Exploring Novel Surface Representations via an Experimental Ray-Tracer in CGA
Publication Date
2021-02-11Journal Title
Advances in Applied Clifford Algebras
ISSN
0188-7009
Publisher
Springer International Publishing
Volume
31
Issue
2
Language
en
Type
Article
This Version
VoR
Metadata
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Hadfield, H., Achawal, S., Lasenby, J., Lasenby, A., & Young, B. (2021). Exploring Novel Surface Representations via an Experimental Ray-Tracer in CGA. Advances in Applied Clifford Algebras, 31 (2) https://doi.org/10.1007/s00006-021-01117-8
Description
Funder: Natural Environment Research Council; doi: http://dx.doi.org/10.13039/501100000270
Abstract
Abstract: Conformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds significant promise in the field of computer graphics. In this paper we implement a simple ray tracer in CGA with a Blinn–Phong lighting model, before putting it to use to examine ray intersections with surfaces generated from the direct interpolation of geometric primitives. General surfaces formed from these interpolations are rendered using analytic normals. In addition, special cases of point-pair interpolation, which might find use in graphics applications, are described and rendered. A closed form expression is found for the derivative of the square root of a scalar plus 4-vector element with respect to a scalar parameter. This square root derivative is used to construct an expression for the derivative of a pure-grade multivector projected to the blade manifold. The blade manifold projection provides an analytical method for finding the normal line to the interpolated surfaces and its use is shown in lighting calculations for the ray tracer and in generating vertex normals for exporting the evolved surfaces as polygonal meshes.
Keywords
Article, T.C. : ENGAGE 2019- Geometric Algebra for Computing, Graphics & Engineering, Conformal geometric algebra, Ray-tracing, Direct object interpolation, Surface curvature, Mesh geometry, Primary 99Z99, Secondary 00A00
Sponsorship
Engineering and Physical Sciences Research Council (1949701)
Identifiers
s00006-021-01117-8, 1117
External DOI: https://doi.org/10.1007/s00006-021-01117-8
This record's URL: https://www.repository.cam.ac.uk/handle/1810/317496
Rights
Attribution 4.0 International (CC BY 4.0)
Licence URL: https://creativecommons.org/licenses/by/4.0/
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