Exponential/Logarithm

group exp_log

The group of rotations, translations, and screws (combined rotatation and translation) is nonlinear. This means, given say, a rotor \(\mathbf{r}\), the rotor \(\frac{\mathbf{r}}{2}\) does not correspond to half the rotation. Similarly, for a motor \(\mathbf{m}\), the motor \(n \mathbf{m}\) is not \(n\) applications of the motor \(\mathbf{m}\). One way we could achieve this is through exponentiation; for example, the motor \(\mathbf{m}^3\) will perform the screw action of \(\mathbf{m}\) three times. However, repeated multiplication in this fashion lacks both efficiency and numerical stability.

The solution is to take the logarithm of the action which maps the action to a linear space. Using log(A) where A is one of rotor , translator , or motor , we can apply linear scaling to log(A) , and then re-exponentiate the result. Using this technique, exp(n * log(A)) is equivalent to \(\mathbf{A}^n\).

Summary

Members	Descriptions
public line KLN_VEC_CALL log(motor m) noexcept	Takes the principal branch of the logarithm of the motor, returning a bivector. Exponentiation of that bivector without any changes produces this motor again. Scaling that bivector by \(\frac{1}{n}\), re-exponentiating, and taking the result to the \(n\)th power will also produce this motor again. The logarithm presumes that the motor is normalized.
public motor KLN_VEC_CALL exp(line l) noexcept	Exponentiate a line to produce a motor that posesses this line as its axis. This routine will be used most often when this line is produced as the logarithm of an existing rotor, then scaled to subdivide or accelerate the motor's action. The line need not be a simple bivector for the operation to be well-defined.
public ideal_line log(translator t) noexcept	Compute the logarithm of the translator, producing an ideal line axis. In practice, the logarithm of a translator is simply the ideal partition (without the scalar \(1\)).
public translator KLN_VEC_CALL exp(ideal_line il) noexcept	Exponentiate an ideal line to produce a translation.
public branch log(rotor r) noexcept	Returns the principal branch of this rotor's logarithm. Invoking exp on the returned [kln::branch](../../api/undefined#classkln_1_1branch) maps back to this rotor.
public rotor exp(branch b) noexcept	Exponentiate a branch to produce a rotor.
public rotor sqrt(rotor r) noexcept	Compute the square root of the provided rotor \(r\).
public rotor sqrt(branch b) noexcept
public translator sqrt(translator t) noexcept	Compute the square root of the provided translator \(t\).
public motor sqrt(motor m) noexcept	Compute the square root of the provided motor \(m\).

Members

line KLN_VEC_CALL log(motor m) noexcept

Takes the principal branch of the logarithm of the motor, returning a bivector. Exponentiation of that bivector without any changes produces this motor again. Scaling that bivector by \(\frac{1}{n}\), re-exponentiating, and taking the result to the \(n\)th power will also produce this motor again. The logarithm presumes that the motor is normalized.

motor KLN_VEC_CALL exp(line l) noexcept

Exponentiate a line to produce a motor that posesses this line as its axis. This routine will be used most often when this line is produced as the logarithm of an existing rotor, then scaled to subdivide or accelerate the motor's action. The line need not be a simple bivector for the operation to be well-defined.

ideal_line log(translator t) noexcept

Compute the logarithm of the translator, producing an ideal line axis. In practice, the logarithm of a translator is simply the ideal partition (without the scalar \(1\)).

translator KLN_VEC_CALL exp(ideal_line il) noexcept

Exponentiate an ideal line to produce a translation.

The exponential of an ideal line \(a \mathbf{e}_{01} + b\mathbf{e}_{02} + c\mathbf{e}_{03}\) is given as:

\[\exp{\left[a\ee_{01} + b\ee_{02} + c\ee_{03}\right]} = 1 +\ a\ee_{01} + b\ee_{02} + c\ee_{03}\]

branch log(rotor r) noexcept

Returns the principal branch of this rotor's logarithm. Invoking exp on the returned [kln::branch](../../api/undefined#classkln_1_1branch) maps back to this rotor.

Given a rotor \(\cos\alpha + \sin\alpha\left[a\ee_{23} + b\ee_{31} +\ c\ee_{23}\right]\), the log is computed as simply \(\alpha\left[a\ee_{23} + b\ee_{31} + c\ee_{23}\right]\). This map is only well-defined if the rotor is normalized such that \(a^2 + b^2 + c^2 = 1\).

rotor exp(branch b) noexcept

Exponentiate a branch to produce a rotor.

rotor sqrt(rotor r) noexcept

Compute the square root of the provided rotor \(r\).

rotor sqrt(branch b) noexcept

translator sqrt(translator t) noexcept

Compute the square root of the provided translator \(t\).

motor sqrt(motor m) noexcept

Compute the square root of the provided motor \(m\).