Articulating Space: Geometric Algebra for Parametric Design - Symmetry, Kinematics, and Curvature
@inproceedings{Colapinto2015ArticulatingSG,
title={Articulating Space: Geometric Algebra for Parametric Design - Symmetry, Kinematics, and Curvature},
author={Pablo Colapinto},
year={2015}
}Author(s): Colapinto, Pablo | Advisor(s): Peljhan, Marko | Abstract: To advance the use of geometric algebra in practice, we develop computational methods for parameterizing spatial structures with the conformal model. Three discrete parameterizations – symmetric, kinematic, and curvilinear – are employed to generate space groups, linkage mechanisms, and rationalized surfaces. In the process we illustrate techniques that directly benefit from the underlying mathematics, and demonstrate how they…
Figures and Tables from this paper
table 1.1 
figure 1.1 
figure 1.2 
table 2.1 
figure 2.10 
figure 2.11 
figure 2.12 
figure 2.13 
figure 2.14 
figure 2.15 
figure 2.16 
figure 2.17 
figure 2.18 
figure 2.19 
table 2.2 
table 2.3 
figure 2.4 
table 2.4 
figure 2.5 
table 2.5 
figure 2.6 
table 2.6 
figure 2.7 
figure 2.8 
figure 2.9 
figure 3.1 
table 3.1 
figure 3.10 
table 3.2 
figure 3.2 
table 3.3 
figure 3.3 
table 3.4 
figure 3.4 
table 3.5 
figure 3.5 
figure 3.6 
figure 3.7 
figure 3.8 
figure 3.9 
figure 4.1 
figure 4.10 
figure 4.11 
figure 4.12 
figure 4.13 
figure 4.14 
figure 4.15 
figure 4.16 
figure 4.2 
figure 4.3 
figure 4.4 
figure 4.5 
figure 4.6 
figure 4.7 
figure 4.8 
figure 4.9 
figure 5.1 
figure 5.10 
figure 5.11 
figure 5.12 
figure 5.13 
figure 5.14 
figure 5.2 
figure 5.3 
figure 5.4 
figure 5.5 
figure 5.6 
figure 5.7 
figure 5.8 
figure 5.9 
figure 6.1 
figure 6.10 
figure 6.11 
figure 6.12 
figure 6.13 
figure 6.14 
figure 6.2 
figure 6.3 
figure 6.4 
figure 6.5 
figure 6.6 
figure 6.7 
figure 6.8 
figure 6.9 
table 7.1 
table A.1
8 Citations
The Construction of 3D Conformal Motions
- MathematicsMath. Comput. Sci.
- 2016
This paper exploits the fact that any 3D conformal motion is governed by two well-chosen point pairs: the motion is composed of (or decomposed into) two specific orthogonal circular motions in planes determined by those point pairs.
The Construction of 3 D Conformal Motions
- Mathematics
- 2016
This paper exposes a very geometrical yet directly computational way of working with conformal motions in 3D. With the increased relevance of conformal structures in architectural geometry, and their…
An All-in-One Geometric Algorithm for Cutting, Tearing, and Drilling Deformable Models
- Computer ScienceAdvances in Applied Clifford Algebras
- 2021
This work demonstrates the merits of multivector usage with a novel, integrated rigged character simulation framework based on CGA, and presents three novel algorithms which enables cutting, tearing and drilling of the input rigged model, where the output model can be further re-deformed in interactive frame rates.
Conformal Geometric Algebra
- Mathematics
- 2018
This chapter is devoted to a presentation of conformal geometric algebra (CGA) targeted to the sort of applications dealt with in chapters 4 and 5, and is designed so that it can encode all conformal transformations of E3 in spinorial form.
Composing Surfaces with Conformal Rotors
- Mathematics
- 2017
Conformal immersions are harmonic and numerically stable surfaces whose tangents scale isometrically, providing many elegant geometric properties of use in design. This paper maps these forms within…
An Extended Implementation Framework for Geometric Algebra Operations on Systems of Coordinate Frames of Arbitrary Signature
- Computer Science
- 2018
If properly implemented, the extended framework can significantly reduce the memory requirements for implementing Geometric Algebras with larger dimensions, especially for systems based on the symbolic processing of multivector scalar coefficients.
Massive Geometric Algebra: Visions for C++ Implementations of Geometric Algebra to Scale into the Big Data Era
- Computer Science
- 2017
This work discusses the “wish list” that an optimal C++ implementation should provide and finds that to cover the various constraints an hybrid approach benefiting from multiple programming paradigms, ranging from generic to object-oriented programming, will be needed.
References
SHOWING 1-10 OF 149 REFERENCES
The Construction of 3D Conformal Motions
- MathematicsMath. Comput. Sci.
- 2016
This paper exploits the fact that any 3D conformal motion is governed by two well-chosen point pairs: the motion is composed of (or decomposed into) two specific orthogonal circular motions in planes determined by those point pairs.
A covariant approach to geometry using geometric algebra
- Mathematics, Computer Science
- 2004
Using the mathematical framework of conformal geometric algebra – a 5-dimensional representation of 3-dimensional space – is shown to provide an elegant covariant approach to geometry, thus enabling us to deal simply with the projective and non-Euclidean cases.
3D Oriented Projective Geometry Through Versors of R
- Mathematics
- 2015
It is possible to set up a correspondence between 3D space and R , interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become…
Applications of Conformal Geometric Algebra in Mesh Deformation
- Computer Science2013 XXVI Conference on Graphics, Patterns and Images
- 2013
It is illustrated the suitability of Conformal Geometric Algebra for representing deformable mesh models by showing how these mesh representations conduct to a fast and easy formulation for the Spline-aligned deformation and a formulation for linear surface deformation based on generalized barycentric coordinates.
Tutorial Appendix: Structure Preserving Representation of Euclidean Motions Through Conformal Geometric Algebra
- MathematicsGuide to Geometric Algebra in Practice
- 2011
Using conformal geometric algebra, Euclidean motions in n-D are represented as orthogonal transformations of a representational space of two extra dimensions, and a well-chosen metric, which becomes a high-level programming language which naturally integrates quantitative computation with the automatic administration of geometric data structures.
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.
Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups
- Mathematics
- 2010
A new interactive software tool is described, that visualizes 3D space group symmetries and relates the choice of symmetry vectors and the origin of cells in the geometric algebra description and its implementation in the SGV to the conventional crystal cell choices in the International Tables of Crystallography.
Geometric modeling with conical meshes and developable surfaces
- Computer ScienceSIGGRAPH '06
- 2006
This work shows how to optimize a quad mesh such that its faces become planar, or the mesh becomes even conical, making subdivision attractive for architecture design and providing an elegant way of modeling developable surfaces.
Motor algebra approach for computing the kinematics of robot manipulators
- MathematicsJ. Field Robotics
- 2000
The flexible method presented here is new, and it widens the current standard point or line representation-based approaches for the treatment of problems related to robot manipulators.
Generalization of rigid foldable quadrilateral mesh origami
- Biology
- 2009
This work generalizes the geometric condition for enabling rigid motion in general quadrilateral mesh origami without the trivial repeating symmetry and derives the identity of functions from the formula for degree-4 single-vertex origami to ensure the existence of a finite motion.