# Arbitrary Subdivision of Bezier Curves

### Brian A. Barsky

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-86-265

November 1985

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1986/CSD-86-265.pdf

Subdivision is a powerful technique that has many useful applications. The fundamental concept is the splitting of a curve or surface into smaller pieces whose union is identical to the original curve or surface. Standard Bezier subdivision splits the curve at the midpoint of the curve, in parametric space.

This paper generalizes midpoint subdivision to arbitrary subdivision, enabling the subdivision to be performed at any parametric value, not solely at the midpoint. This allows for subdivision that would adapt to regions of varying curvature or correlate with the curve length in geometric space.

After explaining the original development of Bezier curves, the mathematical theory for arbitrary subdivision is developed, and finally an illustration of the subdivision process that shows the recursive procedure in a step-by-step manner is given.

BibTeX citation:

@techreport{Barsky:CSD-86-265, Author = {Barsky, Brian A.}, Title = {Arbitrary Subdivision of Bezier Curves}, Institution = {EECS Department, University of California, Berkeley}, Year = {1985}, Month = {Nov}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1985/6091.html}, Number = {UCB/CSD-86-265}, Abstract = {Subdivision is a powerful technique that has many useful applications. The fundamental concept is the splitting of a curve or surface into smaller pieces whose union is identical to the original curve or surface. Standard Bezier subdivision splits the curve at the midpoint of the curve, in parametric space. <p> This paper generalizes midpoint subdivision to arbitrary subdivision, enabling the subdivision to be performed at any parametric value, not solely at the midpoint. This allows for subdivision that would adapt to regions of varying curvature or correlate with the curve length in geometric space. <p> After explaining the original development of Bezier curves, the mathematical theory for arbitrary subdivision is developed, and finally an illustration of the subdivision process that shows the recursive procedure in a step-by-step manner is given.} }

EndNote citation:

%0 Report %A Barsky, Brian A. %T Arbitrary Subdivision of Bezier Curves %I EECS Department, University of California, Berkeley %D 1985 %@ UCB/CSD-86-265 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1985/6091.html %F Barsky:CSD-86-265