Thread: 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!

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!

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 ....letting 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 then where r is the radius

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 ....letting 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 then 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

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

Originally Posted by Mathstud28

that would work but that is a lot of work

Do you think that there is an easier solution?