# Module: tfg.geometry.transformation.dual_quaternion

This module implements TensorFlow dual quaternion utility functions.

A dual quaternion is an extension of a quaternion with the real and dual parts and written as

$$q = q_r + epsilon q_d$$

, where

$$epsilon$$

is the dual number with the property

$$e^2 = 0$$

. It can thus be represented as two quaternions, and thus stored as 8 numbers. We define the operations in terms of the two quaternions

$$q_r$$

and

$$q_d$$

, which are stored as 8-dimensional tensor.

Dual quaternions are extensions of quaternions to represent rigid transformations (rotations and translations). They are in particular important for deforming geometries as linear blending is a very close approximation of closest path blending, which is not the case for any other representation.

$$|q_r| = 1$$

.

## Functions

conjugate(...): Computes the conjugate of a dual quaternion.

from_rotation_translation(...): Converts a rotation matrix and translation vector to a dual quaternion.

inverse(...): Computes the inverse of a dual quaternion.

is_normalized(...): Determines if a dual quaternion is normalized or not.

multiply(...): Multiplies two dual quaternions.

norm(...): Computes the norm, which is in general a dual number.

to_rotation_translation(...): Converts a dual quaternion into a quaternion for rotation and translation.

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