Intersection Of Quadric Surfaces Research Articles

Critical configurations for three projective views

The problem of structure from motion is concerned with recovering the 3-dimensional structure of an object from a set of 2-dimensional images taken by unknown cameras. Generally, all information can be uniquely recovered if enough images and point correspondences are provided, yet there are certain cases where unique recovery is impossible; these are called \emph{critical configurations}. We use an algebraic approach to study the critical configurations for three projective cameras. We show that all critical configurations lie on the intersection of quadric surfaces, and classify exactly which intersections constitute a critical configuration.

A post-quantum key exchange protocol from the intersection of quadric surfaces

In this paper, we present a new key exchange protocol in which Alice and Bob have secret keys given by quadric surfaces embedded in a large ambient space by means of the Veronese embedding and public keys given by hyperplanes containing the embedded quadrics. Both of them reconstruct the isomorphism class of the intersection which is a curve of genus 1, and is uniquely determined by the j-invariant. An eavesdropper, to find this j-invariant, has to solve problems which are conjecturally quantum-resistant.

Quadric Conformal Geometric Algebra of $${\mathbb {R}}^{9,6}$$ R 9 , 6

Geometric Algebra can be understood as a set of tools to represent, construct and transform geometric objects. Some Geometric Algebras like the well-studied Conformal Geometric Algebra constructs lines, circles, planes, and spheres from control points just by using the outer product. There exist some Geometric Algebras to handle more complex objects such as quadric surfaces; however in this case, none of them is known to build quadric surfaces from control points. This paper presents a novel Geometric Algebra framework, the Geometric Algebra of $${\mathbb {R}}^{9,6}$$ , to deal with quadric surfaces where an arbitrary quadric surface is constructed by the mere wedge of nine points. The proposed framework enables us not only to intuitively represent quadric surfaces but also to construct objects using Conformal Geometric Algebra. Our proposed framework also provides the computation of the intersection of quadric surfaces, the normal vector, and the tangent plane of a quadric surface.

The Pairing Computation on Edwards Curves

We propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric interpretation of the group law, we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curves. Our formulae for the doubling step are a little faster than that proposed by Arène et al. Finally, to improve the efficiency of pairing computation, we present twists of degrees 4 and 6 on twisted Edwards curves.

Enhancing Levin's method for computing quadric-surface intersections

Levin's method produces a parameterization of the intersection curve of two quadrics in the form p (u)= a (u)± d (u) s(u) , where a (u) and d (u) are vector valued polynomials, and s( u) is a quartic polynomial. This method, however, is incapable of classifying the morphology of the intersection curve, in terms of reducibility, singularity, and the number of connected components, which is critical structural information required by solid modeling applications. We study the theoretical foundation of Levin's method, as well as the parameterization p (u) it produces. The following contributions are presented in this paper: (1) It is shown how the roots of s( u) can be used to classify the morphology of an irreducible intersection curve of two quadric surfaces. (2) An enhanced version of Levin's method is proposed that, besides classifying the morphology of the intersection curve of two quadrics, produces a rational parameterization of the curve if the curve is singular. (3) A simple geometric proof is given for the existence of a real ruled quadric in any quadric pencil, which is the key result on which Levin's method is based. These results enhance the capability of Levin's method in processing the intersection curve of two general quadrics within its own self-contained framework.

Computing quadric surface intersections based on an analysis of plane cubic curves

Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves.

Computing quadric surface intersections based on an analysis of plane cubic curves

Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves.

Construction of feature volumes using intersection of adjacent surfaces

An algorithm is presented for the construction of feature volumes using the topology and geometry found in the 3D boundary representation of a solid object. The concavity of faces, edges and vertices is used to detect features and any interactions between them. Faces adjacent to the feature faces are intersected to create new edges which can be used to create totally new construction faces. These construction faces are then used to complete the feature volume. Simple rules have been developed which allow the intersection of quadric surfaces. This algorithm is modular in nature and can be used in conjunction with any feature detection module. The basic principle is established using the simplest of examples. The technique is then tested on several benchmark parts taken from the NIST design repository.

Surface design using cyclide patches

Cyclide surfaces have low algebraic degree, exact nurbs representations and circular lines of curvature. Geometric modelling applications of the cyclide include blending of quadric surface intersections, piping layouts and cable harnesses. In this paper, we describe a scheme for smoothly composing a topologically rectangular network of cyclide patches. It is a two-step process. First, a cyclide quadrilateral mesh is constructed to approximate the target shape. Then, cyclide patches corresponding to the quadrilateral mesh are constructed. There is tangent plane continuity across the patch boundaries. Computations are based on the geometry of the cyclide and it is easy to detect degenerate cyclide patches, i.e. parabolic cyclide, cone, cylinder and sphere. We discuss issues related to global and local shape modification, and portability. Implemented examples are also included. Finally, we describe a simple strategy for NC machining of composite cyclide surfaces. Advantages are highlighted and implemented examples included.

Quadric shell intersections

The paper considers intersections of natural quadric shells (surfaces with thickness). A natural quadric shell can be thought of as a solid defined by a natural quadric surface and its offset (which is again a quadric). Thus, the intersection of quadric shells is essentially the same as the set of intersections of quadric surfaces and their offsets. An approach is presented to obtain the intersection of any two offset surfaces (of the initial quadrics) in terms of the offset distances. For the intersection of any two shells (which contain the initial quadrics as one of their surfaces), the intersection curve can be obtained directly by substituting the appropriate shell thickness (offset distances) in the general solution.

On the Formulation of Numerical Control Algorithms Using Computational Geometry

Algorithms have been derived for computing 3D tool paths for planar and non-planar machined surfaces. Concepts from computational geometry such as vector solutions of line intersections with planes, spheres, cones and cylinders allow users the freedom to develop interfacing routines for complex NC machining which are not available in standard AFT packages. Using these algorithms, intersections of quadric surfaces are obtained as analytic solutions of 3D tool coordinates to serve as command inputs to NC processors. The algorithms must be guarded against cases of singularities and degeneracies and extensions to more general free-form machining are also addressed.

Piecewise-circular curves for geometric modeling

Modern solid modelers must be able to represent a wide class of objects, and must support Boolean operations on solids. These operations are very useful for defining solids, detecting interferences, and modeling fabrication processes. Computing the boundaries of solids defined through Boolean operations requires algorithms for surface/surface and curve/surface intersection. Many of the currently available modelers use closed-form parametric expressions for the curves of intersection of quadric surfaces, and compute intersections of these curves with other surfaces by finding the roots of low-degree polynomials. Because the curves that result from intersections involving tori or more complex surfaces generally cannot be expressed in closed form, modelers typically approximate these curves by cubic splines that interpolate points lying on the true intersections. Cubic splines exhibit second-degree continuity, but they are expensive to process in solid modeling computations. In this paper, we trade second-degree continuity for computational simplicity, and present a method for interpolating three-dimensional points and associated unit tangent vectors by smooth space curves composed of straight line segments and circular arcs. These curves are designated as PCCs (for piecewise-circular curves) and have continuous unit tangents. PCCs can be used in efficient algorithms for performing fundamental geometric computations, such as the evaluation of the minimal distance from a point to a curve or the intersection of a curve and a surface. Formulae and algorithms are presented for generating and processing PCCs in solid modelers. We also show that PCCs are useful for incorporating toroidal primitives, as well as sweeping, growing, shrinking, and blending operations in systems that model solids bounded by the natural quadric surfaces—planes, cylinders, cones, and spheres.

Algebraic methods for intersections of quadric surfaces in GMSOLID

The algebraic geometry of quadric surfaces is applied to the construction of boundary representations for computer models of solid objects. A version of this approach is currently operational in the GMSOLID system developed at General Motors. A general method is presented for finding critical points on the intersection curves of quadric surfaces and for determining whether the curves are planar. This information is needed for GMSOLID's computer representation of a designed solid. The method is also used to find intersection points of a plane and two quadrics. The method is sufficiently general to handle all quadric surfaces but has been implemented only for the quadrics of GMSOLID: cylinders, spheres, and cones. The numerical methods used in the implementation are also discussed.

Mathematical models for determining the intersections of quadric surfaces

Intersections of two or three quadric surfaces may be computed by using matrix algebra and parametric equations. Methods are described for generating “line-drawing” pictures of quadric-surface bodies.

A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces

An algorithm for drawing pictures of three-dimensional objects, with surfaces made up of patches of quadric surfaces, is described. The emphasis of this algorithm is on calculating the intersections of quadric surfaces. A parameterization scheme is used. Each quadric surface intersection curve (QSIC) is represented as a set of coefficients and parameter limits. Each value of the parameter represents at most two points, and these may easily be distinguished. This scheme can find the coordinates of points of even quartic (fourth-order) intersection curves, using equations of no more than second order. Methods of parameterization for each type of QSIC are discussed, as well as surface bounding and hidden surface removal.

A parametric algorithm for drawing pictures of solid objects bounded by quadric surfaces

An algorithm for drawing pictures of three-dimensional objects, with surfaces made up of patches of quadric surfaces, is described.The emphasis of this algorithm is calculating the intersections of quadric surfaces. A parameterization scheme is used. Each quadric surface intersection curve (QSIC) is represented as a set of coefficients and parameter limits. Each value of the parameter represents at most two points, and these may easily be distinguished. This scheme can find the co-ordinates of points of even quartic (fourth-order) intersection curves, using equations of no more than second order.Methods of parameterization for each type of QSIC are discussed, as well as surface bounding and hidden surface removal.