# An Adaptive Subdivision Method With Crack Prevention for Rendering Beta-spline Objects

### Brian A. Barsky, Anthony D. DeRose and Mark D. Dippe

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-87-348

March 1987

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1987/CSD-87-348.pdf

Adaptive subdivision is a method of creating polygonal approximations to spline surfaces. An adaptive subdivision algorithm takes as input a spline surface and a tolerance
*epsilon*, and outputs a piecewise planar approximation to the surface that is guaranteed to differ from the actual surface by a distance no greater than
*epsilon*. These algorithms proceed by recursively "splitting" the surface into smaller subsurfaces, ultimately approximating subsurfaces with planar polyhedra. These algorithms are therefore characterized by the mathematics behind the splitting of a surface, the test that is used to determine when to stop the recursion, and the method by which a subsurface is approximated by polyhedra. Algorithms of this type are currently known for spline techniques such as Bezier and B-splines.

The purpose of this paper is two-fold. First, we describe the Beta-spline curve and surface technique and derive the equations governing the splitting of Beta-spline curves and surfaces. Second, we present a very general adaptive subdivision algorithm that can be used with a variety of surface techniques. It incorporates splitting criteria based on flatness and prevents cracks from occurring between approximating polyhedra. The testing required to determine when surface splitting can stop can be very expensive and methods for reducing these costs are examined. The tolerance controlling the splitting process may itself be adaptive, so that as an object moves farther away the tolerance is automatically increased. These ideas have all been used in the implementation of a surface design and rendering system.

BibTeX citation:

@techreport{Barsky:CSD-87-348, Author = {Barsky, Brian A. and DeRose, Anthony D. and Dippe, Mark D.}, Title = {An Adaptive Subdivision Method With Crack Prevention for Rendering Beta-spline Objects}, Institution = {EECS Department, University of California, Berkeley}, Year = {1987}, Month = {Mar}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1987/5992.html}, Number = {UCB/CSD-87-348}, Abstract = {Adaptive subdivision is a method of creating polygonal approximations to spline surfaces. An adaptive subdivision algorithm takes as input a spline surface and a tolerance <i>epsilon</i>, and outputs a piecewise planar approximation to the surface that is guaranteed to differ from the actual surface by a distance no greater than <i>epsilon</i>. These algorithms proceed by recursively "splitting" the surface into smaller subsurfaces, ultimately approximating subsurfaces with planar polyhedra. These algorithms are therefore characterized by the mathematics behind the splitting of a surface, the test that is used to determine when to stop the recursion, and the method by which a subsurface is approximated by polyhedra. Algorithms of this type are currently known for spline techniques such as Bezier and B-splines. <p> The purpose of this paper is two-fold. First, we describe the Beta-spline curve and surface technique and derive the equations governing the splitting of Beta-spline curves and surfaces. Second, we present a very general adaptive subdivision algorithm that can be used with a variety of surface techniques. It incorporates splitting criteria based on flatness and prevents cracks from occurring between approximating polyhedra. The testing required to determine when surface splitting can stop can be very expensive and methods for reducing these costs are examined. The tolerance controlling the splitting process may itself be adaptive, so that as an object moves farther away the tolerance is automatically increased. These ideas have all been used in the implementation of a surface design and rendering system.} }

EndNote citation:

%0 Report %A Barsky, Brian A. %A DeRose, Anthony D. %A Dippe, Mark D. %T An Adaptive Subdivision Method With Crack Prevention for Rendering Beta-spline Objects %I EECS Department, University of California, Berkeley %D 1987 %@ UCB/CSD-87-348 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1987/5992.html %F Barsky:CSD-87-348