# Applications of Geometric Algebra in Mathematical Engineering

## Calculating the rotor between conformal objects

• This chapter introduces the concept of mirror objects, projectors and finding the rotor that takes one object to another
• Here is the first time we give an expression for the mirror object in terms of the addition of two object blades

## Direct linear interpolation of geometric objects in conformal geometric algebra

• Here we explore in detail the properties of the blade reprojection and linear interpolant objects
• This is the first time we touch on screw theory, although in a very limited way, in the context of the interpolation of lines
• Finally we look at some basic examples of applications

## Exploring Novel Surface Representations via an Experimental Ray-Tracer in CGA

• In this chapter we describe higher order interpolations between objects and how surfaces made of these objects would interact with rays of light
• We then build a basic ray-tracer that can image a range of different objects and show how we can do texture mapping for these objects

## REFORM: Rotor Estimation From Object Resampling and Matching

• In this chapter we make use of the rotor between objects that we described in Chapter 2 to construct cost functions between objects that we can use as an optimisation target for finding unknown transformations between noisy objects in an ICP+Ransac like scheme
• We then also leverage the rotor between objects to design fast heuristic algorithms to solve the same problem

## Screw Theory in Geometric Algebra for Constrained Rigid Body Dynamics

• This chapter is about statics, dynamics, and screw theory
• First we describe forces and moments as lines and compare and contrast this force and moment representation with other examples from the literature
• Then we work through how the operators of algebraic Screw Theory embed into GA
• We then do dynamics, showing how the inertia tensor maps into CGA and PGA and how to represent non-axis-aligned inertia tensors
• We show how to impose constraints on our dynamics via traditional virtual power techniques and then via novel multivector pinning
• Finally we show how integration could be done in the bivector domain with a range of lie algebra mappings

## The Kinematics of Multi-body Systems in Geometric Algebra

• This chapter is about multi-body systems: joints, articulated robots etc.
• We describe how to use the richness of CGA to model the various kinematic joints commonly used in robotic mechanisms. Through a mechanism of shared geometry and invariant bilinear operations.
• We show how to form a collection of joint constraints into a matrix and use this matrix to solve articulated body problems in robotics
• We compare and contrast this screw theory based constraint matrix approach vs a direct calculation for the forward and inverse kinematics of the delta robot

## Conclusions

• We added blades together and it was, in general, a really good idea!
• We showed how to use interpolated objects in a range of applications in graphics, vision and robotics
• We described how Screw Theory embeds into CGA and PGA
• We showed how to do statics, kinematics and constrained dynamics in CGA and PGA
• We derived the kinematic equations for several mappings from se3 to SE3
• We showed how to model the kinematics of jointed robots in CGA and PGA using a screw theory inspired method