How to find the point on a circle when a point within it is projected out on to it?

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I need to find the coordinates of a point on a circle (point b in the picture) using the variables shown in the picture.

I know this is quite a maths related problem but i'm writing the program which this is going to be part of in python. I've tried the following code and had no luck, i've check the angle its passing though and that's correct. I've also tried the angle in radians and degrees both with no luck.

    int_x = r * math.cos(angle)
    int_y = r * math.sin(angle)

Please ask any questions about the problem

Thank you

Given the position of angle in the diagram, if you were to draw a triangle that encloses angle, you'd find that int_x is opposite the angle and int_y is adjacent, which means you have your equations flipped (i.e. int_x = r * sin.cos(angle) )

Ex: Find the Point on a Circle Given an Angle and the Radius , Ex: Find the Point on a Circle Given an Angle and the Radius they were teaching this in Duration: 3:06 Posted: 30 Jun 2012 The technique involved is to draw a line starting at the point where the line through the circle center intersects the circle circumference (Point A in Figure 2), and to extend that line until it touches the circle at any point on its’ circumference (Figure 2, Point C).

If the circle center is known to be (c_x, c_y) and the point a is at (a_x, a_y). Then we simply construct a line from the center through point a of length r. This is simply a similar triangle. We compute the hypotenuse of the triangle to be

h = sqrt((a_x - c_x)^2 + (a_y-c_y)^2)

and then we know that

(b_x, b_y) = (c_x + (a_x - c_x) * r/h, c_y + (a_y - c_y) * r/h).

Then you don't need to worry about angles at all! Hope that helps.

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I cannot comment (not enough rep), so this answer is only to indicate that in Changming's answer, it should be r/h instead of h/r (twice).

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  • Are the coordinates of O at (0, 0)?
  • The question is confusing. Either theta or a are unnecessary to find b
  • Thank you very much i've spotted my error and corrected it and now the code is working a charm! Thank you.
  • @FinnHunt Be careful that theta is calculated by atan2, otherwise you can get surprises when a is on the Y-axis.
  • This answer makes most sense, as the problem description with only a picture doesn't tell whether theta is between 0 and 2 pi, nor whether it's sign changes if it would be to the left, nor whether theta would always be measured starting from the up-direction.
  • Thank you very much for your answer i'm going to try it now!