A Beginners Guide to Dual-Quaternions: What They Are, How They Work, and How to Use Them for 3D Character Hierarchies

Abstract

In this paper, we give a beginners guide to the practicality of using dual-quaternions to represent the rotations and translations in character-based hierarchies. Quaternions have proven themselves in many fields of science and computing as providing an unambiguous, un-cumbersome, computationally efficient method of representing rotational information. We hope after reading this paper the reader will take a similar view on dual-quaternions. We explain how dual number theory can extend quaternions to dual-quaternions and how we can use them to represent rigid transforms (i.e., translations and rotations). Through a set of examples, we demonstrate exactly how dual-quaternions relate rotations and translations and compare them with traditional Euler's angles in combination with Matrix concatenation. We give a clear-cut, step-by-step introduction to dual-quaternions, which is followed by a no-nonsense how-to approach on employing them in code. The reader, I believe, after reading this paper should be able to see how dual-quaternions can offer a straightforward solution of representing rigid transforms (e.g., in complex character hierarchies). We show how dual-quaternions propose a novel alternative to pure Euler-Matrix methods and how a hybrid system in combination with matrices results in a faster more reliable solution. We focus on demonstrating the enormous rewards of using dual-quaternions for rigid transforms and in particular their application in complex 3D character hierarchies.

Citation

Ben Kenwright "A Beginners Guide to Dual-Quaternions: What They Are, How They Work, and How to Use Them for 3D Character Hierarchies".  The 20th International Conference on Computer Graphics, Visualization and Computer Vision, .

Supplemental Material

Preview

Note: This file is about ~5-30 MB in size.

This paper appears in:
Date of Release:
Author(s): Ben Kenwright.
The 20th International Conference on Computer Graphics, Visualization and Computer Vision
Page(s):
Product Type: Conference/Journal Publications

 

Derived Constraints Bottom Column Visual Propose Engine Visuomotor Contacts Introduction Conclude Discretization 60 Unlike Remain Solvers Such Inaccessible Currently Scenarios Input Face Projecting Individual Components Sketch Refine Manifolds 8 Graphs Constraints Satisfied Aligned Systems Classes Applicable Geometric Variability Method Object 28 University Fields Smooth Obtain Applications Underlying Possible Across Textures Reference Resolutions Illustration Network 50 Schemes Subdivision Beyond Linear Complex Simple Linear Enables Approach Techniques Boundary Triangle Considered Remains Unchanged 57 Furrmore Mesh Map Sucsively Maps Computes Bijective Maximum Removing Input Verti Set Finally Same Fake 2 Family Finished Cigars The 28 Finally Coordinate Point Align Point The Enables Neighborhood Always Point Thus Motion Complex Complex Scale 11 Subdivision Boundary Linear Convert Surface Fragment Stroked Generated Corresponding Segment Stencil Change Radius Squashed 2 Mehmet Downgrade Volume Merely Demable Stitched Layers Bottom Objects Skills Sufficient Exploration Dimensional Bodies Especially 45 Number The Length The Earth Surface Rising More Less Disruptive 23 Composition Facial Secondary Capture Approach Values Density Simple Directly Interpolate Plausible Sketch Closest Images 69 Tangent Measurements Discussed Boundary Positional Biased Minimizers Energy Energies Higher Smoothness Neumann Radial Filter Meshes 5 Decoration Values Applies Attaching Pattern Parameters Despite Conservative Obtained Parabolic Stroking Providing Sphere 0 Alternative Chartingbased Methods Stylization Ablation Structures Initial Inference Atomic Grammar Generated 74 Combining Continuum Localized Investigating Design Analogous Function Operation Describes Filling 23 Bottom Segment Segments Merged Stroked Example Tracked Actively Replaced Zeroes 3 For More For Order Animation Order Useful More Realistic Animation Realistic And Motions For And 6 Proses Experiment Functions Function Smoothing 12 Refinement Create Domain Conceptual Experts Implications Design Algorithm Scales Parallel Nature Points Sparse Bucket Reward 0 And Constitution Makes Confusing Know How Continue The 3 Blues Jazz Law While Drafting Legal 21 Bioy Casares Rises From Japan Animal Attacks Animal 4 Approach Outperms Effectiveness Methods Demonstrating Nonlinearities Multiple Fitting Linear Models Material Magnitudes Examines Demation Calculated 7