Computational Conformal Geometry Behind Modern Technologies

Abstract

Conformal geometry has deep roots in pure mathematics fields, such as Riemann surface theory, complex analysis, differential geometry, algebraic topology, partial differential equations and others. Historically, conformal geometry has been broadly used in many engineering applications [1], such as electro-magnetics, vibrating membranes, acoustics, elasticity, heat transfer and fluid flow. Most of these applications depend on conformal mappings between planar domains. Recently, with the rapid development of 3D scanning and medical imaging technologies, 3D geometric data has become ubiquitous. Figure 2 shows a human facial surface acquired using a scanning system based on structured light.

Prescribing discrete Gaussian curvature on polyhedral surfaces

“…If j∈V (u j,n − u i * ,n ) n∈N converges, then the sequence (u j,n − u i * ,n ) n∈N converges for all j ∈ V by Convention 4.3, which implies that j∈V R j e α(u j,n −u i * ,n ) converges to a finite negative number by R ≤ 0 and R ≡ 0. Combining this with α < 0 and χ(S) < 0, we have {u i * ,n } n∈N is bounded from above by (20). As {u n } n∈N is unbounded, then similar to the arguments above, the sequence {u i * ,n } n∈N diverges properly to −∞.…”

“…Combining this with χ(S) < 0 and Lemma 4.5, we have lim n→+∞ E(u n ) = +∞. If j∈V R j e α(u j,n −u i * ,n ) tends to a finite negative number, we have 2πχ(S)e −αu i * ,n has a negative lower bound by (20), which implies u i * ,n is bounded from above by α < 0 and χ(S) < 0. Combining this with χ(S) < 0 and j∈V (u j,n − u i * ,n ) n∈N diverges properly to +∞, we have lim n→+∞ E(u n ) = +∞ by Lemma 4.5.…”

“…Therefore, j∈V R j e α(u j,n −u i * ,n ) either tends to zero or to a finite negative number by α < 0, R ≤ 0 and R ≡ 0. If j∈V R j e α(u j,n −u i * ,n ) tends to zero, then 2πχ(S)e −αu i * ,n tends to zero by (20), which implies {u i * ,n } n∈N diverges properly to −∞ by α < 0 and χ(S) < 0. Combining this with χ(S) < 0 and Lemma 4.5, we have lim n→+∞ E(u n ) = +∞.…”

“…For Thurston's circle packings on surfaces, the solution of prescribing discrete Gaussian curvature problem gives rise to the famous Koebe-Andreev-Thurston Theorem, which plays a fundamental role in the study of geometry and topology of 3-dimensional manifolds [36]. For vertex scaling of piecewise linear and piecewise hyperbolic metrics on surfaces, the recent perfect solution of prescribing discrete Gaussian curvature problem by Gu-Luo-Sun-Wu [19] and Gu-Guo-Luo-Sun-Wu [18] provides a new constructive proof of the classical uniformization theorem on closed surfaces with genus g ≥ 1, which is computable and has lots of applications [10,20,21,28,35,45]. In the framework of vertex scaling, Springborn [34] recently proved that the discrete uniformization theorem on the sphere is equivalent to Rivin's realization theorem for ideal hyperbolic polyhedra [33].…”

Harmonic branched coverings and uniformization of CAT($k$) spheres

“…More recently, the harmonic maps approach with an applied mathematics bent has been studied by several authors. For more detail on the work in this area, we refer to the survey paper of X. Gu, F. Luo and S. T. Yau [16] and the references therein. Finally, we point out the more general problem of quasiconformal equivalency of the sphere, i.e.…”