Bézier curves are a cornerstone of computer-aided geometric design, yet their classical formulations operate within Euclidean or projective function spaces. We show that rational Bézier curves arise intrinsically from the algebraic structure of PGA (Projective Geometric Algebra), providing a unified, coordinate-free treatment that naturally extends classical constructions by allowing negative weights and points at infinity. Negative weights may create singularities that correspond to points at infinity, admitting a consistent geometric interpretation. This enables a natural representation of complementary curve sections and conic types. We then generalize the Bézier construction beyond points to higher geometric objects of PGA, including lines, planes, and higher-dimensional primitives. Finally, we extend the framework to versors, the algebraic encodings of rigid body transformations in PGA, yielding smooth Bézier curves of such transformations, suitable for geometric animation. All constructions share the same blending equations and normalization procedure, demonstrating that PGA provides a genuinely unified foundation for curve modeling across geometric objects and transformations.
@article{harquin2026bezier,title={Projective Geometric Algebra powers up B{\'e}zier Curves},author={Harquin, Enzo and Breuils, St{\'e}phane and Biri, Venceslas and Nozick, Vincent},journal={Computer Science Research Notes},volume={CSRN 3601},number={1},pages={121--130},year={2026},month=may,publisher={Union Agency, Science Press},issn={2464-4628},note={Presented at WSCG 2026, the 34th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision},notebook={https://eharquin.github.io/pga-bezier/lab/index.html},}
WIP
Conic Conformal Geometric Algebra: Foundation for Two-Dimensional Conic Geometry
Enzo Harquin, Stéphane Breuils, Venceslas Biri, and 1 more author
2026
Presented at ENGAGE 2026, the Workshop on Empowering Novel Geometric Algebra for Graphics and Engineering, held at Computer Graphics International (CGI) 2026, London
Preprint
Generalize cross-ratios in n-dimensional Plane-Based Geometric Algebra
Enzo Harquin, Stéphane Breuils, Pascal Monasse, and 2 more authors
May 2026
Presented at AGACSE 2026, the International Conference on Applied Geometric Algebras in Computer Science and Engineering, Beijing
We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, we establish an explicit cross-ratio formula, prove that it recovers the appropriate classical invariant, and identify the canonical pairwise measurement operator. A systematic duality analysis further revealed that all eight configurations organize into four dual pairs under the Hodge dual, and that all measurement operators reduce to either the commutator or the commutator dual, depending solely on the geometric configuration rather than on object grade. In each case the formula recovers the appropriate classical invariant: signed distance ratios for parallel configurations and sine cross-ratios for secant ones. These results establish the cross-ratio as a grade-agnostic projective invariant within PGA, and provide a constructive foundation for defining n-dimensional homographies directly from prescribed invariants.
WIP
Homography in Plane-Based Geometric Algebra
Enzo Harquin, Stéphane Breuils, Pascal Monasse, and 2 more authors
2026
Presented at AGACSE 2026, the International Conference on Applied Geometric Algebras in Computer Science and Engineering, Beijing