## texture atlas – Calculate UV Mapping of a Smooth, Convex Patch on a Grid

I have a dataset of 3D points and normals. It is generated from fitting a polynomial function z=poly(x,y) on an evenly spaces grid in x-y. The shape of this dataset is convex and very smooth. It resembles a small patch of the upper half of an ellipsoid or cone (“pointing” to the y-axis direction), with small local variations.

I want to calculate a UV map with the smallest distortion in areas (for texturing).
In addition, I need to keep the x-y scale constant. There is no need for any boundary constraints.

Mathematically speaking, how should I approach it? Is there a way to use the fact that it is very smooth, convex, and set on an evenly spaced x-y grid for UV mapping?

Practically speaking the data is represented by a NumPy array size (n, m, 3) with equal spacing in x, y and another boolean (n, m) array as a mask for the shape.

## ag.algebraic geometry – Definition of smooth morphism of schemes

In Wedhorn’s and Görtz’s book, a morphism $$f: X to Y$$ is said to be smooth of relative dimension $$d$$ when for every $$x in X$$, there exists affine open neighborhoods $$U$$ of $$x$$ and $$V = operatorname{Spec} R$$ of $$y=f(x)$$ such that $$f(U) subseteq V$$, and an open immersion of $$R$$-schemes $$j: U hookrightarrow operatorname{Spec} R(T_1, dots, T_n)/(g_1, dots, g_{n-d})$$ such that the matrix $$left (frac{partial g_i}{partial T_j}(x)right )$$ has rank $$n-d$$.

In Bosch’s book, $$f$$ is smooth of relative dimension $$d$$ if for every $$x in X$$, there exists an open neighborhood $$U$$ of $$x$$ and an $$Y$$-morphism $$j:U to W subseteq mathbb{A}_Y^n$$ (here, $$mathbb{A}^n_Y = underline{operatorname{Spec}}_Y mathscr{O}_Y(T_1, dots, T_n)$$) giving rise to a closed immersion from $$U$$ over an open subscheme $$W subseteq mathbb{A}^n_Y$$ satisfying:

If $$mathcal{I} subseteq mathscr{O}_W$$ is the ideal correspondent to $$j$$, then there are $$n-d$$ sections $$g_1, dots g_{n-d}$$ generating $$mathcal{I}$$ in a neighborhood of $$z = j(x)$$ and such that the matrix $$left(frac{partial g_i}{partial t_j}(z)right)$$ has rank $$n-d$$, where $$t_i$$ are the coordinate functions of $$mathbb{A}^n_Y$$.

To be honest, I can’t quite make sense of Bosch’s definition: what does $$frac{partial g_i}{partial t_j}$$ even mean, given that the $$t_j$$ are global sections and the $$g_i$$ are sections on a neighborhood of $$z$$, and, thus, not necessarily polynomials. Am I missing something? How can I prove that these two definitions are equivalent? I can sort of see how Bosch’s definition imply Wedhorn’s and Görtz’s, if I squint. Can someone help me?

## reference request – Approximation of \$C^1\$-smooth equivariant maps by infinitely smooth ones

Let $$M,N$$ be smooth closed manifolds acted by a finite group $$G$$. Let $$fcolon Mto N$$ be a $$C^1$$-smooth $$G$$-equivariant map.

Is it true that for any $$varepsilon>0$$ there exists a $$C^infty$$-smooth $$G$$-equivariant map
$$f_varepsiloncolon Mto N$$ such that $$||f-f_{varepsilon}||_{C^1}, where the $$C^1$$-norm is taken with respect to some $$G$$-invariant Riemannian metrics on $$M$$ and $$N$$?

A reference would be most helpful.

## algebraic geometry – A question about a nondegenerate smooth curve in \$Bbb P^n\$ of degree \$n\$

Suppose that $$Xsubset Bbb P^n$$ is a nondegenerate smooth projective curve of degree $$n$$. Let $$H$$ be a hyperplane in $$Bbb P^n$$, $$D=text{div}(H)$$, and $$Qsubset |D|$$ be the linear subsystem of hyperplane divisors. Then we have the equalities $$n=dim (Q)leq dim |D|=dim L(D)-1leq n$$
Therefore $$Q=|D|$$ and $$dim L(D)=1+deg (D)$$. The latter equality implies that $$X$$ has genus zero, and the first equality implies that any divisor of degree $$n$$ is a hyperplane divisor.

This is a paragraph in p.217 of Miranda’s book Algebraic Curves and Riemann Surfaces, and I can’t understand the last sentence.

1. How do we know that the genus of $$X$$ is zero? I can only see that by Riemann-Roch $$g=dim L(K-D)$$.

2. How do we know that any divisor of degree $$n$$ is a hyperplane divisor? Is any positive divisor of degree $$n$$ contained in $$|D|$$?

## Fitting a non-decreasing sequence of bivariate data with a non-decreasing smooth function using BSplineCurve or another built-in function

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## real analysis – Smoothness of a unique level set of a smooth function

Suppose that $$f:mathbb{R}^2to mathbb{R}$$ is smooth and, for all $$xin mathbb{R}$$, there exists a unique $$y(x)$$ such that $$f(x,y(x))=0$$. In other words, the graph of a continuous function $$y=y(x)$$ is the $$0$$ level set of $$f$$. Is it true that $$y(x)$$ is differentiable $$x$$-almost everywhere?

## plotting – How to make these lines smooth when using ContourPlot inside of a region?

Consider a region in the $$(R,T)$$ plane defined by the conditions $$0 < Rleq pi,quad |T|+R

Define $$t(R,T)$$ to be $$t(R,T)=dfrac{1}{2}left(tanleft(dfrac{T-R}{2}right)+tanleft(dfrac{T+R}{2}right)right)tag{2}.$$

I want to plot level sets of $$t(R,T)$$ inside the region defined by conditions (1). To do so I used the following code

``````t(R_, T_) := 1/2 (Tan((T - R)/2) + Tan((T + R)/2))
cond1 = 0 <= R <= Pi && Abs(T) + R < Pi && R (Element) Reals && T (Element) Reals
cond2 = Reduce(cond1, {R, T})
reg1 = ImplicitRegion(cond2, {{R, -(Infinity), (Infinity)}, {T, -(Infinity), (Infinity)}})
Show(RegionPlot(reg1, PlotStyle -> None), ContourPlot(t(R, T) == -1, {R, T} (Element) reg1), ContourPlot(t(R, T) == 1, {R, T} (Element) reg1))
``````

The resulting plot, however, does not show smooth curves.

What is the issue here? Why isn’t Mathematica plotting these curves smoothly? How can I correct that?

## at.algebraic topology – Images of smooth endomorphisms and laminations

This question replaces an earlier question of mine, stating it in a better way and with more context.

Let $$M$$ be a compact Riemannian manifold with Riemannian metric $$g$$, and let $$f : Mto M$$ be a smooth endomorphism.

Let $$Nsubset M$$ be a closed submanifold. Fix a positive constant $$C>0$$.

In $$Mtimes mathbf{R}_{ge 0}$$, we consider the collection of subsets
$$Y_n := f^n(N)times (n C, (n+1) C)$$
with $$nge 0$$, $$f^0(N) = N$$, and $$f^n(N)$$ the image of $$N$$ in $$M$$ under $$f$$.

Call $$Y$$ the union of the $$Y_n$$ for all $$nge 0$$ in $$Mtimes mathbf{R}_{ge 0}$$.

Question 0. Under what conditions on $$f$$ is $$Y$$ a smooth manifold, possibly with corners.

Question 1. Is there any way to regard the collection of the $$Y_n$$ as a foliation/lamination on $$Y$$, perhaps under some conditions on $$f$$?

I’d content myself of analogous situations in the literature. I found myself wondering about this while thinking about torus-valued Morse functions, but it’s not so relevant to the question.

## 2D side-scroller game : smooth noise transition between biomes

I’m creating a little terraria-like 2D side-scroller game with TypeScript. Currently, I divide parts of the world into biomes, which each have their own properties (flat, mountainous terrain, etc.). I am using a one dimensional Perlin noise to generate the height of the terrain for each biome. Currently this is a problem for me because the borders between biomes are not at the same height. How could I make the transitions between biomes smoother?

let $$Usubseteq mathbb{R}^n$$ be open, bounded and convex and $$(f_i)$$ is a sequence in $$C^{infty}(bar{U})$$. Suppose that for each $$minmathbb{N}$$, there is a constant $$C_m>0$$ such that for all $$iinmathbb{N}$$, the inequality $$sup_{bar{U}}|D^mf_i|leq C_m$$ holds.
I want to show that $$(f_i)$$ has uniformly convergent subsequence in $$C^{infty}(bar{U})$$.
My attempt is that by Arzela-Ascoli theorem, $$(f_i)$$ should have a uniformly convergent subsequence
(call it $$(f_{j^prime})$$ and the convengent to $$f$$). This is what I can do.
But what I can not do yet is that how to prove $$f$$ is $$C^{infty}$$ (although it should be obversly)
and how to prove is subsequence satisfies that $$D^{alpha}f_{j^prime}to D^{alpha}f$$ uniformly on $$bar{U}$$ for every multiindex $$alpha$$