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.
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