Re: Silly question

In article <deu6pj$bqg$1@xxxxxxxxxxxxxxxxxx>,
Jaco van Niekerk <sparky@xxxxxxxxxx> wrote:
>Hello
>
>I have a torus, say F(u,v) = X(u,v)i + Y(u,v)j + Z(u,v)k
>
>with
> X(u,v) = Acos(u) + Bcos(u)cos(v)
> Y(u,v) = Asin(u) + Bsin(u)cos(v)
> Z(u,v) = Csin(v)
>
> A, B and C are constants

.... with, I presume, A > B > 0 and C > 0. Yes, this is a torus
for, say, -pi <= u <= pi and -pi <= v <= pi.

>Now, for the silly part. I have a point P that definitely lies on the
>surface. Now I need to find a pair (u,v) for the point P. Now this seems
>trivial, but on my graphics display (I'm writing a program) it does not
>work! This is what I've done:
>
>Pz = Csin(v)
>sin(v) = Pz/C
>v = arcsin(Pz/C)

That gives you a v in [-pi/2, pi/2];
OK if cos(v) >= 0, which is true if Px^2 + Py^2 >= A^2.
Otherwise v = pi - arcsin(Pz/C) (if Pz >= 0), or -pi - arcsin(Pz/C)
(if Pz < 0)

>Px = Acos(u) + Bcos(u)cos(v)
>Px = cos(u)[A + Bcos(v) ]
>since I have v, I can calculate u
>cos(u) = Px / (A + Bcos(v))
>u = arccos(Px / (A + Bcos(v)))

OK if Py >= 0. Otherwise -that.

>Yet, my u, v pair is not what it should be. What am I doing wrong?

Looking in the wrong intervals.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

.

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