# Line between circles in space?

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• Jun 10th 2007, 10:39 PM
Malkalypse
Line between circles in space?
Say I have 2 circles which are rotating in space, sometimes parallel, sometimes perpendicular to each other. I have a line segment which is attached to a point on one circle, and must remain of a constant length while also touching the other circle. This is better illustrated by the following animation:

www.versatileartist.com/temp/circles.avi

I need to find a way to describe the point shown by the yellow cube in terms of the point shown by the green one.

Can anyone show me how to do this??
• Jun 11th 2007, 02:17 AM
JakeD
Quote:

Originally Posted by Malkalypse
Say I have 2 circles which are rotating in space, sometimes parallel, sometimes perpendicular to each other. I have a line segment which is attached to a point on one circle, and must remain of a constant length while also touching the other circle. This is better illustrated by the following animation:

www.versatileartist.com/temp/circles.avi

I need to find a way to describe the point shown by the yellow cube in terms of the point shown by the green one.

Can anyone show me how to do this??

Your description of the problem leaves a lot of freedom in how to set it up. Here is one of many ways to do it.

Let the first circle with the yellow box be the set of points

$C_1 = \{ x \in R^3 |\ x_1^2 + x_2^2 = r^2,\ x_3 = 0 \}.$

Let the second circle with the green box be the set of points

$C_2 = \{x' \in R^3 |\ x' = M\bar{x} + b,\ \bar{x}_1^2 + \bar{x}_2^2 = s^2,\ \bar{x}_3 = 0 \}$

where M is a rotation matrix (orthogonal, determinant = 1).

Let L be the length of the line connecting the boxes and let x be the coordinates of the yellow box and x' be the coordinates of the the green box. Then for a given $x' \in C_2,$ the solution(s) x to

$|| x' - x || = L,\ x \in C_1$

"describes" the yellow box x in terms of the green box x'.
• Jun 11th 2007, 11:35 AM
Malkalypse
Hm, this is definitely beyond anything I've ever studied. The furthest I ever got was high school trigonometry, and that was over 10 years ago.. I don't even know what field of math you are using, so that makes it a bit harder for me to understand!

Beyond that, I not only have to understand this myself, but understand it well enough to to write in terms my program (3DS Max) will understand.

What resources can you point me to that might help me understand your notation?
• Jun 11th 2007, 02:15 PM
Malkalypse
Okay, here is maybe a clearer description:

www.versatileartist.com/temp/circles.avi

(program coordinate system: x = right, y = back, z = up)

I need to find either

1) The position of point A or
2) angle B

static variables:
length of side b = 20.485
length of side c = 10.887

radius of circle 1 = 10.877
radius of circle 2 = 12.863

position of point B = [0, 0, 7.384]
center of circle 2 = [0, -14.737, 0]

known, changing variables:
The position of point C is known [x,y,z]
The length of side a is known
• Jun 11th 2007, 07:19 PM
JakeD
Quote:

Originally Posted by Malkalypse
Okay, here is maybe a clearer description:

www.versatileartist.com/temp/circles.avi

(program coordinate system: x = right, y = back, z = up)

I need to find either

1) The position of point A or
2) angle B

static variables:
length of side b = 20.485
length of side c = 10.887

radius of circle 1 = 10.877
radius of circle 2 = 12.863

position of point B = [0, 0, 7.384]
center of circle 2 = [0, -14.737, 0]

known, changing variables:
The position of point C is known [x,y,z]
The length of side a is known

Three sides a, b and c of a triangle are given. Angle B can be calculated using the Law of sines.
• Jun 11th 2007, 10:04 PM
Malkalypse
I just realized I made a mistake in describing what I needed, but as a result I am now a bit closer.

I don't even need the circles, and I can show what I need in a single picture.

http://www.versatileartist.com/temp/circles.jpg

I need the angle ABD.

I have the lengths for all sides described by triangles ABC and BCD. The only length I do not know is AD.

Lengths:
AB = 10.887, AC = 20.485
BC changes, but is known at any time

Right Angles:
ADC and ADB are right angles

Angles which change but are known
BCD, BDC, CDB, and CBD

unknown angles:
all others
• Jun 11th 2007, 11:49 PM
JakeD
Quote:

Originally Posted by Malkalypse
I just realized I made a mistake in describing what I needed, but as a result I am now a bit closer.

I don't even need the circles, and I can show what I need in a single picture.

http://www.versatileartist.com/temp/circles.jpg

I need the angle ABD.

I have the lengths for all sides described by triangles ABC and BCD. The only length I do not know is AD.

Lengths:
AB = 10.887, AC = 20.485
BC changes, but is known at any time

Angles:
ADC, ADB, and BDC are right angles

Angles which change but are known
BCD, CDB, and CBD

unknown angles:
all others

Only very basic trig is needed here. When you have right angles, use them. ADB is a right triangle, so you can use the Pythagorean Theorem to calculate the unknown side. Then use sin(ABD) = ratio of two of the sides.