Equation of an Ellipse in 3-Space

Date: 07/02/2003 at 02:15:07
From: Clark Keith
Subject: Equation of an Ellipse in 3-space

I am looking for the equation of an ellipse in 3-dimensional space.  
It can be a parametric formulation (e.g., x(t), y(t), z(t)) or a more 
canonical form (e.g., the 3D analog to the 2D form 
((X*X)/a)+((Y*Y)/b)=1). I am assuming that an ellipse can be oriented 
in any arbitrary way in space so that it has components in all three 
principal directions.

We are working an astrodynamics problem that involves relative motion 
between two objects (a resident space object, RSO, around which a 
servicing satellite is moving. The RSO is the center of the coordinate 
frame and its local horizontal and local vertical define the principal 
axes such that +X points along the vector from the center of the earth 
to the RSO, +Y points in the direction of motion of the RSO, and +Z 
completes a right-handed coordinate frame). The path, or trajectory, 
of the servicer in this coordinate frame is often described as an 
ellipse and we are trying to develop a set of parameters that 
characterize the trajectory. 

We want to develop terms that specify the major and minor axes and 
orientation of those axes in 3D space and want to show that they 
really do represent an ellipse. We have looked everywhere and can't 
find any specification of the equation of an ellipse in 3-space. We 
have considered deriving it but are not sure where to start.  

I thought you might be aware of a formulation or give us some guidance 
on how to derive it ourselves (not wanting to spend inordinate amounts 
of time on this we would, of course, prefer getting the equations or 
being pointed to someone or some publication that would have them. 
Deriving the result doesn't really gain us anything in terms of trying 
to work the real problem we're trying to solve).

Date: 07/02/2003 at 09:51:27
From: Doctor George
Subject: Re: Equation of an Ellipse in 3-space

Hi Clark,

Thanks for writing to Doctor Math.

There is probably more than one way to do this. I'm not sure just what 
information you have as inputs. For now I will assume that you can 
describe the ellipse on some plane, and are looking for how to express 
it in 3-space.

In 2-space we have

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

Now for some angle 'theta', it is apparent that the point

               (a cos(theta), b sin(theta))

is on the ellipse. Thus we have a parametric description for it.

If we know that the center of the ellipse is at C, with unit vectors 
U and V for the major and minor axes, then we can describe the ellipse 
in 3-space as...

             C + a cos(theta) U + b sin(theta) V

Does that help?

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

Date: 07/04/2003 at 20:59:50
From: Clark Keith
Subject: Equation of an Ellipse in 3-space

Thank you so much for responding.

I'm not clear about what you mean when you say:

"...with unit vectors U and V for the major and minor axes..."

Do you mean unit vectors in the direction of those two axes? If that 
is, indeed, what you're saying then I presume that there is a further 
relationship between U and V; i.e., they are orthogonal to one 
another. So does that mean that if I know one then I automatically 
know the other? I guess only partially since there would be some 
ambiguity about its orientation. In 3-space one could rotate that 
perpendicular unit vector, V, through 360 deg about U, and still be 
orthogonal. To resolve that ambiguity one would need to know the 
specific rotation angle (i.e., orientation) of the V vector with 
respect to V.

I'll work on it. I think you may have given me the start that I need.

Again, thanks so very much.

Date: 07/05/2003 at 14:29:54
From: Doctor George
Subject: Re: Equation of an Ellipse in 3-space
Hi Clark,

Yes, U and V are unit vectors in the directions of the major and 
minor axes, and they are perpendicular. Without understanding all 
that you are doing, I assumed that you know the essential properties 
of your ellipse (such as its plane, foci, axes lengths, etc.) and 
simply needed a parametric representation.

If you know the major axis direction, and a normal vector for the 
plane of the ellipse, then the minor axis will be their cross product.

Write again if you need more help.

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

Date: 07/06/2003 at 20:51:14
From: Clark Keith
Subject: Equation of an Ellipse in 3-space

I thank you very much.  I actually don't exactly have the basic 
elements of the ellipse. What we have are parameters that describe 
the relative motion of one artificial satellite (call it the deputy) 
around another reference artificial satellite (call it the chief). 
The coordinate frame describing the relative motion is centered on 
the chief (i.e., the origin of the coordinate frame (0,0,0) is at the 
center of the chief).  

The +X axis is radially outward from the center of the earth through 
the center of the satellite, the +Y axis is orthogonal to the X-axis 
and point roughly in the direction of motion of the chief in its 
orbit (I say "roughly" because unless the chief is in a circular orbit 
then it doesn't exactly coincide with the chief's velocity vector), 
and the +Z-axis completes an orthogonal, right-handed coordinate 
frame).

The relative motion is describe roughly by Hill's Equations (also 
known Clohessy-Wilshire Equations). Again, "roughly" because Hill's 
doesn't take into account such real-life issues as drag, 
non-circular orbits, complex gravitational fields, etc.  

We are, in fact, in the process of extending and expanding upon 
Hill's work to add some of those complexities. But our team has been 
working on a preliminary aspect which is a better set of terms to 
allow better visualization of the relative motion.

In this visualization formulation we have created a set of parameters 
that describe the motion given any initial conditions. The motion of 
the deputy is, generally (though not always) elliptical around the 
chief. The parameters describe the semi-major axis, the eccentricity, 
the angle around the center of the ellipse that the deputy is at any 
given moment around the ellipse, and an angle that describes the 
z-component.  

The issue that we're trying to deal with is that we intuitively know 
that the motion is elliptical, but we're trying to prove that it is 
elliptical. We have an equation of the motion in x(t), y(t), and z (t) 
and we "assert" that they are the equations of an ellipse - but we're 
trying to prove it.

Hence, we're looking for the parametric form of an ellipse in 3-space 
(i.e., x(t), y(t), and z(t)...where t is the paramteric parameter).

Clark

Date: 07/07/2003 at 08:11:18
From: Doctor George
Subject: Re: Equation of an Ellipse in 3-space
Hi Clark,

This is an interesting problem. Let's go back to this equation.

         C + a cos(theta) U + b sin(theta) V

If we switch from 'theta' to 't' and break down the vectors into 
their components we get

         x(t) = Cx + a cos(t) Ux + b sin(t) Vx
         y(t) = Cy + a cos(t) Uy + b sin(t) Vy
         z(t) = Cz + a cos(t) Uz + b sin(t) Vz

If your equations for x(t), y(t) and z(t) can be transformed into the 
above form, then you will have proven that your equations describe an 
ellipse.

Write again if you need more help.

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

Date: 07/09/2003 at 01:43:21
From: Clark Keith
Subject: Thank you (Equation of an Ellipse in 3-space)

Thank you so much for your time. I believe you have just broken 
through the barrier for us and we greatly appreciate your time and 
help. With that last formulation I believe you have, indeed, gotten us 
over the hump and I think we can take it from here.  Your patience and 
time and encouragment have been wonderful. Again, thank you.