# equation of a circle from 2 circles and a point?

• Apr 8th 2008, 07:19 PM
erinspice
equation of a circle from 2 circles and a point?
My husband is trying to build a robot that follows you around. We are trying to triangulate the position of a point with three sensors, and we have only gotten this far.

This image is not at all to scale. In fact, circle C should be much much larger than circle D. Oops. :(
B is the person.
The origin is the lower left corner of the image.
A, C, and D are sensors on the robot.
We know the points A, C, and D.
We know the radii of circles C and D by measuring the difference in time it took our ultrasonic signal to travel from B to A, C, and D.
We know the equations of circles C and D because we know the centerpoints and the radii.

We do not know the equation of circle B.
We do not know the points of intersection of circle B and circles C and D.

Can we calculate the equation of circle B given the equations of C and D and the point A? I suspect there is some way to do this, perhaps by solving these equations simultaneously, perhaps using the equation of the curvature of the arc made by A and the intersections of circle B and circles C and D.

Any ideas? I know there is somebody out there smarter than me who can figure this out!

http://erin.thespicelands.com/triangulator.gif
• Apr 9th 2008, 03:40 PM
Mathstud28
Ok
Quote:

Originally Posted by erinspice
My husband is trying to build a robot that follows you around. We are trying to triangulate the position of a point with three sensors, and we have only gotten this far.

This image is not at all to scale. In fact, circle C should be much much larger than circle D. Oops. :(
B is the person.
The origin is the lower left corner of the image.
A, C, and D are sensors on the robot.
We know the points A, C, and D.
We know the radii of circles C and D by measuring the difference in time it took our ultrasonic signal to travel from B to A, C, and D.
We know the equations of circles C and D because we know the centerpoints and the radii.

We do not know the equation of circle B.
We do not know the points of intersection of circle B and circles C and D.

Can we calculate the equation of circle B given the equations of C and D and the point A? I suspect there is some way to do this, perhaps by solving these equations simultaneously, perhaps using the equation of the curvature of the arc made by A and the intersections of circle B and circles C and D.

Any ideas? I know there is somebody out there smarter than me who can figure this out!

http://erin.thespicelands.com/triangulator.gif

Unless I am misunderstanding your question if you know the length from the center of circe B to the center of circle D/C ...and you know the centers of either of those circles then just do this $d=\sqrt{(x-x_0)^2+(y-y_0)^2}$....letting $(x_0,y_0)$ be the center of the circle you know...then solving this simultaneously with the fact that you know the radius...now here is where it gets tricky...since you know that circle C and circle D intersect but you dont know the intersections make a very good guess then with that guess say its $(x_{00},y_{00})$ then $r=\sqrt{(x-x_{00})^2+(y-y_{00})^2}$ where r is the radius
• Apr 9th 2008, 04:38 PM
erinspice
Quote:

Originally Posted by Mathstud28
Unless I am misunderstanding your question if you know the length from the center of circe B to the center of circle D/C ...and you know the centers of either of those circles then just do this $d=\sqrt{(x-x_0)^2+(y-y_0)^2}$....letting $(x_0,y_0)$ be the center of the circle you know...then solving this simultaneously with the fact that you know the radius...now here is where it gets tricky...since you know that circle C and circle D intersect but you dont know the intersections make a very good guess then with that guess say its $(x_{00},y_{00})$ then $r=\sqrt{(x-x_{00})^2+(y-y_{00})^2}$ where r is the radius

I made a typo in the image -- we don't know anything about circle B including the center and radius. However, I think we can apply the solution to Apollonius' Tangency Problem to solve this. I haven't tried it yet though. This solution was suggested to me by someone on another forum.

Apollonius' Tangency Problem
• Apr 9th 2008, 04:46 PM
Mathstud28
Yeah
Quote:

Originally Posted by erinspice
I made a typo in the image -- we don't know anything about circle B including the center and radius. However, I think we can apply the solution to Apollonius' Tangency Problem to solve this. I haven't tried it yet though. This solution was suggested to me by someone on another forum.

Apollonius' Tangency Problem

that would work but that is a lot of work (Surprised)
• Apr 9th 2008, 05:09 PM
erinspice
Quote:

Originally Posted by Mathstud28
that would work but that is a lot of work (Surprised)

Do you think that there is an easier solution?