Find a specific point on a circle by a point & angle between them

Question

I use C# WPF and im stuck with that for couple of days.

How do I find point M?

Answer 1

Basic trigonometry or application of a rotation matrix (not to forget: translate the center to the origin and after rotation back to its inital position):

XM=a+cos(C°)*(X1-a)-sin(C°)*(Y1-b)
YM=b+sin(C°)*(X1-a)+cos(C°)*(Y1-b)

The input to the trigonometric functions has to be in radians, 1°=pi/180, C°=C*1°.

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In screen coordinates, you have to first replace all y components with their negatives to get a properly oriented cartesian coordinate system. And then back for screen coordinates. Combined, this leads to replacing Y1-b and YM-b with b-Y1 and b-YM, resulting in the formulas

XM=a+cos(C°)*(X1-a)-sin(C°)*(b-Y1)
YM=b-sin(C°)*(X1-a)-cos(C°)*(b-Y1)

which can be rearranged to

XM=a+cos(C°)*(X1-a)+sin(C°)*(Y1-b)
YM=b-sin(C°)*(X1-a)+cos(C°)*(Y1-b)

or

XM=a+cos(-C°)*(X1-a)-sin(-C°)*(Y1-b)
YM=b+sin(-C°)*(X1-a)+cos(-C°)*(Y1-b)

which is also geometrically sensible since a reflection on the x axis changes the direction of rotations.