Question

I want to calculate the angle between two vectors a and b. Lets assume these are at the origin. This can be done with

theta = arccos(a . b / |a| * |b|)

However arccos gives you the angle in [0, pi], i.e. it will never give you an angle greater than 180 degrees, which is what I want. So how do you find out when the vectors have gone past the 180 degree mark? In 2D I would simply let the sign of the y-component on one of the vectors determine what quadrant the vector is in. But what is the easiest way to do it in 3D?

EDIT: I wanted to keep the question general but here we go. I’m programming this in c and the code I use to get the angle is theta = acos(dot(a, b)/mag(a)*mag(b)) so how would you programmatically determine the orientation?

Answer 1

I came up with the following solution that takes advantage of the direction change of the cross product of the two vectors:

1. Make a vector n = a X b and normalize it. This vector is normal to the plane spanned by a and b.

2. Whenever a new angle is calculated compare it with the old normal. In the comparison, treat the old and the current normals as points and compute the distance between them. If this distance is 2 the normal (i.e. the cross product a X b has flipped).

You might want to have a threshold for the distance as the distance after a flip might be shorter than 2, depending on how the vectors a and b are oriented and how often you update the angle.