Exponential Map Guide

Specification

Location	Author	Maintained by
isl/math/expmap.py	Kenny Erleben	DIKU

The Exponential Map

The purpose of the exomap.py module is to make exponential mapping possible. We used the code examples and theory from the paper of F. Sebastian Grossia to implement this module.

Definition

Let $\Theta = |v|$ , where Theta is in radians and $\hat{\Theta} = v/|v|$ then

$q = e^{v} = [cos(\frac{1}{2}\Theta ), \frac{sin(\frac{1}{2} \Theta )}{\Theta} v]^T$

This module helps parameterize rotations matrices in group SO(3). Rotations with these parameters avoid singularities by converting every rotation to be in the range of 0 and $\pi$ .

Examples

First load the module

    import isl.math.expmap

Example 1

This example shows how to initialize a rotation matrix from v.

        # Rotates with pi/2
        v_rot_const = (np.pi/2) * (1/(np.sqrt(3)))
        v           = vec3.make(v_rot_const,v_rot_const,v_rot_const)
        exp_map     = exp(v)
        to_matrix(exp_map)

Output

array([[ 0.84066355, -0.21922386,  0.49520268],
       [ 0.49520268,  0.6813271 , -0.5390433 ],
       [-0.21922386,  0.69837975,  0.6813271 ]])

Example 2

Finding the inverse of exp_map

    log(exp_map), v

Output

(array([0.90689968, 0.90689968, 0.90689968]),
 array([0.90689968, 0.90689968, 0.90689968]))

Example 3

If you want to rotate with more than pi you need to reparmeterize.

    # Rotate with 1.5 * pi
    v_rot_const = (1.5*np.pi) * (1/(np.sqrt(3)))
    v           = vec3.make(v_rot_const,v_rot_const,v_rot_const)
    v           = reparameterization(v)
    exp_map     = exp(v)
    print(f' Theta_hat: {2*np.pi -  vec3.norm(v)}, Theta: {1.5*np.pi}')

Output

Theta_hat: 4.7123889803846915, Theta: 4.71238898038469

Example 4

Find the derivative of v given an angle omega (See paper section 4.1).

    rot_const_omega     =  np.pi + np.pi / 4
    rot_const_v         =  np.pi + np.pi / 2 
    quad_const          = 1/np.sqrt(3)
    rot_1               = rot_const_omega * quad_const
    rot_2               = rot_const_v * quad_const
    omega     = vec3.make(rot_1,rot_1,rot_1)
    v         = vec3.make(rot_2,rot_2,rot_2)
    dvdt(omega, v)

Output

array([2.26724921, 2.26724921, 2.26724921])

Example 5

Finding the derivative of a rotation matrix with respect to a axis i (See section 3.1 in the paper). Let i = 0 meaning the first axis:

    const1     = np.pi +  np.pi / 4
    quad_const = 1/np.sqrt(3)
    rot        = const1 * quad_const
    v          = vec3.make(rot,rot,rot)
    v          = reparameterization(v)
    dRdv_i(v, 0)

Output

    array([[-0.28556917, -0.35808005, -0.3111484 ],
           [-1.02955486, -0.64055624, -0.23988447],
           [-0.23988447, -0.3111484 , -0.64055624]])