From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Looking for 3D transformation resources
Date: 21 Apr 1998 21:31:29 GMT

In article <6gv38j$39p$1@nnrp1.dejanews.com>,  <radevenz@ix.netcom.com> wrote:
>I am trying to generate coordinates for the surface of a toroidial treifoil
>knot.  My big hurdle is how to generate coordinates at a cross sectional
>slice.
>
>The problem I am looking to solve is:
>Given a normal N, Ai+Bj+Ck, at a point x0,y0,z0 I know that the plane is
>defined as (x,y,z) such that A(x-x0) + B(y-y0) + C(z-z0) = 0.  This however,
>to me, suggests a z=f(grid over x & y) mapping, which is not conducive to a
>circular cross section.

Is this then what you want: given a parameterization (x,y,z) = F(t) of the
underlying knot, you know that at each point  F(t)  the vector  F'(t)  is
tangent to the curve. Now let  v(t)  be any vector at each point which
is not parallel to  F'(t) -- for example, in your  setting you can
probably select a point  p  not on the curve (perhaps the center of your
picture) and use  v(t) = F(t) - p. Then you can project  v  to the normal
plane at each point  (i.e. let v2 = v - (<v,F'>/<F',F'>)F'  ) and scale
to get a unit vector (i.e. let v3=v2/||v2||). You can then get an
orthonormal basis for the normal plane by using cross-products
(let v4 = v3 x (F'/||F'||) ). Then you have a parameterization for the
circle in the normal plane at each point:  cos(u) v3(t) + sin(u) v4(t).
Letting  t  vary as well will give you your surface.

dave