Efficient solution of rational conics

Efficient Solution of Rational

We present eecient algorithms for solving Legendre equations over Q (equivalently, for nding rational points on rational conics) and parametriz-ing all solutions. Unlike existing algorithms, no integer factorization is required , provided that the prime factors of the discriminant are known.

Eecient Solution of Rational Conics

Necessary and sufficient conditions for rational quartic representation of conic sections

Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bézier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than . However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as bein...

Rational Families of Conics and Quadrics

A surface generated by a one parameter family of conics c(t) is called conic surface. If c(t) can be described by rational functions, the generated conic surface is rational. An algorithm to construct real rational parametrizations for such surfaces and some examples will be given.

Arithmetic Progressions on Conics.

In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose x-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We als...