Find coordinates for points on circle given R, 2 Points, and angle or 2 points and center?

Question

I would like to find coordinates for points on a circle given:

1. Radius of circle
2. Coordinates of 2 points on the circle
3. Angle of point 1, center, and point 2.

Ultimately, I would like to write a formula in excel to calculate points on circle to stake out coordinates for surveying.

Thanks again for your help. Please let me know if I can provide any more information or rewrite question using better terminology.

Answer 1

Refer to the diagram below:-

(1) From given A and B, draw the corresponding perpendicular bisector such that P is their point of intersection.

(1.1) The co-ordinates of P can be found (by midpoint formula).

(2) From the given angle (say, 2θ), bisect it such that it is θ + θ.

(3) Find y where y = R cos θ.

(4) From P, draw arcs with radius = R cos θ cutting the brown line at Q. (Note that there are two positions of Q.)

(5) Q is now the center and QA (or QB) is the radius of the circle required.

Answer 2

Your problem is overdetermined since you only need either the radius or the angle since they are related by $R = \frac{d}{2} \cot\frac{\alpha}{2}$ where $d$ is the distance between the two points on the circle and $\alpha$ is the angle. The position of the center of the circle is the solution to this equation: $$(x_C-x_A)^2 + (y_C-y_A)^2 = (x_C-x_B)^2 + (y_C-y_B)^2 = R^2$$ $C$ is the center. Solving $(x_C-x_A)^2 + (y_C-y_A)^2 = (x_C-x_B)^2 + (y_C-y_B)^2$ gives you the linear relation between $x_C$ and $y_C$. Substitute $x_C$ or $y_C$ in $(x_C-x_A)^2 + (y_C-y_A)^2 = R^2$ then solve the quadratic equation. This gives you two possible coordinates for $C$. Then you can use the standard formula for a circle with given center and radius