3.1. Rodrigues’ Rotation Formula

Suppose, a is a unit vector denoting the axis of rotation and θ is the angle of rotation about this axis. Then, the rotation tensor is given by R(a,θ)=Icosθ+asinθ+a⊗ a(1−cosθ).

Rodrigues' rotation formula provides a way to represent rotation in three-dimensional space. Here, 'a' is a unit vector that indicates the axis around which the rotation occurs, while 'θ' denotes the angle of rotation. The expression defines a rotation tensor (R) which can be constructed using the identity matrix (I) and trigonometric functions of the rotation angle. This formula is powerful because it captures the three-dimensional nature of rotations, making it applicable even for large angles.

Imagine you're spinning a pencil around a finger. The pencil represents the axis of rotation (the unit vector 'a'), and how much you spin it represents the angle 'θ'. The Rodrigues' rotation formula is like a set of instructions on exactly how to calculate the new position of the pencil after you spin it.

Here, a denotes the skew symmetric tensor corresponding to a. Stated differently, the vector a is the axial vector of the skew symmetric tensor. This is a general formula for rotation which is valid even when the angle of rotation θ is large. Let’s consider the case where the angle of rotation (θ) is very small. We can then approximate the trigonometric functions using their Taylor’s expansion.

For very small angles, the formulas for cos(θ) and sin(θ) can be approximated by using their Taylor series expansions. For example, cos(θ) can be approximated to 1, and sin(θ) can be approximated to θ. Substituting these approximations into the Rodrigues’ formula simplifies the expression, allowing us to represent an infinitesimal rotation with less complexity. This means that for small rotations, we can think of the rotation as nearly identical to just adding a little shift to the current position.

Think about turning the steering wheel of a car a tiny bit. If you turn it just a little bit, the car moves slightly in the direction you're steering. In this small-angle scenario, the complex operations of steering can be simplified to just moving the wheel a bit, similar to how we can simplify calculations of rotation for small angles using Rodrigues’ formula.

Thus, an arbitrary small/infinitesimal rotation can be represented using the above formula which can now be compared with the expression I + W in (22). This proves that I + W indeed represents local infinitesimal rotation.

The reduction of Rodrigues' formula for small angles leads to a simpler expression I + θa that accurately describes the rotation. Here, 'I' represents the identity, and 'W' represents the skew-symmetric part related to the small rotation. The results imply that for small rotations, the combination of the identity matrix and the small rotation term 'W' accurately captures the nature of the rotation without the need for complex calculations. This relationship affirms that the approach using 'I + W' is valid for understanding local rotations in material deformations.

Consider a door that you're slightly pushing open. The position of the door just slightly shifts when you push it, which is similar to how we use a simple formula to describe the rotation of the door. Even if you pushed it just a little, you can categorize that movement easily using our simplified rotation formula.

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