Path: utzoo!utgpu!watmath!clyde!att!rutgers!mailrus!eecae!cps3xx!usenet
From: usenet@cps3xx.UUCP (Usenet file owner)
Newsgroups: comp.graphics
Subject: Re: Superquadrics
Keywords: superquadrics
Message-ID: <1245@cps3xx.UUCP>
Date: 6 Dec 88 14:37:14 GMT
References: <3270@entire.UUCP> <1665@hp-sdd.HP.COM> <1988Nov16.234857.2372@cs.rochester.edu> <1679@hp-sdd.HP.COM> <146@terminus.Morgan.COM>
Reply-To: flynn@pixel.cps.msu.edu (Patrick J. Flynn)
Distribution: na
Organization: Pattern Rec. & Img. Processing Lab, CS, Mich. State U.
Lines: 36

In article <146@terminus.Morgan.COM> chuck@Morgan.COM () writes:
>Could someone explain the nature of superquadrics along with mathematical
>representations and interpretations.  I am especially interested in
>rendering techniques.  I have heard that there are a number of articles
>around but no one wants to provide a specific reference.  Thanks.
            ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Shame on them.

A point on a `basic' superquadric surface is given as follows.  Let
eta and omega be latitude and longitude parameters (angles), respectively;
C(eta), S(eta), C(omega), S(omega) are the sine and cosine of eta and omega, 
epsilon1 and epsilon2 are the `shape' parameters of the SQ.  Then a point
on the SQ is given by
x(eta,omega)= (C(eta)**epsilon1)*(C(omega)**epsilon2)
y(eta,omega)= (C(eta)**epsilon1)*(S(omega)**epsilon2)
z(eta,omega)= S(eta)**epsilon1

Vary eta between -pi/2 and pi/2, and omega between -pi and pi, and you
sweep out a closed surface.  Therefore, SQs are volumetric primitives.
You can build other shapes by bending, twisting, and tapering the basic form.

Some references:
Barr, Superquadrics and angle-preserving transformations, IEEE CG&A 1, 1-20.
Pentland, Perceptual Organization and the Representation of Natural Form,
Artificial Intell. J. 28, 293-331.
Bacjsy and Solina, Three Dimensional Object Repreentation Revisited,
Proc. 1st Int. Conf. on Computer Vision (ICCV-87), London, 231-240.
Boult and Gross, Recovery of superquadrics from depth information, Proc. 1987
Workshop on Spatial Reasioning and Multisensor Fusion, St. Charles, IL, 128-137.

These, and the references in them, should be a decent introduction.  Pentland's
paper is the best of the four (IMHO).
--
Patrick Flynn, Dept. of Computer Science, Michigan State University
flynn@cpsvax.cps.msu.edu flynn@eecae.UUCP FLYNN@MSUEGR.BITNET
"First we break 'em in half.... then we mash 'em to a pulp."