Path: utzoo!mnetor!uunet!lll-winken!lll-lcc!ames!mailrus!umix!utah-gr!jcobb
From: jcobb@utah-gr.UUCP (Jim Cobb)
Newsgroups: comp.graphics
Subject: Re: Defining a sphere with Bezier patches
Message-ID: <2424@utah-gr.UUCP>
Date: 1 Apr 88 21:20:15 GMT
References: <2390@saturn.ucsc.edu> <1632@pixar.UUCP> <1639@pixar.UUCP> <3183@csli.STANFORD.EDU> <2552@saturn.ucsc.edu>
Reply-To: jcobb@gr.utah.edu.UUCP (Jim Cobb)
Organization: University of Utah CS Dept
Lines: 35
Keywords: sphere, Bezier, REYES

It is impossible to exactly parametrize a sphere with polynomial
patches.  I will describe here a simple rational quadratic Bezier patch
that IS exact.  This patch covers an octant of the sphere.

Let sq = sqrt(2)/2.  Define homogeneous control points P_ij as

    P_00 =	[   0   0   -1   1 ]
    P_01 =	[  sq   0  -sq  sq ]
    P_02 =	[   1   0    0   1 ]

    P_10 =	[   0   0  -sq  sq ]
    P_11 =	[ 0.5 0.5 -0.5 0.5 ]
    P_12 =	[  sq  sq    0  sq ]

    P_20 =	[   0   0   -1   1 ]
    P_21 =	[   0  sq  -sq  sq ]
    P_22 =	[   0   1    0   1 ]

A note about the interpretation of these coefficients: There are two
conflicting conventions for the meaning of control points for a rational
Bezier curve or surface.  I am using the convention that denotes the
components of the points as [ X Y Z W ], with resulting surface
evaluation given by

    	    	 sum [X_ij Y_ij Z_ij] theta_i(u) theta_j(v)
    S( u, v ) =  ------------------------------------------
    	    	      sum W_ij theta_i(u) theta_j(v)


There is a slight problem with this parametrization: the patch is
degenerate along one of its edges (an entire edge curve collapses into a
single point).  There are alternative schemes to avoid this difficulty,
and there are ways of dealing with the degeneracy (see Faux & Pratt p.
235 ff.).

Jim