Ellipse in 3D Defined with Parametric Equations

Date: 12/23/2003 at 13:18:49
From: Rick
Subject: Parametric equation for an ellipse in 3 dimensions

Given the lengths of an ellipse's semi-axes, and their directions in
3-space, how do I calculate the amplitudes (Ai) and phases (Bi) of the
corresponding parametric equations?

Date: 12/23/2003 at 14:27:45
From: Doctor George
Subject: Re: Parametric equation for an ellipse in 3 dimensions

Hi Rick,

The directions of the ellipse semi-axes give you the unit vectors U 
and V, where 'a' and 'b' have their usual meanings in the standard 
ellipse equation.  If C is the center then P is on the ellipse when

  P = C + a*cos(Z)*U + b*sin(Z)*V,        0 <= Z < 2pi

Now break down the points and vectors into components, and isolate 
Px.  Py and Pz will be handled the same way.  (Note that I am using y 
and z for components, and Y and Z for angles.)

  Px = Cx + a*ux*cos(Z) + b*vx*sin(Z)

Now define 

  sin(Y) = a*ux/sqrt[(a*ux)^2 + (b*vx)^2]
  cos(Y) = b*vx/sqrt[(a*ux)^2 + (b*vx)^2]

You can see that this is valid by looking at sin^2(Y) + cos^2(Y).

This give us

  Px = Cx + sqrt[(a*ux)^2 + (b*vx)^2]*[sin(Y)cos(Z) + cos(Y)*sin(Z)]

  Px = Cx + sqrt[(a*ux)^2 + (b*vx)^2]*sin(Y + Z)

The magnitude is sqrt[(a*ux)^2 + (b*vx)^2], and the phase is
Y = atan2[sin(Y),cos(Y)].

Is this what you are looking for?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/28/2003 at 12:49:47
From: Rick
Subject: Thank you (Parametric equation for an ellipse in 3 dimensions)

Dear Dr. George --

Yes, this is exactly what I was looking for.  Problem solved!  Thank
you very much.

--Rick