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!