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Computing real inflection points of cubic algebraic curves

2003
*
Computer Aided Geometric Design
*

In this paper we present an algorithm for

doi:10.1016/s0167-8396(03)00022-0
fatcat:wzxfoplrajdhnoonrq6qlrd2om
*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*. ...##
###
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

2020
*
KoG
*

All distance product

doi:10.31896/k.24.3
fatcat:46akfkh6ijajza6nyxx5glorg4
*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). ...##
###
On algebraically integrable outer billiards
[article]

2007
*
arXiv
*
pre-print

We prove that if the outer billiard map around a plane oval is

arXiv:0708.0255v1
fatcat:4o2iakyxprelponyt2mbd2nfqi
*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. ...##
###
Characterization of Planar Cubic Alternative curve
[article]

2013
*
arXiv
*
pre-print

In this paper, we analyze the planar

arXiv:1304.7848v1
fatcat:uqoh7cfmrvg6jf5jnun6y6wrg4
*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. ...##
###
On algebraically integrable outer billiards

2008
*
Pacific Journal of Mathematics
*

We prove that if the outer billiard map around a plane oval is

doi:10.2140/pjm.2008.235.89
fatcat:ijmszoeu3bdcpc6tfeednfg26a
*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*). ...##
###
Real Polynomials with a Complex Twist
[article]

2018
*
arXiv
*
pre-print

In order to fully appreciate the Fundamental Theorem

arXiv:1805.05321v1
fatcat:fijjhfixqbcgza6ofl45na6yti
*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 . ...##
###
Hermite Interpolation by Pythagorean Hodograph Quintics

1995
*
Mathematics of Computation
*

"

doi:10.2307/2153373
fatcat:p45qeoad2rawfhktxr7nej3q5m
*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*. ...##
###
Self-conjugate vectors of immersed 3-manifolds in R6

2012
*
Topology and its Applications
*

We classify the possible generic

doi:10.1016/j.topol.2011.09.019
fatcat:nmxo4drn4vgshbdaqn73xfu4yq
*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. ...##
###
Auxetic deformations and elliptic curves

2018
*
Computer Aided Geometric Design
*

A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants

doi:10.1016/j.cagd.2018.02.003
pmid:30643352
pmcid:PMC6329461
fatcat:mbnqxbrtxzbptkjh7mx54filvy
*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. ...##
###
Real inflections of hinged planar four-bar coupler curves

1995
*
Mechanism and Machine Theory
*

Plücker's and Klein's equations provide an upper bound on the number

doi:10.1016/0094-114x(95)00047-3
fatcat:rficqayynjftlikumj3lut4q3i
*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*. ...##
###
Auxetic deformations and elliptic curves
[article]

2016
*
arXiv
*
pre-print

A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants

arXiv:1612.02100v1
fatcat:lfl2hjzihzf3lm6chn5pif5k4y
*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. ...##
###
Questions and Discussions

1927
*
The American mathematical monthly
*

A StmpLE Way To Discuss

doi:10.1080/00029890.1927.11986741
fatcat:7mfb2i366fgepfgrh5kd6qdvoy
*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*. ...##
###
Hermite interpolation by Pythagorean hodograph quintics

1995
*
Mathematics of Computation
*

"

doi:10.1090/s0025-5718-1995-1308452-6
fatcat:ew7spddaojaonhnawe7bbsahqa
*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*. ...##
###
Elliptic curves in honeycomb form
[article]

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*. ...

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