# Modeling 3D Geometry in the Clifford Algebra R(4, 4)

@article{Du2017Modeling3G, title={Modeling 3D Geometry in the Clifford Algebra R(4, 4)}, author={Juan Du and Ron Goldman and Stephen Mann}, journal={Advances in Applied Clifford Algebras}, year={2017}, volume={27}, pages={3039-3062} }

We flesh out the affine geometry of $${{\mathbb {R}}^3}$$R3 represented inside the Clifford algebra $${\mathbb {R}}(4,4)$$R(4,4). We show how lines and planes as well as conic sections and quadric surfaces are represented in this model. We also investigate duality between different representations of points, lines, and planes, and we show how to represent intersections between these geometric elements. Formulas for lengths, areas, and volumes are also provided.

## 18 Citations

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## References

SHOWING 1-10 OF 21 REFERENCES

### R(4, 4) As a Computational Framework for 3-Dimensional Computer Graphics

- Mathematics
- 2015

We investigate the efficacy of the Clifford algebra R(4, 4) as a computational framework for contemporary 3-dimensional computer graphics. We give explicit rotors in R(4, 4) for all the standard…

### Conic and Cyclidic Sections in Double Conformal Geometric Algebra G_{8,2}

- Mathematics
- 2016

The G_{8,2} Geometric Algebra, also called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar…

### 3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$R3,3

- Mathematics
- 2015

It is possible to set up a correspondence between 3D space and $${\mathbb{R}^{3,3}}$$R3,3, interpretable as the space of oriented lines (and screws), such that special projective collineations of the…

### Geometric Algebras for Euclidean Geometry

- Mathematics
- 2014

AbstractThe discussion of how to apply geometric algebra to euclidean $${n}$$n-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a…

### Line Geometry in Terms of the Null Geometric Algebra over ℝ3, 3, and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms

- MathematicsGuide to Geometric Algebra in Practice
- 2011

In this chapter, the classical line geometry is modeled in ℝ3,3, where lines are represented by null vectors, and points and planes by null 3-blades. The group of 3D special projective…

### On the Homogeneous Model of Euclidean Geometry

- MathematicsGuide to Geometric Algebra in Practice
- 2011

An elementary account of Euclidean kinematics and rigid body motion within this framework is concluded, focusing on the cases of n=2 and n=3 in detail, enumerating the geometric products between k-blades and m-blade.

### Geometric algebra for computer graphics

- Mathematics
- 2008

John Vince tackles complex numbers and quaternions; the nature of wedge product and geometric product; reflections and rotations; and how to implement lines, planes, volumes and intersections in this accessible and very readable introduction to geometric algebra.

### Double Conformal Geometric Algebra for Quadrics and Darboux Cyclides

- MathematicsCGI
- 2016

We introduce Clifford geometric algebra based multivector modeling of quartic and general quadric surfaces, Darboux cyclides, Dupin cyclies, tori and pairs of standard CGA objects. A computer…

### New algebraic tools for classical geometry

- Mathematics
- 2001

Classical geometry has emerged from efforts to codify perception of space and motion. With roots in ancient times, the great flowering of classical geometry was in the 19th century, when Euclidean,…

### Geometric Algebra with Applications in Engineering

- MathematicsGeometry and Computing
- 2009

Students will profit from the detailed introduction to geometric algebra, while the text is supported by the author's visualization software, CLUCalc, freely available online, and a website that includes downloadable exercises, slides and tutorials.