# Modeling Implicit Quadrics and Free-form Surfaces With Trimmed Rational Quadratic Bezier Patches

### Seth J. Teller and Carlo H. Séquin

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-90-577

June 1990

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/CSD-90-577.pdf

Designers require familiar, well-behaved surfaces for solid modeling tasks. Implementers desire modeling primitives that can be specified compactly, computed efficiently, and rendered quickly. Implicit surface representations are cumbersome in these contexts.

We present a method for constructing a rational quadratic Bezier patch that interpolates a portion of a quadric surface, and clarify the geometric and parametric degrees of freedom inherent in any such construction. The surface to be interpolated is specified implicitly, along with a (possibly empty) set of halfspaces in R^3 whose intersection bounds the desired region of the surface.

We demonstrate a novel equivalence between familiar stereographic maps in two dimensions and rational quadratic Bezier curves, and extend this equivalence to an important subset of Bezier surfaces -- namely, those that interpolate quadrics. This equivalence can be exploited to produce trivially invertible parametric curves and surfaces, with no loss of representational power. We describe a new method of altering control weights that, given a triangular or quadrilateral subpatch of a quadric, produces the entire quadric.

These techniques are demonstrated for a collection of common modeling situations, and frequently occurring surface fragments, such as hyperbolic and toroidal fillets, cylindrical joins, and rounded corners. We argue that current heterogeneous representations of implicit quadrics can be replaced with a single trimmed surface representation based on the stereographic map correspondence. Finally, we discuss the prospects for integration of these new representational techniques into existing modeling environments.

BibTeX citation:

@techreport{Teller:CSD-90-577, Author = {Teller, Seth J. and Séquin, Carlo H.}, Title = {Modeling Implicit Quadrics and Free-form Surfaces With Trimmed Rational Quadratic Bezier Patches}, Institution = {EECS Department, University of California, Berkeley}, Year = {1990}, Month = {Jun}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/5528.html}, Number = {UCB/CSD-90-577}, Abstract = {Designers require familiar, well-behaved surfaces for solid modeling tasks. Implementers desire modeling primitives that can be specified compactly, computed efficiently, and rendered quickly. Implicit surface representations are cumbersome in these contexts. <p>We present a method for constructing a rational quadratic Bezier patch that interpolates a portion of a quadric surface, and clarify the geometric and parametric degrees of freedom inherent in any such construction. The surface to be interpolated is specified implicitly, along with a (possibly empty) set of halfspaces in R^3 whose intersection bounds the desired region of the surface. <p>We demonstrate a novel equivalence between familiar stereographic maps in two dimensions and rational quadratic Bezier curves, and extend this equivalence to an important subset of Bezier surfaces -- namely, those that interpolate quadrics. This equivalence can be exploited to produce trivially invertible parametric curves and surfaces, with no loss of representational power. We describe a new method of altering control weights that, given a triangular or quadrilateral subpatch of a quadric, produces the entire quadric. <p>These techniques are demonstrated for a collection of common modeling situations, and frequently occurring surface fragments, such as hyperbolic and toroidal fillets, cylindrical joins, and rounded corners. We argue that current heterogeneous representations of implicit quadrics can be replaced with a single trimmed surface representation based on the stereographic map correspondence. Finally, we discuss the prospects for integration of these new representational techniques into existing modeling environments.} }

EndNote citation:

%0 Report %A Teller, Seth J. %A Séquin, Carlo H. %T Modeling Implicit Quadrics and Free-form Surfaces With Trimmed Rational Quadratic Bezier Patches %I EECS Department, University of California, Berkeley %D 1990 %@ UCB/CSD-90-577 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/5528.html %F Teller:CSD-90-577