Angle between 2 3D straight lines

Hi everybody,

i have two 3D straight lines as follow:

the line "S1" passing through the points A=[3,7,9] and B=[4,5,8]

and the line "S2" passing through the points A=[3,7,9] and C=[4,5,0].

How can i calculate the angle between S1 and S2.

Thank you very much!

Riccardo

Answer 1

[Jan]

A = [3,7,9];
B = [4,5,8];
C = [4,5,0];

S1 = B - A;
S2 = C - A;

Theta = atan2(norm(cross(S1, S2)), dot(S1, S2));

W. Kahan suggested in his paper "Mindless.pdf":

2 * atan(norm(S1 * norm(S2) - norm(S1) * S2) / ...
         norm(S1 * norm(S2) + norm(S1) * S2))

Both methods are more stable than appraoches using ACOS(DOT) or ASIN(CROSS).

[Riccardo Rossi]

Thank you for your answer. Result is in degree or radians?

[Bruno Luong]

This Kahan's formula seems to be clever.

[Anthony Dave]

Thanks, Jan. The result angle would change if I change the vector's direction. For example, S2 = A - C. And the two angles I got are mutual supplementary angles. But if it is a polygon with several lines, how can I accurately get its internal angles?

[Jan]

@Anthony Dave: The unique definition of a polygon requires the defined order of its vertices. If you e.g. swap two neighboring vertices of a square, the angles are 45° instead of 90°.

This means, that you have to stay at the same order of vertices and changing the view by something like "S2 = A - C" is not allowed. But even then, the included angle is replied and this is < 180 in every case. In a polygon you can have angles > 180 between oriented edges. Then you have to include the information about the polygon's normal N also:

S1xS2 = cross(S1, S2);
Theta = atan2(norm(S1xS2), dot(S1, S2)) * sign(dot(S1xS2, N));