Identification of inflection points and cusps on rational curves

Inflection points and singularities on C-curves

We show that all so-called C-curves are affine images of trochoids or sine curves and use this relation to investigate the occurrence of inflection points, cusps, and loops. The results are summarized in a shape diagram of C-Bézier curves, which is useful when using C-Bézier curves for curve and surface modeling.  2003 Elsevier B.V. All rights reserved.

On the singularity of a class of parametric curves

We consider parametric curves that are represented by combination of control points and basis functions. We let a control point vary while the rest is held fixed. We show that the locus of the moving control point that yields a zero curvature point on the curve is a developable surface, the regression curve of which is the locus that guarantees a cusp on the curve. We also specify the surface t...

Detecting Cusps and In ection Points

In many applications it is desirable to analyze parametric curves for undesirable features like cusps and innection points. Previously known algorithms to analyze such features are limited to cubics and in many cases are for planar curves only. We present a general purpose method to detect cusps in polynomial or rational space curves of arbitrary degree. If a curve has no cusp in its deening in...

Rational Points on Atkin-Lehner Quotients of Shimura Curves

We study three families of Atkin-Lehner quotients of quaternionic Shimura curves: X, X 0 (N), and X D+ 1 (N), which serve as moduli spaces of abelian surfaces with potential quaternionic multiplication (PQM) and level N structure. The arithmetic geometry of these curves is similar to, but even richer than, that of the classical modular curves. Two important differences are the existence of a no...

How Many Rational Parametric Cubic Curves Are There? Part 1: Inflection Points