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
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