Huanyu Chen · University of Southern California · UCLA, October 26, 2026
Mercator: round, but sizes explode
Equal-area: same size, but squashed
2. How do we minimize it, fast and safely?
3. Why are all six cells one problem?
zoom in: locally, every map is linear
x↦F x,F=∂ map∂x∈R2×2x \mapsto F\,x, \qquad F = \frac{\partial\,\text{map}}{\partial x}\in\mathbb{R}^{2\times 2} x↦Fx,F=∂x∂map∈R2×2
F=U Σ V⊤,U=[ u1 u2 ],Σ=diag(σ1,σ2),V=[ v1 v2 ]F = U\,\Sigma\,V^\top, \qquad U = [\,u_1\;u_2\,],\quad \Sigma = \operatorname{diag}(\sigma_1, \sigma_2),\quad V = [\,v_1\;v_2\,] F=UΣV⊤,U=[u1u2],Σ=diag(σ1,σ2),V=[v1v2]
ψ(R F Q)=ψ(F) ⟹ ψ(F)=ψ^(σ1,σ2),ψ^ symmetric\psi(R\,F\,Q) = \psi(F) \;\;\Longrightarrow\;\; \psi(F) = \hat\psi(\sigma_1, \sigma_2), \quad \hat\psi \text{ symmetric} ψ(RFQ)=ψ(F)⟹ψ(F)=ψ^(σ1,σ2),ψ^ symmetric
angles: distance from the diagonal
area: distance from the hyperbola
both: distance from (1, 1) (ARAP)
E(x)=∑tAt ψ(Ft) ≈ ∫Ωψ(F) dXE(x) = \sum_t A_t\,\psi(F_t) \;\approx\; \int_\Omega \psi(F)\,dX E(x)=t∑Atψ(Ft)≈∫Ωψ(F)dX
minx E(x)=∑tAt ψ(Ft(x)),ψARAP=(σ1−1)2+(σ2−1)2\min_{x}\; E(x) = \sum_t A_t\,\psi(F_t(x)), \qquad \psi_{\text{ARAP}} = (\sigma_1-1)^2 + (\sigma_2-1)^2 xminE(x)=t∑Atψ(Ft(x)),ψARAP=(σ1−1)2+(σ2−1)2
x←x−α (∇2E)−1 ∇Ex \leftarrow x - \alpha\,\big(\nabla^2 E\big)^{-1}\,\nabla E x←x−α(∇2E)−1∇E
∂ψ∂F=U (∂1ψ^00∂2ψ^)V⊤ = ∂1ψ^ u1v1⊤+∂2ψ^ u2v2⊤\frac{\partial \psi}{\partial F} = U\,\begin{pmatrix}\partial_1\hat\psi & 0\\ 0 & \partial_2\hat\psi\end{pmatrix} V^\top \;=\; \partial_1\hat\psi\; u_1 v_1^\top + \partial_2\hat\psi\; u_2 v_2^\top ∂F∂ψ=U(∂1ψ^00∂2ψ^)V⊤=∂1ψ^u1v1⊤+∂2ψ^u2v2⊤
Q⊤∂2ψ∂F2 Q = ∇σ2ψ^⏟scaling ⊕ (abba)⏟twist, flip,Q=[vec(u1v1⊤) vec(u2v2⊤) vec(u1v2⊤) vec(u2v1⊤)]Q^\top \frac{\partial^2\psi}{\partial F^2}\, Q \;=\; \underbrace{\nabla^2_\sigma\hat\psi}_{\text{scaling}} \;\oplus\; \underbrace{\begin{pmatrix} a & b\\ b & a\end{pmatrix}}_{\text{twist, flip}}, \qquad Q = \big[\operatorname{vec}(u_1v_1^\top)\;\; \operatorname{vec}(u_2v_2^\top)\;\; \operatorname{vec}(u_1v_2^\top)\;\; \operatorname{vec}(u_2v_1^\top)\big] Q⊤∂F2∂2ψQ=scaling∇σ2ψ^⊕twist, flip(abba),Q=[vec(u1v1⊤)vec(u2v2⊤)vec(u1v2⊤)vec(u2v1⊤)]
Di=uivi⊤,T12=12(u1v2⊤−u2v1⊤),L12=12(u1v2⊤+u2v1⊤)D_i = u_i v_i^\top, \qquad T_{12} = \tfrac{1}{\sqrt2}\big(u_1 v_2^\top - u_2 v_1^\top\big), \qquad L_{12} = \tfrac{1}{\sqrt2}\big(u_1 v_2^\top + u_2 v_1^\top\big) Di=uivi⊤,T12=21(u1v2⊤−u2v1⊤),L12=21(u1v2⊤+u2v1⊤)
λDi=eig(∇σ2ψ^),λT=∂1ψ^+∂2ψ^σ1+σ2,λL=∂1ψ^−∂2ψ^σ1−σ2\lambda_{D_i} = \operatorname{eig}\big(\nabla^2_{\sigma}\hat\psi\big), \qquad \lambda_{T} = \frac{\partial_1\hat\psi+\partial_2\hat\psi}{\sigma_1+\sigma_2}, \qquad \lambda_{L} = \frac{\partial_1\hat\psi-\partial_2\hat\psi}{\sigma_1-\sigma_2}λDi=eig(∇σ2ψ^),λT=σ1+σ2∂1ψ^+∂2ψ^,λL=σ1−σ2∂1ψ^−∂2ψ^
Ht=∑kmax(λk,0) qkqk⊤H_t = \sum_k \max(\lambda_k, 0)\; q_k q_k^\top Ht=k∑max(λk,0)qkqk⊤
ψARAP=∑i(σi−1)2,ψARAP(σ1,0)=(σ1−1)2+1<∞\psi_{\text{ARAP}} = \sum_i(\sigma_i-1)^2, \qquad \psi_{\text{ARAP}}(\sigma_1, 0) = (\sigma_1 - 1)^2 + 1 < \infty ψARAP=i∑(σi−1)2,ψARAP(σ1,0)=(σ1−1)2+1<∞
ψSD=∑iσi2+σi−2 → ∞as σi→0\psi_{\text{SD}} = \sum_i \sigma_i^2 + \sigma_i^{-2} \;\to\; \infty \quad\text{as } \sigma_i \to 0 ψSD=i∑σi2+σi−2→∞as σi→0
(σ1−1)2+(σ2−1)2(\sigma_1-1)^2 + (\sigma_2-1)^2(σ1−1)2+(σ2−1)2
∑iσi2+σi−2\sum_i \sigma_i^2 + \sigma_i^{-2}∑iσi2+σi−2
σ1σ2+σ2σ1\dfrac{\sigma_1}{\sigma_2} + \dfrac{\sigma_2}{\sigma_1}σ2σ1+σ1σ2
μ2(∑iσi2−2)−μlogJ\tfrac{\mu}{2}\big(\sum_i \sigma_i^2 - 2\big) - \mu \log J2μ(∑iσi2−2)−μlogJ+ λ2log2J, J=σ1σ2+\, \tfrac{\lambda}{2}\log^2 J, \;\; J = \sigma_1\sigma_2+2λlog2J,J=σ1σ2
xn+1=argminx 12h2∥x−x~∥M2⏟inertia+∑tAt ψ(Ft)⏟E(x): elastic energy+κ∑kb(dk(x))⏟contact barrier (IPC)x_{n+1} = \arg\min_x\; \underbrace{\frac{1}{2h^2}\|x - \tilde x\|_M^2}_{\text{inertia}} + \underbrace{\textstyle\sum_t A_t\,\psi(F_t)}_{E(x):\ \text{elastic energy}} + \underbrace{\kappa \textstyle\sum_k b\big(d_k(x)\big)}_{\text{contact barrier (IPC)}} xn+1=argxmininertia2h21∥x−x~∥M2+E(x): elastic energy∑tAtψ(Ft)+contact barrier (IPC)κ∑kb(dk(x))
minx ∑tAt ψ(Ft)s.t. ∥xi∥=1, no triangle flips\min_{x}\; \sum_t A_t\,\psi(F_t) \quad \text{s.t.}\;\; \|x_i\| = 1, \;\; \text{no triangle flips} xmint∑Atψ(Ft)s.t.∥xi∥=1,no triangle flips
Ft∈R3×3F_t \in \mathbb{R}^{3\times3}Ft∈R3×3, detFt∝\det F_t \proptodetFt∝ tet volume
Symmetric Dirichlet ⇒ no spherical flips
Eigensystem: the 3D column of the table
Minimize E(P(x))E(P(x))E(P(x)), PPP = radial projection
Max step: first root of det(si+αdi)=0\det(s_i + \alpha d_i) = 0det(si+αdi)=0, a cubic
the texture: 127 flat patches (white is wasted)
painted giraffe, seams in black
xn+1=argminx 12h2 ∥x−x~∥M2+E(x)+κ Bcontact(x)x_{n+1} = \arg\min_x\; \frac{1}{2h^2}\,\|x - \tilde x\|_M^2 + E(x) + \kappa\,B_{\text{contact}}(x) xn+1=argxmin2h21∥x−x~∥M2+E(x)+κBcontact(x)
Near rest, every ψ is quadratic:its quadratic part is Linear Corotational
λ=ψ12(1,1,1)\lambda = \psi_{12}(1,1,1)λ=ψ12(1,1,1)μ=12(ψ11(1,1,1)−ψ12(1,1,1))\mu = \tfrac12\big(\psi_{11}(1,1,1) - \psi_{12}(1,1,1)\big)μ=21(ψ11(1,1,1)−ψ12(1,1,1))
E=μ(3λ+2μ)λ+μ,ν=λ2(λ+μ)E = \dfrac{\mu(3\lambda + 2\mu)}{\lambda + \mu}, \qquad \nu = \dfrac{\lambda}{2(\lambda + \mu)}E=λ+μμ(3λ+2μ),ν=2(λ+μ)λ
ψα(σ)=1α2 ψ(σ1α,σ2α,σ3α)\psi_\alpha(\sigma) = \frac{1}{\alpha^2}\,\psi(\sigma_1^\alpha, \sigma_2^\alpha, \sigma_3^\alpha) ψα(σ)=α21ψ(σ1α,σ2α,σ3α)
Same E and ν for every α
α < 1: softens under stretch (rubber)α > 1: stiffens under stretch (foam)
∫−h/2h/2ψ(1+κ ξ) dξ ≈ ψ′′24 κ2 h3\int_{-h/2}^{h/2} \psi\big(1 + \kappa\,\xi\big)\,d\xi \;\approx\; \tfrac{\psi''}{24}\,\kappa^2\, h^3 ∫−h/2h/2ψ(1+κξ)dξ≈24ψ′′κ2h3
bending stiffness ∝ h³
ψshell:=∫−h/2h/2ψ(F(ξ)) (1−2Hξ+Kξ2) dξ = h ψ(λ1,λ2,ρ) + h312(ψbend+ψ∇ρ)+O(h5)\psi_{\text{shell}} := \int_{-h/2}^{h/2} \psi\big(F(\xi)\big)\,\big(1 - 2H\xi + K\xi^2\big)\,d\xi \;=\; h\,\psi(\lambda_1, \lambda_2, \rho) \;+\; \frac{h^3}{12}\big(\psi_{\text{bend}} + \psi_{\nabla\rho}\big) + O(h^5) ψshell:=∫−h/2h/2ψ(F(ξ))(1−2Hξ+Kξ2)dξ=hψ(λ1,λ2,ρ)+12h3(ψbend+ψ∇ρ)+O(h5)
ψrod:=1Γ∫Sψ(F(ξ1,ξ2)) ∣G∣ dξ1dξ2 = πh2ab ψ(λ,ρ,ρ) + π8h4ab (ψbend+ψtwist)+O(h6)\psi_{\text{rod}} := \frac{1}{\Gamma}\int_{\mathcal S} \psi\big(F(\xi_1, \xi_2)\big)\,|G|\,d\xi_1 d\xi_2 \;=\; \pi h^2 ab\;\psi(\lambda, \rho, \rho) \;+\; \frac{\pi}{8} h^4 ab\,\big(\psi_{\text{bend}} + \psi_{\text{twist}}\big) + O(h^6) ψrod:=Γ1∫Sψ(F(ξ1,ξ2))∣G∣dξ1dξ2=πh2abψ(λ,ρ,ρ)+8πh4ab(ψbend+ψtwist)+O(h6)
ψ(σ)=∑if(σi)+λ2log2(σ1σ2σ3)\psi(\sigma) = \sum_i f(\sigma_i) + \tfrac{\lambda}{2}\log^2(\sigma_1\sigma_2\sigma_3)ψ(σ)=∑if(σi)+2λlog2(σ1σ2σ3), fff = cubic spline in σ\sigmaσ
2. Minimize: closed-form Hessian modes + clamping + safe steps, any dimension
3. Same problem: maps, deformations and dynamics, 2D and 3D, share one machine
F=U Σ V⊤F = U\,\Sigma\,V^\top F=UΣV⊤
F=U Σ V⊤F = U\,\Sigma\,\color{#b5005b}{V^\top} F=UΣV⊤
F=U Σ V⊤F = U\,\color{#b5005b}{\Sigma}\,V^\top F=UΣV⊤
F=U Σ V⊤F = \color{#b5005b}{U}\,\Sigma\,V^\top F=UΣV⊤
ψ=σ1σ2+σ2σ1\psi = \frac{\sigma_1}{\sigma_2} + \frac{\sigma_2}{\sigma_1} ψ=σ2σ1+σ1σ2
ψ=(σ1σ2−1)2\psi = (\sigma_1\sigma_2 - 1)^2 ψ=(σ1σ2−1)2
ψ=(σ1−1)2+(σ2−1)2\psi = (\sigma_1 - 1)^2 + (\sigma_2 - 1)^2 ψ=(σ1−1)2+(σ2−1)2
λtwist=ψ1+ψ2σ1+σ2\lambda_{\text{twist}} = \frac{\psi_1+\psi_2}{\sigma_1+\sigma_2} λtwist=σ1+σ2ψ1+ψ2
x←x+αd,α=1,12,14,… until E(x+αd)<E(x)x \leftarrow x + \alpha d, \qquad \alpha = 1, \tfrac12, \tfrac14, \dots \;\text{ until } E(x + \alpha d) < E(x) x←x+αd,α=1,21,41,… until E(x+αd)<E(x)
x ← start (Tutte map / previous frame), flip-free repeat for each triangle t: ▹ independent: parallel F_t = D_s(x) D_m⁻¹, F_t = U Σ Vᵀ (signed SVD) g_t = U diag(∂ψ̂/∂σ) Vᵀ (gradient) λ, q = closed-form eigenpairs (D₁, D₂, T₁₂, L₁₂) H_t = Σ_k max(λ_k, 0) q_k q_kᵀ (projected Hessian) g = Σ_t A_t Bᵀ vec(g_t), H = Σ_t A_t Bᵀ H_t B solve H d = −g (sparse Cholesky) α = 1, or 0.9 · first α with det F_t(x + αd) = 0 (barrier) while E(x + αd) > E(x) + c α gᵀd: α ← α / 2 x ← x + α d until |gᵀd| or the change in E is tiny
λtwistARAP=2(σ1+σ2−2)σ1+σ2\lambda_{\text{twist}}^{\text{ARAP}} = \frac{2(\sigma_1+\sigma_2-2)}{\sigma_1+\sigma_2} λtwistARAP=σ1+σ22(σ1+σ2−2)
ψi=2σi−2σi−3\psi_i = 2\sigma_i - 2\sigma_i^{-3} ψi=2σi−2σi−3
λscale=2+6σi4,λtwist=2−2 σi2−σiσj+σj2σi3σj3,λflip=2+2 σi2+σiσj+σj2σi3σj3\lambda_{\text{scale}} = 2 + \frac{6}{\sigma_i^4}, \qquad \lambda_{\text{twist}} = 2 - 2\,\frac{\sigma_i^2 - \sigma_i\sigma_j + \sigma_j^2}{\sigma_i^3\sigma_j^3}, \qquad \lambda_{\text{flip}} = 2 + 2\,\frac{\sigma_i^2 + \sigma_i\sigma_j + \sigma_j^2}{\sigma_i^3\sigma_j^3}λscale=2+σi46,λtwist=2−2σi3σj3σi2−σiσj+σj2,λflip=2+2σi3σj3σi2+σiσj+σj2
Barrier: the minimizer has no flips
Max step + backtracking: no iterate flips
d2dσ2log2σ=2(1−logσ)σ2<0\dfrac{d^2}{d\sigma^2}\log^2\sigma = \dfrac{2(1-\log\sigma)}{\sigma^2} < 0dσ2d2log2σ=σ22(1−logσ)<0 for σ>e\sigma > eσ>e
A few triangles stretch 700× instead of many stretching a little
∂2ψ∂F2=∑i,j∂2Ψ∂Ii ∂Ij gigj⊤+∑i∂Ψ∂Ii Hi,∥F−1∥2=(I12−I2)24I32−2I1I3\frac{\partial^2\psi}{\partial F^2} = \sum_{i,j} \frac{\partial^2\Psi}{\partial I_i\,\partial I_j}\, g_i g_j^\top + \sum_i \frac{\partial \Psi}{\partial I_i}\, H_i, \qquad \|F^{-1}\|^2 = \frac{(I_1^2 - I_2)^2}{4 I_3^2} - \frac{2 I_1}{I_3} ∂F2∂2ψ=i,j∑∂Ii∂Ij∂2Ψgigj⊤+i∑∂Ii∂ΨHi,∥F−1∥2=4I32(I12−I2)2−I32I1
hii=2Ψ2+Ψ11+4σi2Ψ22+I32σi2Ψ33+4σiΨ12+4I3Ψ23+2I3σiΨ13h_{ii} = 2\Psi_{2} + \Psi_{11} + 4\sigma_i^2\Psi_{22} + \tfrac{I_3^2}{\sigma_i^2}\Psi_{33} + 4\sigma_i\Psi_{12} + 4 I_3\Psi_{23} + \tfrac{2I_3}{\sigma_i}\Psi_{13} hii=2Ψ2+Ψ11+4σi2Ψ22+σi2I32Ψ33+4σiΨ12+4I3Ψ23+σi2I3Ψ13
hij=σkΨ3+Ψ11+4I3σkΨ22+σkI3Ψ33+2σk(I2−σk2)Ψ23+(I1−σk)(σkΨ13+2Ψ12)h_{ij} = \sigma_k\Psi_3 + \Psi_{11} + \tfrac{4I_3}{\sigma_k}\Psi_{22} + \sigma_k I_3\Psi_{33} + 2\sigma_k(I_2-\sigma_k^2)\Psi_{23} + (I_1-\sigma_k)(\sigma_k\Psi_{13} + 2\Psi_{12}) hij=σkΨ3+Ψ11+σk4I3Ψ22+σkI3Ψ33+2σk(I2−σk2)Ψ23+(I1−σk)(σkΨ13+2Ψ12)
∇σ2 ψ^=[ hij ],hij=∂2ψ^∂σi ∂σj\nabla^2_{\sigma}\,\hat\psi = [\,h_{ij}\,], \qquad h_{ij} = \frac{\partial^2\hat\psi}{\partial\sigma_i\,\partial\sigma_j} ∇σ2ψ^=[hij],hij=∂σi∂σj∂2ψ^
λtwist=2σx+σy ∂Ψ∂I1+2 ∂Ψ∂I2+σz ∂Ψ∂I3\lambda_{\text{twist}} = \frac{2}{\sigma_x+\sigma_y}\,\frac{\partial\Psi}{\partial I_1} + 2\,\frac{\partial\Psi}{\partial I_2} + \sigma_z\,\frac{\partial\Psi}{\partial I_3} λtwist=σx+σy2∂I1∂Ψ+2∂I2∂Ψ+σz∂I3∂Ψ
⇓ in singular values\Big\Downarrow\; \text{in singular values} ⇓‖in singular values
λtwist=ψx+ψyσx+σy\lambda_{\text{twist}} = \color{#b5005b}{\frac{\psi_x + \psi_y}{\sigma_x + \sigma_y}} λtwist=σx+σyψx+ψy
The same hessian_eigensystem, unchanged, on a random 4 × 4 F:
hessian_eigensystem
central: σ₁/σ₂ up to √3
conformal: σ₁/σ₂ ≤ 1.02 except at 8 corners
squares stay square…
…but shrink below a pixel
ν = 0
ν = 0.49
Quadratic part = Linear Corotational
λ=ψ12(1)\lambda = \psi_{12}(1)λ=ψ12(1), μ=12(ψ11−ψ12)\;\mu = \tfrac12(\psi_{11} - \psi_{12})μ=21(ψ11−ψ12)