Doing Euclidean Plane Geometry Using Projective Geometric Algebra

Conformal Geometry, Euclidean Space and Geometric Algebra

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach — the Euclidean distance between points is not handled in a s...

Three-Dimensional Projective Geometry with Geometric Algebra

The line geometric model of 3-D projective geometry has the nice property that the Lie algebra sl(4) of 3-D projective transformations is isomorphic to the bivector algebra of Cl(3, 3), and line geometry is closely related to the classical screw theory for 3-D rigid-body motions. The canonical homomorphism from SL(4) to Spin(3, 3) is not satisfying because it is not surjective, and the projecti...

Projective Geometric Algebra

Ground Plane Obstacle Detection using Projective Geometry

|The problem addressed in this paper is obstacle detection in the context of mobile robot navigation using visual information. The goal is achieved by analyzing successive pairs of time varying images acquired with the TV camera mounted on the moving robot. Assuming the robot is moving on a at ground, any obstacle is identi ed by any cluster of points not coplanar with the largest number of poi...

Paracatadioptric Geometry using Conformal Geometric Algebra

Computer vision provides non-contact measurements of the world, extending the robot ability to operate in circumstances and environments which can not be accurately controlled. The use of visual observations to control the motions of robots has been extensively studied, this approach is referred in literature as visual servoing. Conventional cameras suffer from a limited field of view. One effe...