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.
[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.
[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.
@article{"International Journal of Information, Control and Computer Sciences:58300", author = "Pi-Chung Hsu", title = "An Implicit Representation of Spherical Product for Increasing the Shape Variety of Super-quadrics in Implicit Surface Modeling", abstract = "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.", keywords = "Implicit surfaces, Soft objects, Super-quadrics.", volume = "6", number = "12", pages = "1690-7", }