Computing minimum distance between two implicit algebraic surfaces

Chen, Xiao-Diao; Yong, Jun-Hai; Zheng, Gangtie; Paul, Jean-Claude; Sun, Jun

doi:10.1016/j.cad.2006.04.012

Cited by 17 publications

(9 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…So far, the case of six common normals has only been observed for nested QSS which are not relevant for distance computation. We compare our results to those obtained by [1], for computing the distance between two quadric surfaces. They use a Newton-type algorithm to find the common normals and need 56.25µs, composed of 40µs preprocessing and 16.25µs "main algorithm" on a 1.7 GHz PC, which corresponds to approximately 47.81µs on our hardware.…”

Section: Example 1: Common Normals Of Two Static Qssmentioning

confidence: 94%

“…However, the boundary of the volume has singular curves, and the orientation of the unit normals n along the boundary may change along these singularities 1 . Now we consider two QSS with the support functions f (see (1)) and…”

Section: Enclosing Moving Objects By Qssmentioning

confidence: 99%

“…In particular, the use of ellipsoids (and more general algebraic surfaces) has been proposed in the literature [1,2,7,10]. Methods for generating collections of bounding ellipsoids for 3D objects have been discussed in [12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fast Distance Computation Using Quadratically Supported Surfaces

Rabl

Jüttler

2009

Computational Kinematics

View full text Add to dashboard Cite

We use the class of surfaces with quadratic polynomial support functions in order to define bounding geometric primitives for shortest distance computation. The common normals of two such surfaces can be computed by solving a single polynomial equation of degree six. Based on this observation, we formulate an algorithm for computing the shortest distance between enclosures of two moving or static objects by surfaces of this type. It is demonstrated that the performance of this algorithm compares favourably with methods for computing the distance between two ellipsoids, which can also be used as bounding primitives for distance computation and collision detection.

show abstract

Section: Example 1: Common Normals Of Two Static Qssmentioning

confidence: 94%

Section: Enclosing Moving Objects By Qssmentioning

confidence: 99%

See 1 more Smart Citation

Fast Distance Computation Using Quadratically Supported Surfaces

Rabl

Jüttler

2009

Computational Kinematics

View full text Add to dashboard Cite

show abstract

“…Ma et al [7] studied the problem for point orthogonal projection onto NURBS curves and surfaces for which they adopted four-step technique: subdividing curve or surface into curve segments or surface patches, finding out the relationship between the control polygon of curve segment or surface patch and the test point, candidate curve segment or surface patch and approximate projection points being found out by comparison and the final projective point being obtained by comparing the distance between the test point and these approximate projective points. Since the minimum distance between two geometries occurred between a pair of special points, they studied the minimum distance between various specific geometry by using the property and obtained some satisfactory results [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Point Orthogonal Projection onto a Spatial Algebraic Curve

Cheng

et al. 2020

Mathematics

View full text Add to dashboard Cite

Point orthogonal projection onto a spatial algebraic curve plays an important role in computer graphics, computer-aided geometric design, etc. We propose an algorithm for point orthogonal projection onto a spatial algebraic curve based on Newton’s steepest gradient descent method and geometric correction method. The purpose of Algorithm 1 in the first step of Algorithm 4 is to let the initial iteration point fall on the spatial algebraic curve completely and successfully. On the basis of ensuring that the iteration point fallen on the spatial algebraic curve, the purpose of the intermediate for loop body including Step 2 and Step 3 is to let the iteration point gradually approach the orthogonal projection point (the closest point) such that the distance between them is very small. Algorithm 3 in the fourth step plays an important double acceleration and orthogonalization role. Numerical example shows that our algorithm is very robust and efficient which it achieves the expected and ideal result.

show abstract

“…Chen et al [1] compute the distance of two implicit algebraic surfaces by using an offsetting technique. They reduce the problem of distance computation between a quadric surface in implicit representation and (a) a cylinder, (b) a cone, (c) an elliptic paraboloid, (d) an ellipsoid and (e) a torus to the problem of solving a univariate polynomial of degree (a) 4, (b) 8, (c) 16, (d) 36 and (e) 16.…”

Section: Introductionmentioning

confidence: 99%

Oriented bounding surfaces with at most six common normals

Rabl

González-Vega

Jüttler

et al. 2009

2009 IEEE International Conference on Robotics and Automation

View full text Add to dashboard Cite

Abstract-We present a new type of oriented bounding surfaces, which is particularly well suited for shortest distance computations. The bounding surfaces are obtained by considering surfaces whose support functions are restrictions of quadratic polynomials to the unit sphere. We show that the common normals of two surfaces of this type -and hence their shortest distance -can be computed by solving a polynomial of degree six. This compares favorably with other existing bounding surfaces, such as quadric surfaces, where the computation of the common normals is known to lead to a polynomial of degree 24.

show abstract

Computing minimum distance between two implicit algebraic surfaces

Cited by 17 publications

References 39 publications

Fast Distance Computation Using Quadratically Supported Surfaces

Fast Distance Computation Using Quadratically Supported Surfaces

Point Orthogonal Projection onto a Spatial Algebraic Curve

Oriented bounding surfaces with at most six common normals

Contact Info

Product

Resources

About