Alyssa Novelia

On geodesics of the rotation group SO(3)

Geodesics on SO(3) are characterized by constant angular velocity motions and as great circles on a three-sphere. The former interpretation is widely used in optometry and the latter features in the interpolation of rotations in computer graphics. The simplicity of these two disparate interpretations belies the complexity of the corresponding rotations. Using a quaternion representation for a rotation, we present a simple proof of the equivalence of the aforementioned characterizations and a straightforward method to establish features of the corresponding rotations.

Bishop Frames and Reference Frames Along the Discretized Curve

Two of the three frames associated with an edge of a discretized curve are discussed in this chapter: a Bishop frame and a reference frame. Both of these frames are continually updated as the rod deforms and a pair of rotation operators are used to perform these updates. The operators are known as a space-parallel transport operator and a time-parallel transport operator, respectively. These operators combined with a rotation about the tangent vector to an edge will be used later to define the torsion of the discrete elastic rod. The chapter concludes with a simple example of a discrete rod with two edges. This example is used to illustrate the transport operators and the concept of reference twist.

Equations of Motion and Energetic Considerations

The purpose of this final chapter is to illuminate how the kinematic results presented in the earlier chapters are used to formulate the governing equations for the discrete elastic rod. In particular, representations of the kinetic and potential energies for the discrete elastic rod formulation are discussed. The gradients of the elastic energies associated with bending, stretching, and torsion are used to establish expressions for the internal forces in the rod. Assigned forces, including non-conservative forces, can also be incorporated into the discrete elastic rod formulation. To this end, we include examples, such as a force couple and an applied non-conservative force, to illuminate how applied forces and applied moments can be accommodated.

Spherical Excess and Reference Twist

The method by which a component of the rotation of the cross-section is computed in the discrete elastic rod formulation is exceptional and exploits a phenomenon in differential geometry known as a holonomy. In this chapter, relevant background from differential geometry and spherical geometry are presented so the reader can understand how the reference twist in the rod can be related to a solid angle enclosed by the trace of a unit tangent vector on a sphere. A derivation of the expression for the variation of the reference twist as a function of the variation of the tangent vector is also discussed in detail. Some of the developments in this chapter are illuminated with the help of a simple example of a rod with three vertices that was also considered in a previous chapter.

Material Frames and Measures of Twists

The material frames associated with an edge of a discretized curve is discussed in this chapter. For the discrete curve, a pair of material vectors play the role of directors in Kirchhoff’s rod theory. The motion of these vectors is calibrated using either a Bishop frame or a reference frame that is continually being updated. The pair of parallel transport operators defined in the previous chapter combined with a rotation about the tangent vector to an edge are used to define the motions the material vectors. Of particular interest is the difference in an angle of twist between two adjacent edges. This angle will be identified with the torsion of the rod-like body that the discrete elastic rod is modeling. In this chapter the bending strains will also be defined. The developments in this chapter and, in particular, the twist of the reference frame are illustrated using the example of a rod uncoiling under its own weight.

The Discretized Curve: Vertices, Edges, and Curvature

Relevant background material on approximating a continuous space curve using a discrete set of lines connected at vertices is assembled in this chapter. The formulation uses concepts from the nascent field of discrete differential geometry. The resulting discretized curve is a central component of the discrete elastic rod formulation. In particular, the discrete curvature vector associated with a vertex is used as a measure of bending strains and the length of the edges are used to account for stretching. For the purposes of illustration, the discretization of a helical space curve is discussed in detail.

Variations, Gradients, and Hessians

In the discrete elastic rod formulation, expressions for the variations, gradients, and Hessians of kinematic variables induced by changes to the vertices are required. The present chapter provides the background and intermediate computations that are needed to establish the desired representations for these gradients and Hessians. Some of resulting expressions are employed to compute the elastic restoring forces in a discrete elastic rod in a later chapter.

Kirchhoff’s Theory of an Elastic Rod

By way of background, a rapid review of Kirchhoff’s theory of an inextensible, unshearable elastic rod is discussed in this chapter. The formulation of the theory incorporates a pair of deformable vectors (or directors) associated with each point of a flexible material curve. In addition, the Frenet and Bishop framings of the material curve are introduced. The latter framing does not feature in classic textbooks on rod theories and is central to understanding the discrete elastic rod formulation.