From: "Earl F. Glynn"
Newsgroups: borland.public.delphi.graphics
References: <3ac14368_1@dnews> <3ac14c7c$1_2@dnews> <3ac15184$1_1@dnews>
Subject: Re: Parametric equations for an ellipsoid (ie. ellipsoid coordinate system)?
Date: Tue, 27 Mar 2001 22:44:37 -0600
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"Paul Nicholls" <phantom@southcom.com.au> wrote in message news:3ac15184$1_1@dnews...
> Sorry about not explaining clearly enough at the start Earl :-/

Sorry I'm being a bit dense on this tonight.  I discovered I'm a bit rusy with my 3D math.  I hope I
get this right:

Let's consider a 2D ellipse first:

x^2/a^2  +  y^2/b^2  =  1

This can be drawn in the XY plane (Z = 0) with a single angular parameter.  Let's use theta:

x = a*cos(theta)
y = b*sin(theta)

Why does this work?

If you plug the expressions for x and y back into the ellipse equation, you get:

cos^2(theta) + sin^2(theta) = 1, which is a well-known trig identity.

We can draw an ellipsoid as a series of elliptical "slices" along the z-axis, from zMin to zMax with
some apprpropriate regular interval.

Start with the ellipsoid formula:

x^2/a^2  +  y^2/b^2  + z^2/c^2  =  1

If we know z, we can easily get the appropriate XY ellipse equation for the "z" slice:

x^2/a^2  +  y^2/b^2  = 1 - z^2/c^2

But we need this equation to look like the equation at the top:

x^2/[a^2 * (1 - z^2/c^2)] + y^2/[b^2 * (1 - z^2/c^2)] = 1

This means the z-slice ellipse equation as a function of theta is:

x = SQRT(a^2 * (1 - z^2/c^2)) * cos(theta)
y = SQRT(b^2 * (1 - z^2/c^2)) * sin(theta)

Draw the z-slice by varying theta from 0 to 2 PI with some appropriate increment.

After drawing the z slices from zMin to zMax, x slices can be drawn that are perpendicular to the z
slices.

This time we assume we know xMin to xMax for the ellipsoid.

Start again with the ellipsoid formula:

x^2/a^2  +  y^2/b^2  + z^2/c^2  =  1

If we know x, we can easily get the appropriate YZ ellipse equation for the "x" slice:

y^2/b^2  +  z^2/c^2  = 1 - x^2/a^2

But we need this equation to look like the equation at the top:

y^2/[b^2 * (1 - x^2/a^2)] + z^2/[c^2 * (1 - x^2/a^2)] = 1

This means the x-slice ellipse equation as a function of theta is:

y = SQRT(b^2 * (1 - x^2/a^2)) * cos(theta)
z = SQRT(c^2 * (1 - x^2/a^2)) * sin(theta)

Again, the theta parameter needs to vary from 0 to 2*PI in some appropriate increment.

And if you want to, you could form y slices with XZ ellipses.

Maybe this approach would get you regularly-spaced coordinates for an ellipsoid.

This is the same approach as drawing a sphere as a series of circles
http://homepages.borland.com/efg2lab/Graphics/SphereInCube.htm

--
efg     efg2@efg2.com     Earl F. Glynn, Overland Park, KS  USA

efg's Computer Lab:  http://www.efg2.com/Lab
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