A projective invariant generalization of the de Casteljau algorithm (original) (raw)

Representation of conics in the oriented projective plane

1997

We present a geometric definition of conic sections in the oriented projective plane and describe some of their nice properties. This definition leads to a very simple and unambiguous representation for affine conics and conic arcs. A conic (of any type) is represented by the homogeneous coordinates of its foci and one point on it, hence, the metric plays a major role in this case as opposed to the traditional algebraic characterization of conics as second degree polynomial curves. This representation is particularly suitable for the implementation of geometric solutions of problems that involve the concept of distance. Furthermore, we discuss point location with respect to conic curves which constitutes an important elementary operation for the solution of many such problems

Projective Geometric Computing

Applications to Projective Geometry This paper applies geometric algebra to the geometry of conics in the plane. Starting from the classical double algebra expression for a conic on 5 points in terms of a running variable, we show how to eliminate this variable (by the use of tensor products) and express the conic on 5 points without resorting to a running variable. Writing , and designating the conic by Q P , the homogene-

On Rational Geometry of Conic Sections

Journal of Symbolic Computation, 2000

Simple geometric objects and transformations appear in representations and algorithms of geometric facilities in computer applications such as modelling, robotics, or graphics. Usually, these applications only support objects and transformations fully describable by rational parameters, and a computer display of points of the objects at least implicitly requires points with rational coordinates. In this setting we investigate some basic questions of the geometry of rational conic sections, when the geometry is defined by the group of rational projective transformations, the group of rational affine transformations, or the group of rational rigid transformations. Some results follow classical results, while others turn out to be quite different. In particular, we obtain a complete classification scheme for nondegenerate rational conics for rational affine geometry and a constructive method for production of a minimal set of representatives of all equivalence classes.

Constructive procedure for determination of absolute conic figure in general collinear spaces

When they are collinear, projective spaces set with five pairs of biunivocally associated points are general. In order to map quadrics (II degree surfaces), in these spaces, the absolute conic was used. Geometrical position of all the absolute points in the infinitely distant plane of one space, i.e. an absolute conic of space cannot be graphically represented. To the infinitely distant planes are associated by the vanishing planes, and the absolute conics are associated by the conic in the vanishing planes, that is, figures of the absolute conics. Prior to mapping the quadrics, it is necessary to constructively determine the characteristics parameters such as the vanishing planes, axes and centers of space, and then the figures of the absolute conics, in the vanishing planes of both spaces. In order to constructively determine the figure of the absolute conic in the second space, a sphere in the first space was used, which maps into a rotating ellipsoid in the second space. The center of the sphere is on the axis of the first space, and the infinitely distant plane intersects it along the absolute conic. The associated rotational ellipsoid, whose center is on the axis of the seconds space is intersected by the vanishing plane of the first space along the imaginary circumference aI , whose real representative is circumference az. The circumference aI is the figure of the absolute conic of the first space. General collinear spaces are presented in a pair of Monge's projections.

The Offset to an Algebraic Curve and an Application to Conics

Computational Science and Its Applications, 2005

Curve offsets are important objects in computer-aided design. We study the algebraic properties of the offset to an algebraic curve, thus obtaining a general formula for its degree. This is applied to computing the degree of the offset to conics. We also compute an implicit equation of the generalised offset to a conic by using sparse resultants and the knowledge of the degree of the implicit equation.

Efficient solution of rational conics

Mathematics of Computation, 2002

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

Constructing a family of conics by curvature-dependent offsetting from a given conic

Three new general properties of conic sections are established, namely: (1) By offsetting from a given conic (ellipse, parabola or hyperbola) perpendicularly to it by a distance proportional to the cube root of its radius of curvature, another conic of the same kind is generated; (2) The cube root (or proportional to it) is the only function for with such a property can be stated; (3) The cube root of the radius of curvature at any point is proportional to its distance to any one of the principal axes of the conic, taken perpendicularly to it. Starting from any particular conic, and taking the proportionality constant k as a parameter, a family of conics of its kind is generated. Piling these conics up in the 3D space, different surfaces can be defined. If one of the Cartesian coordinates is made to be proportional to k, these surfaces are ruled, which greatly facilitates their constructive applications. We derive the parametric equations of these surfaces and represent them graphically, choosing viewpoints for a good visualization. Some ideas of applications are proposed for further development. Ó 1999 Elsevier Science B.V. All rights reserved.

Offsets to conics and quadrics

ACM Communications in Computer Algebra, 2019

A new determinantal presentation of the implicit equation for offsets to non degenerate conics and quadrics is introduced which is specially well suited for intersection purposes.

An Adapted Version of the Bentley-Ottmann Algorithm for Invariants of Plane Curves Singularities

Lecture Notes in Computer Science, 2011

We report on an adapted version of the Bentley-Ottmann algorithm for computing all the intersection points among the edges of the projection of a three-dimensional graph. This graph is given as a set of vertices together with their space Euclidean coordinates, and a set of edges connecting them. More precisely, the three-dimensional graph represents the approximation of a closed and smooth implicitly defined space algebraic curve, that allows us a simplified treatment of the events encountered in the Bentley-Ottmann algorithm. As applications, we use the adapted algorithm to compute invariants for each singularity of a plane complex algebraic curve, i.e. the Alexander polynomial, the Milnor number, the delta-invariant, etc.

On the Implicit Equation of Conics and Quadrics Offsets

Mathematics, 2021

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in “A Treatise on the Analytic Geometry of Three Dimensions” by G. Salmon for offsets to non-degenerate conics and central quadrics.