Intersections of hyperplanes and conic sections in Rn\mathbf{R}^nRn (original) (raw)

On the Intersection of a Hyperboloid and a Plane

2017

The intersection topic is quite popular at an interdisciplinary level. It can be the friends of geometry, geodesy and others. The curves of intersection resulting in this case are not only ellipses but rather all types of conics: ellipses, hyperbolas and parabolas. In text books of mathematics usually only cases are treated, where the planes of intersection are parallel to the coordinate planes. Here the general case is illustrated with intersecting planes which are not necessarily parallel to the coordinate planes. We have developed an algorithm for intersection of a hyperboloid and a plane with a closed form solution. To do this, we rotate the hyperboloid and the plane until inclined plane moves parallel to the XY plane. In this situation, the intersection ellipse and its projection will be the same. This study aims to show how to obtain the center, the semi-axis and orientation of the intersection curve.

On the hyperbox–hyperplane intersection problem

2009

Finding the intersection between a hyperbox and a hyperplane can be computationally expensive specially for high dimensional problems. Naive algorithms have an exponential complexity. A border node is a node (in the graph induced by the hyperbox) at or next to the intersection of the hyperbox and the hyperplane. The algorithm proposed in this paper implements a systematic way to efficiently generate border nodes; given a border node, a subset of its incident edges is explored to determine one or more intersections. This systematic exploration allows us to focus on the border region, discarding the two regions before and after the plane. Pruning those regions produces a computational cost linear on the number of vertices of the hyperpolygon that represents the intersection.

Supporting conic design methods and conic intersection properties

Optical Engineering, 2013

The supporting ellipsoids and linear programming reflector design methods build upon the property of conics to address the inverse problem of finding the freeform surface that directs light from a point source to produce a prescribed target distribution. We review the properties and main computational limitations of the two methods and show that a fast flux estimation method based on contour detection can be used in combination with the supporting ellipsoid algorithm. Once the intersections between neighboring conic patches on the reflector are known, it is possible to estimate the collected flux using the vertices of the intersection boundary. The advantage of using the intersection method to estimate the flux instead of the more common approach-Monte Carlo ray tracing-is that there is no tradeoff between speed and accuracy. Examples of flux estimation with the intersection method for different target configurations are shown.

Vectorial Paraboloid in N-Dimensional Space

A general equation of a paraboloid is stated with three terms: a bilinear positive definite form term, a linear term and a constant term. We propose a resolvent to this equation that calculates one closed curve X1=X2\X_1=\X_2X_1=X2 or two open curves X1\X_1X1 and X2\X_2X2, according to the discriminant dyadic bfDelta=DD{\bf\Delta}=\D\DbfDelta=DD. A procedure to obtain the curve or curves is proposed, finding the intersection of the curves with a nnn-sphere of radius sss, where sss acts as a parameter of the curve, on the hyperplane P2(X)=0P_2(\X)=0P_2(X)=0. With the use of the real eigenvalues of the symmetrical matrix bfA{\bf A}bfA, an orthogonal transformation bfS\bf SbfS is defined, and a global transformation bfQ=bfS.bfL{\bf Q}={\bf S}.{\bf L}bfQ=bfS.bfL also (${\bf L}$ is the inverse of the square root of the diagonal eigenvalues matrix), that permits to obtain D\DD and the solutions of the paraboloid equation. A tuning (n−1)(n-1)(n−1)-dimensional angle tildeTHETA\tilde\THETAtildeTHETA is then refined to find the solution where sss and ∣X∣\|\X\|∣X∣ coincide. Also the centred paraboloid ($\B=\Zero$) solution is analyzed with eigenvalues and eigenvectors. Examples in real2\real^2real2 are resolved to show the features of the procedure for elliptical, hyperbolical, centred or not, paraboloids. Another parameter ttt is necessary in the case of negative eigenvalue, where sss and ttt are almost inversely proportional related.

On families of quadratic surfaces having fixed intersections with two hyperplanes

2011

Abstract We investigate families of quadrics that have fixed intersections with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter. In particular we show how the quadrics are transformed as the parameter changes. This research was motivated by an application in mixed-integer conic optimization.

Near-optimal parameterization of the intersection of quadrics: II. A classification of pencils

Journal of Symbolic Computation, 2008

We present here the first classification of pencils of quadrics based on the type of their intersection in real projective space and we show how this classification can be used to compute efficiently the type of the real intersection. This classification is at the core of the design of the algorithms, presented in Part III, for computing, in all cases

Non-transversal intersection curves of hypersurfaces in Euclidean 4-space

Journal of Computational and Applied Mathematics, 2015

In this paper, we consider the problem of how to compute the Frenet apparatus of the non-transversal intersection curves (hyper-curves) of three hypersurfaces (given by their implicit-implicit-parametric and implicit-parametric-parametric equations) in Euclidean 4-space. The non-transversal intersection (in which the normal vectors of the intersecting hypersurfaces are linearly dependent) includes two different cases. In each case, on the contrary to transversal intersections, we face some difficulties even finding the tangential direction. To overcome such difficulties, we give some algorithms for finding all Frenet vectors and curvatures of the intersection curve. Finally, we demonstrate our methods by giving several examples.

On the Hinge Finding Algorithm for Hinging Hyperplanes-Revised Version

1995

This paper concerns the estimation algorithm for hinging hyperplane (HH) models, a non-linear black box model structure suggested in 3]. The estimation algorithm is analysed and it is shown that it is a special case of a Newton algorithm applied on a quadratic criterion. This insight is then used to suggest possible improvements of the algorithm so that convergence can be guaranteed. In addition the way of updating the parameters in the HH model, is discussed. In 3] a stepwise updating procedure is proposed. In this paper we stress that simultaneous updating of the model parameters can be preferable in some cases.

On the hinge-finding algorithm for hingeing hyperplanes

1998

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.

Intersections of hyperplanes and conic sections in Rn\mathbf{R}^nRn (original) (raw)

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