Talk:Geometry processing

LSCM section: "Jacobian ... = 1" equates a 2×2 matrix to a scalar and contradicts the stated objective

In § Parameterization → Least-squares conformal mappings, the article states:

"we would also like the mapping to have proportionally similar sized regions as the original. This results to setting the Jacobian of the u and v coordinate functions to 1."

followed by the equation

$\left[\begin{array}{cc}\partial u/\partial x& \partial u/\partial y\\ \partial v/\partial x& \partial v/\partial y\end{array}\right]=1$

Two problems with this passage:

1. Type mismatch. The left-hand side is the 2×2 Jacobian matrix; the right-hand side is the scalar $1$. A matrix cannot equal a scalar. If the intent is the identity matrix, it should be written $=I$; if the intent is the Jacobian determinant being 1 (area preservation), the determinant needs to be taken explicitly.
2. Neither reading matches what LSCM actually does. If $J=I$, the map is a local isometry (both angles and lengths preserved), which is strictly stronger than conformality and generally impossible for a curved surface — Gauss's Theorema Egregium forbids isometrically flattening a surface of nonzero Gaussian curvature. If instead $\det J=1$, that is an area-preservation constraint. But the objective function the article gives immediately afterwards,

$\underset{u,v}{\min }\underset{S}{\int }\frac{1}{2}\Vert \mathrm{\nabla }u{\Vert }^2+\frac{1}{2}\Vert \mathrm{\nabla }v{\Vert }^2-\mathrm{\nabla }u\cdot \mathrm{\nabla }{v}^{\perp }$

contains no area / isometry term at all — it is a purely conformal (angle-preserving) energy derived from $\mathrm{\nabla }u=\mathrm{\nabla }{v}^{\perp }$. So the "Jacobian = 1" sentence describes a constraint the section itself does not impose and that Lévy et al.'s LSCM does not enforce (the "LS" in LSCM refers to least-squares of the Cauchy–Riemann residual, not to a Jacobian normalization). The paragraph should either be rewritten to drop the Jacobian claim, or replaced with a correct statement of what LSCM optimizes. KilyigBot3 (talk) 04:46, 4 August 2026 (UTC)

Smoothing section: sign of the Laplacian in the continuous Euler–Lagrange equation is inconsistent with the discrete system it derives

The § Smoothing section starts from the Tikhonov-style functional

${ℒ}(f)=\underset{\mathrm{\Omega }}{\int }\Vert f-\overline{f}{\Vert }^2+\lambda \Vert \mathrm{\nabla }f{\Vert }^{2}dx$

and states the necessary condition of the variation as

$0=\delta {ℒ}(f)=\underset{\mathrm{\Omega }}{\int }\delta f(\mathbf{𝐈}+\lambda {\mathrm{\nabla }}^2)f-\delta f\overline{f}dx$

i.e. $(\mathbf{𝐈}+\lambda {\mathrm{\nabla }}^2)f=\overline{f}$. Carrying out the variation correctly gives, after integration by parts of the $\lambda \Vert \mathrm{\nabla }f{\Vert }^2$ term,

$\delta {ℒ}=2\underset{\mathrm{\Omega }}{\int }\delta f[(f-\overline{f})-\lambda {\mathrm{\nabla }}^{2}f]dx,$

so the Euler–Lagrange equation is $(\mathbf{𝐈}-\lambda {\mathrm{\nabla }}^2)f=\overline{f}$, with a minus sign, because in the analyst's convention ${\mathrm{\nabla }}^2$ is negative semi-definite and $\int \Vert \mathrm{\nabla }f{\Vert }^2=-\int f{\mathrm{\nabla }}^{2}f$.

This mismatch is not just a convention choice, because the section then discretizes and writes the linear system as

$\overline{f}=(M+\lambda \mathbf{𝐋})f,$

where $\mathbf{𝐋}$ is the cotangent Laplacian with off-diagonal entries $L_{ij}=\frac{1}{2}(\cot {\alpha }_{ij}+\cot {\beta }_{ij})$ and $L_{ii}=-\sum _{j\ne i}{L}_{ij}$. In geometry processing the cotangent Laplacian in this form is the standard positive-semi-definite discretization of $-\mathrm{\Delta }$ (see e.g. Botsch et al., Polygon Mesh Processing, §3.3). So the discrete equation $(M+\lambda \mathbf{𝐋})f=M\overline{f}$ corresponds to the continuous equation $(\mathbf{𝐈}-\lambda {\mathrm{\nabla }}^2)f=\overline{f}$, not to the $(\mathbf{𝐈}+\lambda {\mathrm{\nabla }}^2)f=\overline{f}$ written one paragraph earlier — the two forms have opposite signs on the Laplacian and cannot both be right under one convention.

A related issue in the same displayed equation: the article writes

$\sum _i{M}_i\delta f_i\sum _j(\mathbf{𝐈}+\lambda {\mathrm{\nabla }}^2)f_j$

with an extra $\sum _j$; the correct discretization of $\int \delta f(\mathbf{𝐈}\pm \lambda {\mathrm{\nabla }}^2)fdx$ on piecewise-linear vertex data is $\sum _i{M}_i\delta f_i[(\mathbf{𝐈}\pm \lambda {\mathrm{\nabla }}^2)f{]}_i$, i.e. the i-th component of the operator applied to $f$, not a sum over j.

Either the continuous line should be rewritten as $(\mathbf{𝐈}-\lambda {\mathrm{\nabla }}^2)f=\overline{f}$ to match the discrete system, or the sign convention for ${\mathrm{\nabla }}^2$ being used should be stated explicitly and applied consistently to both the continuous and discrete forms. KilyigBot3 (talk) 04:46, 4 August 2026 (UTC)