Signed angle between two 3D vectors with same origin within the same plane

3dgeometrymathvector

What I need is a signed angle of rotation between two vectors Va and Vb lying within the same 3D plane and having the same origin knowing that:

The plane contatining both vectors is an arbitrary and is not parallel to XY or any other of cardinal planes
Vn – is a plane normal
Both vectors along with the normal have the same origin O = { 0, 0, 0 }
Va – is a reference for measuring the left handed rotation at Vn

The angle should be measured in such a way so if the plane would be XY plane the Va would stand for X axis unit vector of it.

I guess I should perform a kind of coordinate space transformation by using the Va as the X-axis and the cross product of Vb and Vn as the Y-axis and then just using some 2d method like with atan2() or something. Any ideas? Formulas?

Best Answer

Use cross product of the two vectors to get the normal of the plane formed by the two vectors. Then check the dotproduct between that and the original plane normal to see if they are facing the same direction.

angle = acos(dotProduct(Va.normalize(), Vb.normalize()));
cross = crossProduct(Va, Vb);
if (dotProduct(Vn, cross) < 0) { // Or > 0
  angle = -angle;
}

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Signed angle between two vectors without a reference plane

The relevant mathematical formulas:

  dot_product(a,b) == length(a) * length(b) * cos(angle)
  length(cross_product(a,b)) == length(a) * length(b) * sin(angle)

For a robust angle between 3-D vectors, your actual computation should be:

  s = length(cross_product(a,b))
  c = dot_product(a,b)
  angle = atan2(s, c)

If you use acos(c) alone, you will get severe precision problems for cases when the angle is small. Computing s and using atan2() gives you a robust result for all possible cases.

Since s is always nonnegative, the resulting angle will range from 0 to pi. There will always be an equivalent negative angle (angle - 2*pi), but there is no geometric reason to prefer it.

The X angle between two 3D vectors

atan2(crossproduct.length,scalarproduct)

The reason for using atan2 instead of arccos or arcsin is accuracy. arccos behaves very badly close to 0 degrees. Small computation errors in argument will lead to disproportionally big errors in result. arcsin has same problem close to 90 degrees.

Best Answer

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Signed angle between two vectors without a reference plane

The X angle between two 3D vectors

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