Find the normal angle of the face of a triangle in 3D, given the co-ordinates of its vertices

Question

As you may be able to tell from this screenshot, I am trying to make a physics engine for a platformer I am working on, but I have run into a definite problem: I need to be able to find out the angle of any one of the triangles that you can see make up this mesh, so that I can work out the rotation and therefore angular acceleration of the player on that triangle.

I can use an algorithm that I created to find the locations of all 3 points of any triangle that the player is in contact with, but I don't know how to use those points to work out the rotation of the triangle.

By the rotation, I mean the direction of the normal away from the centre of the face, i.e., the angle at which a person would be leaning if they stood on that surface. Can someone come up with a series of equations that will allow for this problem to be solved?

Answer 1

If you take the cross product of the two vectors:

p1 - p0

and

p2 - p0

where p0, p1 and p2 are three vertices of the triangle, you'll get the normal. A triangle is considered to be pointing towards you if the vertices are ordered clockwise with respect to its outward normal. This is called the left hand rule. Imagine holding your left hand with your fingers curled from p0 to p1, your thumb sticks out in the direction of the face normal:

Answer 2

The cross product is the right answer. Dont forget to normalise the result, and dont forget that if the triangle has zero area, the result is invalid because there is no well defined normal. Basically, if your three vertices are p0, p1 and p2:

vector temp = cross(p1 - p0, p2 - p0);
if (length(temp) < epsilon) then
    Degenerate_triangle_error;
else
    return normalize(temp);

Also, as the other answer says, whether you get the 'up facing' or 'down facing' normal will depend on the ordering of your vertices.

Answer 3

To finish answering your question, once you have the unit normal vector of your triangle you can work out the angle using a dot product.

The dot product of two unit vectors is equal to the cosine of the angle between them, so if you calculate the arccos of the dot product of your unit normal vector and your unit Up vector you will get the slope angle of your triangle (angle away from horizontal).

Also, note that OpenGL conventionally uses a right-handed coordinate system, so if you are using that then your triangle vertices will have a counter-clockwise ordering.

Answer 4

There are 2 normals to the triangle (of course) and the one you get from standard algorithms depends on the order of ther vertices. Quoting wiki

"For a polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon."

But the direction of the normal depends on the order of points chosen, you can calculate it and decide using some other heuristics whether the reverse vector is the normal you are interested in.