**Author:** Robert B. Easter

**Abstract:** This paper introduces the Double Conformal Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA that adds geometrical entities for all 3D quadric surfaces and a torus surface. More generally, DCGA has an entity for general cyclide surfaces in 3D, which is a class of quartic surfaces that includes the quadric surfaces and toroid and also their inversions in spheres known as Dupin cyclides. All entities representing various geometric surfaces and points can be transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. Versors provide an algebra of spatial transformations that are different than linear algebra transformations. Entities representing the intersections of geometric surfaces can also be created by wedge products. All entities can be intersected with spheres, planes, lines, and circles. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.

**Download:** http://vixra.org/pdf/1508.0086v5.pdf (PDF)

**Comments:** 49 Pages.

*Source:* Email from R. B. Easter, (dirname_AT_yahoo.com), 9/4/2015, http://vixra.org/pdf/1508.0086v5.pdf

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Why was this placed on vixra.org rather than arxiv?

Revision v6 of the paper was posted on Sept. 20, 2015.

http://vixra.org/pdf/1508.0086v6.pdf

52 pages.

Changes in Revision v6:

– Fixed some errors.

– Added more figures and improved many figures (31 figures).

– Added a new section on parabolic cyclides (cubic surfaces).

– Renamed “general cyclide” as “Darboux cyclide” to match other literature.

– Added a few more references.

– Added an Abstract with Keywords and MSC on page 1.

– Improved the discussions on inversions in spheres and intersections.

– Explained why point e_infinity is on the ellipsoid entity.

More revisions are possible, but not planned. If an error is found, then there could be another revision. Revision v6 could be final, and could be cited by others that choose to do so.

– Robert B. Easter