Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-29T12:34:05.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space

Published online by Cambridge University Press:  17 February 2009

H. T. Rathod  and
H. S. Govinda Rao
H. T. Rathod
Affiliation:
Department of Mathematics, Central College Campus, Bangalore University, Bangalore-1, India
H. S. Govinda Rao
Affiliation:
Department of Mathematics, Central College Campus, Bangalore University, Bangalore-1, India
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper concerns with analytical integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space. The volume integration of trivariate polynomials over linear polyhedra is computed as sum of surface integrals in R3 on application of the well known Gauss's divergence theorem and by using triangulation of the linear polyhedral boundary. The surface integrals in R3 over an arbitrary triangle are connected to surface integrals of bivariate polynomials in R2. The surface integrals in R2 over a simple polygon or over an arbitrary triangle are computed by two different approaches. The first algorithm is obtained by transforming the surface integrals in R2 into a sum of line integrals in a one-parameter space, while the second algorithm is obtained by transforming the surface integrals in R2 over an arbitrary triangle into a parametric double integral over a unit triangle. It is shown that the volume integration of trivariate polynomials over linear polyhedra can be obtained as a sum of surface integrals of bivariate polynomials in R2. The computation of surface integrals is proposed in the beginning of this paper and these are contained in Lemmas 1–6. These algorithms (Lemmas 1–6) and the theorem on volume integration are then followed by an example for which the detailed computational scheme has been explained. The symbolic integration formulas presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, for example, the volume, centre of mass, moments of inertia etc., required in engineering design processes.

Type
Research Article
Information
The ANZIAM Journal , Volume 39 , Issue 3 , January 1998 , pp. 355 - 385
Copyright
Copyright © Australian Mathematical Society 1998

References

[1] Bernardini, F., “Integration of polynomials over n-dimensional polyhedra”, Comput. Aided Des. 23 (1) (1991) 51–58.Google Scholar
[2] Cattani, C. and Paoluzzi, A., “Boundary integration over linear polyhedra”, Comput. Aided Des. 22 (1990) 130–135.CrossRefGoogle Scholar
[3] Cattani, C. and Paoluzzi, A., “Symbolic analysis of linear polyhedra”, Engineering with Computers 6 (1990) 17–29.CrossRefGoogle Scholar
[4] Lee, Y. T. and Requicha, A. A. G., “Algorithms for computing the volume and other integral properties of solids I: known methods and open issues”, Comm. ACM 25 (9) (1982) 635–641.CrossRefGoogle Scholar
[5] Lee, Y. T. and Requicha, A. A. G., “Algorithms for computing the volume and other integral properties of solids II: a family of algorithms based on representation conversion and on cellular approximation”, Comm. ACM 25 (9) (1982) 642–650.CrossRefGoogle Scholar
[6] Lien, S. and Kajiya, J. T., “A symbolic method for calculating the integral properties of arbitrary nonconvex polyhedra”, IEEE Comput. Graph. Applic. 4 (9) (1984) 35–41.CrossRefGoogle Scholar
[7] O'Leary, J. R., “Evaluation of mass properties by finite elements”, Guidance and Control 3 (2) (1980) 188–190.CrossRefGoogle Scholar
[8] Timmer, H. G. and Stern, J. M., “Computation of global geometric properties of solids”, Comput. Aided Des. 12 (6) (1980) 301–304.CrossRefGoogle Scholar
[9] Wilson, H. B. Jr and Farrior, D. S., “Computation of geometrical and inertial properties for general areas and volumes of revolution”, Comput. Aided Des. 8 (4) (1976) 257–263.CrossRefGoogle Scholar