mathtriangular

Endpoint of a fillet when radius and one endpoint is known


I'm creating a fillet between two lines, one is vertical and other is at an angle theta, I need to find the coordinates for the endpoint [one of them is cos(theta)*radius]. How would I go about doing that?

Edited the question with this diagram


Solution

  • (Since I don't have a good way to make diagrams, I'll have to do this in words.)

    Take a circle of radius r, centered on the origin. A line, making an angle θ with the horizontal, is tangent to the circle in the (x<0, y>0) quadrant. The coordinates of the tangent point are (-r sin(θ), r cos(θ)). Drop a verttical from the tangent point, and it will intersect the x-axis at (-r sin(θ), 0).

    The distance from that intersection to the leftmost point of the circle is

    d = -r sin(θ) - (-r) = r(1 - sin(θ))