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algorithmgeometryrectangles

Assistance on creating algorithm for creating box around arc

I am looking for some assistance on creating an algorithm that is able to calculate a bounding box for an arc. Typically, this would be a simple problem. However, I am required to have one of the edges of the box to be the line that connects the starting point and the ending point of the arc.

I am having difficulty in developing an algorithm that is able to compute the additional two points of the rectangle.

Here is a picture of what I am attempting to calculate:

arc box

The two purple dots, I need to develop an algorithm that will determine these two locations. The green dots are the known points that can be inputs. Additionally, I do know the radius and the center of the arc.

I would need the algorithm to handle the different cases for when line AB is vertical, has a + slope, and has a - slope.

I have been considering a few directions. For example, I know that the line through point E is parallel to line AB. Since it is parallel, that means they will have the same slopes and the line from point A to the purple point is perpendicular to this line. I can then consider the intersection point of the line through E and this perpendicular line.

This method seems messy because then I would need to consider the cases for when the slope of line AB is infinite and 0. I am wondering if there is some algorithm that could account for that automatically (or not even consider the slope at all for line AB)

Solution

  • enter image description here

    You claim to know points A, B, C, D, E, and that the amplitude of the sustaining angle of the circular arc does not exceed 180° (semi circle).
    Let P1, and P2, the two points that complement the bounding box - (in the drawing, P1 is the purple point above A, and P2 the one above B).

    Using vectors:

    E-C = vector perpendicular to segment AB', i/e subtract C from E; its magnitude is equal to the distance EC.
    P1 = A + (E-C)
    P2 = B + (E-C)

    Bounding Box = Rectangle A, P1, P2, B

    If you need a closer fit, you can replace vector (E-C) with vector (D-C) to place the bounding segment P1P2 tangent to D