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 $||f-f_{varepsilon}||_{C^1}<varepsilon$, 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 $|D|$?

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 $$0 < Rleq pi,quad |T|+R<pitag{1}$$

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.

enter image description here

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?

Thank you in advance for your answers enter image description here

functional analysis – Apply Arzela-Ascoli theorem to smooth function sequence

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 $alpha$