Subject: Re: Morphing magic - NOT!?!?
From: Ernie Wright <[email-masked]>
To: [email-masked]
Date: Tue, 17 Oct 1995 04:06:11 -0400 (EDT)

Mathew Mower wrote:

> Why is it that LW requires exact point correspondence for object
> morphs?
> 
> I'm sure that I have seen other programs that allow arbitrary object
> morphs (at least I did see cubes to spheres).  Is this a limitation
> for all packages or just LW.

A lot could be said about this, but the short answer is that it's a 
limitation of 3D in general.

The cubes to spheres morph was probably modeled as a superellipsoid:

   pow( pow( x/a, n2 ) + pow( y/b, n2 ), n1/n2 ) + pow( z/c, n1 ) - 1

where pow( x, n ) is the C function that raises x to the n power.

The parameters n1 and n2 are sometimes called bulge factors.  As you
vary them, the shape goes from a rounded box (n1 and n2 > 10.0 or so)
to a sphere (n1 = n2 = 2.0) to an octahedron (1.0) to sort of a six-
pointed star shape.  There are also superhyperboloids (of one and two 
sheets) and supertorii.  As a group they're called superquadrics.

The morph is possible because the shape can be completely described by
a single, continuous function.  This is the analytic equivalent of two
polygon-based shapes having the same number of points.  The only good
model representation for completely general morphing is voxels, but
the use of these on a day-to-day basis is still a decade away at least.

- Ernie

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Ernie Wright <[email-masked]> sent this message.
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