Given three coordinate points, how do you detect when the angle between them crosses 180 degrees?

2
votes

This seems like it should be really simple but I'm having trouble with it. Basically, I have three points that keep changing (lets call them p1, p2, and p3). Also, let's define p2 as the vertex point.

Essentially, what I need to do is calculate the angle between the three points. A good example would be if the three angles form a 179 degree angle, then the points change to form a 181 degree angle. So what I really need is a good method for determining if an angle is greater than 180 degrees. I tried using the law of cosines, but it did not give me a good answer because when the points form a 181 degree angle, it simply interprets it as a 179 degree angle in a different direction. Also, I am doing this in Python, if that helps. Thanks!

2
pythonmathgeometrytrigonometryangle
Do you have a consistent angle measurement scheme? Like are all angles measured ccw? – Henry Prickett-Morgan
In this case you must know that the angle is on the clockwise or counterclockwise side of the first segment, correct? Otherwise interpreting 181 as 179 is absolutely correct. – Mark Ransom
All angles are measured consistently. It doesn't really matter whether its counterclockwise or clockwise, all that matters is when the angle crosses the 180 degree boundary. – user3047641

2 Answers

6
votes

What you are trying to decide is whether (p3-p2) is a left or right turn comparing to (p2-p1). This is actually a core part of Graham Scan which is used for computing convex hulls (https://en.wikipedia.org/wiki/Graham_scan). Quoting Wikipedia with slight edits:

...determining whether three points constitute a "left turn" or a "right turn" does not require computing the actual angle between the two line segments, and can actually be achieved with simple arithmetic only. For three points P1=(x1, y1), P2=(x2, y2), and P3=(x3, y3), simply compute the z-coordinate of the cross product of the two vectors (p2-p1) and (p3-p1), which is given by the expression (x2 - x1) * (y3 - y1) - (y2 - y1) * (x3 - x1). If the result is 0, the points are collinear; if it is positive, the three points constitute a "left turn" or counter-clockwise orientation, otherwise a "right turn" or clockwise orientation (for counter-clockwise numbered points).

0
votes

To get signed angle in the full range, use atan2 function with dot and cross product of vectors p2p1 and p2p3

Angle(in radians) = atan2(cross(p2p1,p2p3), dot(p2p1,p2p3))