Computer graphics: principles and practice (2nd ed.)
Computer graphics: principles and practice (2nd ed.)
Animating rotation with quaternion curves
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
On the algebraic and geometric foundations of computer graphics
ACM Transactions on Graphics (TOG)
Visualizing Quaternions (The Morgan Kaufmann Series in Interactive 3D Technology)
Visualizing Quaternions (The Morgan Kaufmann Series in Interactive 3D Technology)
Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics)
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
An Integrated Introduction to Computer Graphics and Geometric Modeling
An Integrated Introduction to Computer Graphics and Geometric Modeling
Rethinking Quaternions: Theory and Computation
Rethinking Quaternions: Theory and Computation
Robotics and Computer-Integrated Manufacturing
Parameterizing rational offset canal surfaces via rational contour curves
Computer-Aided Design
| Hi-index | 0.00 |
Quaternion multiplication can be applied to rotate vectors in 3-dimensions. Therefore in Computer Graphics, quaternions are sometimes used in place of matrices to represent rotations in 3-dimensions. Yet while the formal algebra of quaternions is well-known in the Graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this paper are: i.To provide a fresh, geometric interpretation of quaternions, appropriate for contemporary Computer Graphics; ii.To derive the formula for quaternion multiplication from first principles; iii.To present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in 3-dimensions based on insights from the algebra and geometry of multiplication in the complex plane; iv.To develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection; v.To show how to apply sandwiching to compute perspective projections. In Part I of this paper, we investigate the algebra of quaternion multiplication and focus in particular on topics i and ii. In Part II we apply our insights from Part I to analyze the geometry of quaternion multiplication with special emphasis on topics iii, iv and v.