the oval is an ellipse.  ...  Kharlamov for providing a proof of Lemma 3, to R. Schwartz for interest and criticism, and to the referee for helpful suggestions.  ...  We use the notation C for the complex algebraic curve given by the homogenized polynomialf (x : y : z) = f (x/z, y/z). Unless C is a conic, this curve has inflection points (not necessarily real).  ... 
doi:10.2140/pjm.2008.235.89 fatcat:ijmszoeu3bdcpc6tfeednfg26a

Real Polynomials with a Complex Twist [article]

Michael Warren, John Gresham, Bryant Wyatt
2018 arXiv   pre-print
In order to fully appreciate the Fundamental Theorem of Algebra, and the non-real solutions of a polynomial equation, traditional graphs are inadequate.  ...  Advancements in computer graphics allow us to easily illustrate a more complete graph of polynomial functions that is still accessible to students of many different levels.  ...  All cubics have one inflection point, where the second derivative is equal to zero. This hyperbola is centered at the point of inflection, where = − 3 .  ... 
arXiv:1805.05321v1 fatcat:fijjhfixqbcgza6ofl45na6yti

Hermite Interpolation by Pythagorean Hodograph Quintics

R. T. Farouki, C. A. Neff
1995 Mathematics of Computation  
" cubics.  ...  We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the "good" interpolant (in terms  ...  cubic in its number of inflections.  ... 
doi:10.2307/2153373 fatcat:p45qeoad2rawfhktxr7nej3q5m

Self-conjugate vectors of immersed 3-manifolds in R6

Daniel Dreibelbis
2012 Topology and its Applications  
We classify the possible generic algebraic structures of the asymptotic vectors at a parabolic point or an inflection point, and we classify the generic topological structures of the parabolic surface.  ...  a r t i c l e i n f o a b s t r a c t MSC: 58K30 14E20 53A07 58K05 This paper generalizes the notion of asymptotic vectors, parabolic curves, and inflection points on surfaces in R 4 to n-manifolds in  ...  We find normal forms for these quadratic triples and the cubic curves, and we relate the structure of the cubic to the type of point on the manifold.  ... 
doi:10.1016/j.topol.2011.09.019 fatcat:nmxo4drn4vgshbdaqn73xfu4yq

Auxetic deformations and elliptic curves

Ciprian S. Borcea, Ileana Streinu
2018 Computer Aided Geometric Design  
A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve. A related alternative is also considered.  ...  We show that the existence of auxetic deformations is related to properties of an associated elliptic curve.  ...  The authors are grateful for the hospitality of the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University during the Fall 2016, when this work was completed.  ... 
doi:10.1016/j.cagd.2018.02.003 pmid:30643352 pmcid:PMC6329461 fatcat:mbnqxbrtxzbptkjh7mx54filvy

Real inflections of hinged planar four-bar coupler curves

P.S Donelan, C.P Scott
1995 Mechanism and Machine Theory  
Plücker's and Klein's equations provide an upper bound on the number of real inflections on the coupler curve of a hinged planar four-bar mechanism.  ...  This enables us to locate coupler curves exhibiting the maximum possible number of inflections.  ...  . §5 Computer Graphics The realisation of coupler curves with various numbers of inflection points up to the maximum 12 was completed by computer.  ... 
doi:10.1016/0094-114x(95)00047-3 fatcat:rficqayynjftlikumj3lut4q3i

Auxetic deformations and elliptic curves [article]

Ciprian S. Borcea, Ileana Streinu
2016 arXiv   pre-print
A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve.  ...  We show that the existence of auxetic deformations is related to properties of an associated elliptic curve.  ...  The authors are grateful for the hospitality of the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University during the Fall 2016, when this work was completed.  ... 
arXiv:1612.02100v1 fatcat:lfl2hjzihzf3lm6chn5pif5k4y

Questions and Discussions

J. J. Nassau, Alan D. Campbell, Raymond Garver, Roger A. Johnson
1927 The American mathematical monthly  
A StmpLE Way To Discuss Points OF INFLECTION ON PLANE CuBIC CURVES ALAN D.  ...  CAMPBELL, Syracuse University In this paper we show by means of illustrations a simple but very effective way to discuss points of inflection on plane cubic curves.  ... 
doi:10.1080/00029890.1927.11986741 fatcat:7mfb2i366fgepfgrh5kd6qdvoy

Hermite interpolation by Pythagorean hodograph quintics

R. T. Farouki, C. A. Neff
1995 Mathematics of Computation  
" cubics.  ...  We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the "good" interpolant (in terms  ...  cubic in its number of inflections.  ... 
doi:10.1090/s0025-5718-1995-1308452-6 fatcat:ew7spddaojaonhnawe7bbsahqa

Elliptic curves in honeycomb form [article]

Melody Chan, Bernd Sturmfels
2012 arXiv   pre-print
of the tropical group law on such a curve.  ...  We explicitly compute such representations from a given j-invariant with negative valuation, we give an analytic characterization of elliptic curves in honeycomb form, and we offer a detailed analysis  ...  Acknowledgments This project grew out of discussions we had with Spencer Backman and Matt Baker. We are grateful for their contributions and help with the analytic theory of elliptic curves.  ... 
arXiv:1203.2356v1 fatcat:xz43orkc7veptn3tcmvuug2ks4
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