An Implicit Representation of Spherical Product for Increasing the Shape Variety of Super-quadrics in Implicit Surface Modeling

Super-quadrics can represent a set of implicit surfaces, which can be used furthermore as primitive surfaces to construct a complex object via Boolean set operations in implicit surface modeling. In fact, super-quadrics were developed to create a parametric surface by performing spherical product on two parametric curves and some of the resulting parametric surfaces were also represented as implicit surfaces. However, because not every parametric curve can be redefined implicitly, this causes only implicit super-elliptic and super-hyperbolic curves are applied to perform spherical product and so only implicit super-ellipsoids and hyperboloids are developed in super-quadrics. To create implicit surfaces with more diverse shapes than super-quadrics, this paper proposes an implicit representation of spherical product, which performs spherical product on two implicit curves like super-quadrics do. By means of the implicit representation, many new implicit curves such as polygonal, star-shaped and rose-shaped curves can be used to develop new implicit surfaces with a greater variety of shapes than super-quadrics, such as polyhedrons, hyper-ellipsoids, superhyperboloids and hyper-toroids containing star-shaped and roseshaped major and minor circles. Besides, the newly developed implicit surfaces can also be used to define new primitive implicit surfaces for constructing a more complex implicit surface in implicit surface modeling.

Authors:

• Pi-Chung Hsu

Keywords:

• Implicit surfaces
• Soft objects
• Super-quadrics.

References:

[1] A. H. Barr, "Superquadrics", IEEE Computer Graphics and Applications,
Vol. 1, No 1, pp. 11-23, 1981.
[2] J. F. Blinn, "A generalization of algebraic surface drawing", ACM
Transactions on Graphics, Vol. 1, No 3, pp. 235-256, 1982.
[3] G. Wyvill and B. Wyvill, "Field functions for implicit surfaces", The
Visual Computer, Vol. 5, pp. 78-52, 1989.
[4] A. Ricii, "A Constructive Geometry for Computer Graphics", The
Computer Journal, Vol.16, No 2, pp. 157-160, May 1973.
[5] B Wyvill, A Guy, and E Galin, "Extending the CSG tree: warping,
blending and boolean operations in an implicit surface modeling
systems", in Proc. of Implicit Surfaces-98, pp.128-136, 1998.
[6] P.-C. Hsu and C. Lee, "The scale method for blending operations in
functionally based constructive geometry", Computer Graphics Forum,
Vol. 22, No 2, pp. 143-158, 2003.
[7] Q. Li, "Smooth piecewise polynomial blending operations for implicit
shaped", Computer Graphics Forum, Vol. 26, No 2, pp. 143-158, 2007.
[8] J. Bloomenthal and B. Wyvill, "Interactive techniques for implicit
modeling", in SIGGRAPH Computer Graphics, Vol.24, No 2, pp.
109-116, 1990.
[9] E. Akleman, "Interactive construction of smoothly blended star solids", in
Graphical Interface-96, pp. 159-167, May 1996.
[10] E. Akleman and J. Chen, "Generalized distance functions", in Proc.
Shape Modeling International '99, pp. 72-79, 1999.
[11] B. Crespin, C. Blanc, and C. Schlick, "Implicit sweep objects", in
Eurographics-96, Vol.15, No 3, pp. 165-175, 1996.
[12] A. Hanson, "Hyperquadrics: smoothly deformable shapes with convex
polyhedral bounds", Computer Vision, Graphics and Images Processing,
Vol. 44, No 1, pp. 191-210, 1988.
[13] C. Blanc and C. Schlick, "Ratioquadrics: an alternative model to
superquadrics", The Visual Computer, Vol. 12, pp. 420-428, 1996.