getting bone base and tip positions from a transform matrix?

Question

I need this for a Blender3d script, but you don't really need to know Blender to answer this.

I need to get bone head and tip positions from a transform matrix read from a file. The position of base is the location part of the matrix, length of the bone (distance from base to tip) is the scale, position of the tip is calculated from the scale (distance from bone base) and rotation part of the matrix.

So how to calculate these?

bone.base([x,y,z]) # x,y,z - floats
bone.tip([x,y,z])

Answer 1

You can find the base by multiplying the origin vector (0, 0, 0) by the transform matrix.

The tip question is slightly more in-depth: it really depends on what the canonical direction is that all the bones are defined from - in other words, where does the bone originally rotate from?

One way or another, to get the tip position you multiply a unit vector (for example, in Z this would be (0, 0, 1)) by the transformation matrix.

Bear in mind as well that your bones are probably hierarchical and the position will be relative to its parent joint - you will need to multiply one particular joint's matrix by its parent (and the parent joint by its parent, and so on) to get a global, world position.

Answer 2

If you want to preserve the original transform so that Blender's bones recreate the same transform, you will also need to calculate the bone roll.

Blender has a function to calculate the bone head, tail and roll from a transform matrix in C code. I haven't found a convenient way to access this functionality from the Python API, but fortunately the functions are short and can easily be translated into Python.

Below is a direct translation of the functions you need, for Blender 2.62 or later:

def vec_roll_to_mat3(vec, roll):
    target = Vector((0,1,0))
    nor = vec.normalized()
    axis = target.cross(nor)
    if axis.dot(axis) > 0.000001:
        axis.normalize()
        theta = target.angle(nor)
        bMatrix = Matrix.Rotation(theta, 3, axis)
    else:
        updown = 1 if target.dot(nor) > 0 else -1
        bMatrix = Matrix.Scale(updown, 3)
    rMatrix = Matrix.Rotation(roll, 3, nor)
    mat = rMatrix * bMatrix
    return mat

def mat3_to_vec_roll(mat):
    vec = mat.col[1]
    vecmat = vec_roll_to_mat3(mat.col[1], 0)
    vecmatinv = vecmat.inverted()
    rollmat = vecmatinv * mat
    roll = math.atan2(rollmat[0][2], rollmat[2][2])
    return vec, roll

pos = transform.to_translation()
axis, roll = mat3_to_vec_roll(transform.to_3x3())

bone.head = pos
bone.tail = pos + axis
bone.roll = roll