Typically we do not add objects in conformal geometric algebra (CGA), rather we apply operations that preserve grade, usually via rotors, such as rotation, translation, dilation, or via reflection and inversion. However, here we show that direct linear interpolation of conformal geometric objects can be both intuitive and of practical use. We present a method that generates useful interpolations of point pairs, lines, circles, planes and spheres and describe algorithms and proofs of interest for computer vision applications that use this direct averaging of geometric objects. * Corresponding author. 1 There are a range of different notations in use in the literature to define the basis of CGA: e is sometimes referred to in other works as e + ,ē as e − , n∞ as e∞ and an eo is sometimes defined which is equal to −n 0 .

The Delta robot is one of the most popular parallel robots in industrial use today. In this paper we analyse the forward and inverse kinematics of the robot from a geometric perspective using Conformal Geometric Algebra. We calculate explicit formulae for all joints in both the forward and inverse kinematic problems as well as explicit forward and inverse Jacobians to allow for velocity and force control. Finally we verify the kinematics in Python and simulate a physical model in the Unity3D game engine to act as a test-bed for future development of control algorithms.

In this paper we will address the problem of recovering covariant transformations between objects-specifically; lines, planes, circles, spheres and point pairs. Using the covariant language of conformal geometric algebra (CGA), we will derive such transformations in a very simple manner. In CGA, rotations, translations, dilations and inversions can be written as a single rotor, which is itself an element of the algebra. We will show that the rotor which takes a line to a line (or plane to a plane etc) can easily be formed and we will investigate the nature of the rotors formed in this way. If we can recover the rotor between one object and another of the same type, a useable metric which tells us how close one line (plane etc) is to another, can be a function of how close this rotor is to the identity. Using these ideas, we find that we can define metrics for a number of common problems, specifically recovering the transformation between sets of noisy objects.

In this paper we tackle the problem of constrained rigid body dynamics in the Conformal and Projective Geometric Algebras (CGA, PGA). First we construct a screw-theory based formulation of dynamics in CGA and note the equivalence between this and the PGA dynamics presented by Gunn in [1]. After verifying the formulation via simulation, we move on to the challenge of adding constraints. First we apply the standard mechanical engineering technique of virtual power to the constraint problem in our Geometric Algebra (GA) framework. We then discuss a novel technique for 'pinning' dynamic rigid bodies to geometric primitives, a technique that relies on the invariance of certain multivectors and functions of multivectors to specific rotor transformations.

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