Quaternions, Spinors, and Surfaces

Kamberov, G.; Norman, Peter; Pedit, Franz; Pinkall, Ulrich

doi:10.1090/conm/299

Cited by 32 publications

(13 citation statements)

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“…Continuous Dirac Operators. In the continuous setting, an extrinsic version of the Dirac operator D f , namely one that depends on the three‐dimensional immersion of a surface, was defined by Kamberov et al [KPP98, Kam02]. An intrinsic Dirac operator D , depending only on the Riemannian metric and algebraic structure of the manifold and not on embedding coordinates, has been known in the mathematical community since Atiyah and Singer [AS63].…”

Section: Related Workmentioning

confidence: 99%

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Diamanti

Tang

et al. 2018

Computer Graphics Forum

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Spectral mesh analysis and processing methods, namely ones that utilize eigenvalues and eigenfunctions of linear operators on meshes, have been applied to numerous geometric processing applications. The operator used predominantly in these methods is the Laplace‐Beltrami operator, which has the often‐cited property that it is intrinsic, namely invariant to isometric deformation of the underlying geometry, including rigid transformations. Depending on the application, this can be either an advantage or a drawback. Recent work has proposed the alternative of using the Dirac operator on surfaces for spectral processing. The available versions of the Dirac operator either only focus on the extrinsic version, or introduce a range of mixed operators on a spectrum between fully extrinsic Dirac operator and intrinsic Laplace operator. In this work, we introduce a unified discretization scheme that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds. In this discretization, both operators are very closely related, and preserve their key properties from the smooth case. We showcase various applications of our operators, with improved numerics over prior work.

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Section: Related Workmentioning

confidence: 99%

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Diamanti

Tang

et al. 2018

Computer Graphics Forum

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“…We rely on the treatment of smooth isothermic surfaces by means of quaternionic analysis as developed in [25,26,33]. …”

Section: Smooth Analoguesmentioning

confidence: 99%

Isothermic triangulated surfaces

Lam

Pinkall

2016

Math. Ann.

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Abstract. We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes.A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.

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“…Quaternions have ample use in diverse areas of mathematics like computation, geometry and algebra; see, e.g. [5][6][7][8]. The researcher in [9] used quaternion in the field of computer graphics.…”

Section: Introductionmentioning

confidence: 99%

The General Solution of Quaternion Matrix Equation Having η-Skew-Hermicity and Its Cramer’s Rule

Rehman

Kyrchei

Ali

et al. 2019

Mathematical Problems in Engineering

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We determine some necessary and sufficient conditions for the existence of the η-skew-Hermitian solution to the following system AX-(AX)η⁎+BYBη⁎+CZCη⁎=D,Y=-Yη⁎,Z=-Zη⁎ over the quaternion skew field and provide an explicit expression of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we derive its explicit determinantal representation formulas that are an analog of Cramer’s rule. A numerical example is also provided to establish the main result.

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Quaternions, Spinors, and Surfaces

Cited by 32 publications

References 0 publications

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Isothermic triangulated surfaces

The General Solution of Quaternion Matrix Equation Having η-Skew-Hermicity and Its Cramer’s Rule

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