The Convex Hull of Rational Plane Curves

Advances in Geometry, 2000

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present methods for computing their defining polynomials, and we exhibit a wide range of examples.

2016

Given two rational, properly parametrized space curves ${\mathcal C}_1$ and ${\mathcal C}_2$, where $\CCC_2$ is contained in some plane $\Pi$, we provide an algorithm to check whether or not there exist perspective or parallel projections mapping $\CCC_1$ onto $\CCC_2$, i.e. to recognize $\CCC_2$ as the projection of $\CCC_1$. In the affirmative case, the algorithm provides the eye point(s) of the perspective transformation(s), or the direction(s) of the parallel projection(s). The problem is mainly discussed from a symbolic point of view, but an approximate algorithm is also included.

Lecture Notes in Computer Science, 1989

We present a variety of computational techniques dealing with algebraic curves both in the plane and in space. Our main results are polynomial time algorithms (1) to compute the genus of plane algebraic curves, (2) to compute the rational parametric equations for implicitly defined rational plane algebraic curves of arbitrary degree, (3) to compute birational mappings between points on irreducible space curves and points on projected plane curves and thereby to compute the genus and rational parametric equations for implicitly defined rational space curves of arbitrary degree, (4) to check for the faithfulness (one to one) of parameterizations.

2018

We present a subdivision based technique for finding the intersections of two algebraic curves inside a convex region. Even though it avoids computing resultants, the technique is guaranteed to find all intersections with bounded backwards error. The subdivision, called an encasement, also encodes the arrangement structure of the curves. We implement the encasement algorithm using adaptive precision interval arithmetic. We compare its performance to the CGAL library implementation of resultant based curve intersection techniques. We provide CPU and CPU/GPU versions of the algorithm and implementation. On the CPU, encasement generates all curve intersections, to accuracy 10−8, 10 to 30 times faster than CGAL for degrees 8 to 18, and it handles degrees up to 20 that CGAL cannot handle. The GPU speeds up the calculation by a factor of 3 to 4.

Computer Aided Geometric Design, 2019

Given a real rational parametrization P(t) of a plane curve C, we present an algorithm to compute polynomial curves to approximate C for the whole parameter domain. In this case, the denominators often have real roots in the whole interval. We decompose the interval as the union of finitely many intervals according to the real roots of the denominators. The key technique of the paper is to approximate the given curve by their asymptotes and error analysis at each interval is also presented. The asymptotes are associated with the infinity points corresponding to the real roots of the denominators. Numeric algorithms and examples are proposed to illustrate our results.

ACM Transactions on Graphics, 1999

In this paper, we give a simple method for drawing a closed rational curve speci ed in terms of control points as two B ezier segments. The main result is the following:

Computer Aided Geometric Design, 2010

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or use the implicit equation of the curve (in the case of planar curves) or of any projection (in the case of space curves). Moreover, these algorithms have been implemented in Maple; the examples considered and the timings obtained show good performance skills.

Lecture Notes in Computer Science, 1997

A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There are several approaches to deciding whether an algebraic curve is parametrizable and if so computing a parametrization. In all these approaches we ultimately need some simple points on the curve. The eld in which we can nd such points crucially in uences the coe cients in the resulting parametrization. We show how to nd such simple points over some practically interesting elds. Consequently, we are able to decide whether an algebraic curve de ned over the rational numbers can be parametrized over the rationals or the reals. Some of these ideas also apply to parametrization of surfaces. If in the term geometric reasoning we do not only include the process of proving or disproving geometric statements, but also the analysis and manipulation of geometric objects, then algorithms for parametrization play an important role in this wider view of geometric reasoning.

Computer Aided Geometric Design, 2010

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance ǫ > 0 and an ǫ-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of ǫ-rationality, and we provide an algorithm to parametrize approximately affine ǫ-rational plane curves, without exact singularities at infinity, by means of linear systems of (d − 2)-degree curves. The algorithm outputs a rational parametrization of a rational curve C of degree at most d which has the same points at infinity as C. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that C and C are close in practice.