R – Signed Angle in 3D Vectors

3dmathopenglvector

This is a purely math question, I believe.

I have a camera object in an arbitrary coordinate system. I have the direction vector of the camera, and I have a vector that points in the direction of north along the surface of the sphere.

I wish to calculate the angle of the camera in regards to the north vector. Is there a simple way to calculate the final sign of the angle? I'm aware than angle = acos(dir dot north), but when I implement it, the angle is clamped to [0,180] degrees. Is there a way to figure out whether the camera is painting eastward or not?

Best Answer

So, N is the north vector, F is the forward vector, V is the vector from the middle of sphere out to the current location of the camera.

N cross V should get you a vector that points eastward (E). Then, I think you just want to project (just dot, since we only care about the sign) F onto E and check the sign. Positive means it's pointing east-ish, zero means it's pointing north or south, negative means it's pointing west-ish.

Depending on what exact question you're asking of the data, you can fiddle with it, but knowing how to grab a vector that means 'east from where I am' should help.

Does this sound right? I'mma bit rusty with this stuff.

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I'm assuming you're trying to compute a vector lying in the plane of the path, not perpendicular to it (since you've already got one - namely the vector from the origin to your point).

You first need to compute vectors lying in that plane that point due north and due east. To do this, let's call P your point, O the origin, and N = (0, 0, R) is the point at the top of your sphere. Then

e = cross(N - P, P - O)

is a vector that points due east, and is tangent to the sphere because it's perpendicular to P - O, a radius of the sphere.

For similar reasons

n = cross(e, P - O)

will point due north, and will be tangent to the sphere.

Now normalize n and e, and you've got an orthonormal basis for the tangent space at your point. To find a vector in a direction theta (say, counterclockwise from the positive east axis, to simplify the math), just take a little of e and a little of n:

v = cos(theta) * e + sin(theta) * n

Best Answer

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