A rational (resp. integral) distance set is a subset S of the plane R such that for all s, t ∈ S, the distance between s and t is a rational number (resp. is an integer). Huff [4] considered rational distance sets S of the following form: given distinct a, b ∈ Q∗, S contains the four points (0,±a) and (0,±b) on the y-axis, plus points (x, 0) on the x-axis, for some x ∈ Q∗. Such a point (x, 0) m...

In this paper, we propose a linear method for C approximation of rational Bézier curve with arbitrary degree polynomial curve. Based on weighted least-squares, the problem be converted to an approximation between two polynomial curves. Then applying Bernstein-Jacobi hybrid polynomials, we obtain the resulting curve. In order to reduce error, degree reduction method for Bézier curve is used. A e...

A cuspidal curve is a curve whose singularities are all cusps, i.e. unibranched singularities. The article describes computations which lead to the following conjecture: A rational cuspidal plane curve of degree greater or equal to six has at most three cusps. The curves with precisely three cusps occur in three series. Assuming the Flenner–Zaidenberg rigidity conjecture the above conjecture is...

This paper presents a simple and robust method for computing the bisector of two planar rational curves. We represent the correspondence between the foot points on two planar rational curves C1ðtÞ and C2ðrÞ as an implicit curve F(t,r) 1⁄4 0, where F(t,r) is a bivariate polynomial B-spline function. Given two rational curves of degree m in the xy-plane, the curve F(t,r) 1⁄4 0 has degree 4m 1 2, ...

This paper presents a simple and robust method for computing the bisector of two planar rational curves. We represent the correspondence between the foot points on two planar rational curves C1(t) and C2(r) as an implicit curve F(t; r) = 0, where F(t; r) is a bivariate polynomial B-spline function. Given two rational curves of degree m in the xy-plane, the curve F(t; r) = 0 has degree 4m 2, whi...

The method of moving curves and moving surfaces is a new, eeective tool for implicitizing rational curves and surfaces. Here we investigate a relationship between the moving line coeecient matrix and the moving conic coeecient matrix for rational curves. Based on this relationship, we present a new proof that the method of moving conics always produces the implicit equation of a rational curve ...

We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability of hyperelliptic curves. We also discuss app...