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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.
```

```
Date: 01/24/2017 at 11:55:27
From: Tamas
Subject: equation of parabola in space

Dear Doctor Math,

I'm writing you because of your good description, above.

I want to collect equations for all conic sections in space, in general
position. I used Mathematica to come up with the equation for an ellipse
(and for a circle). Now I am looking for the parametric equation of
parabola in space.

My question: Would you write me the parametric equation for a parabola in
3-space, the same way as you wrote the equation for a 3D ellipse?

Tamas

```

```
Date: 01/30/2017 at 08:53:55
From: Doctor George
Subject: Re: equation of parabola in space

Hi Tamas,

Thanks for writing to Doctor Math.

In 2 dimensions, a parabola with vertex (0, 0) has this equation:

y = a * x^2

We can write this parametrically with parameter t, as follows:

x = t
y = a * t^2

Now in 3 dimensions, if we let C be an arbitrary vertex, and let U and V
be unit vectors corresponding to the x and y directions, we can describe
the points on the parabola like this:

P = C + t U + at^2 V

Does that make sense? Write again if you need more help.

- Doctor George, The Math Forum at NCTM

```
Associated Topics:
College Conic Sections/Circles
College Higher-Dimensional Geometry

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