For questions about surfaces.

learn more… | top users | synonyms

0
votes
0answers
24 views

Orientation of surfaces

From the book: Fixing a parametrization $x(u,v)$ of a neighborhood of a point $p$ of a regular surface $S$, we determine an orientation of the tangent plane $T_p (S)$, namely, the orientation of the ...
3
votes
1answer
48 views

Inverse mapping for a simple $\mathbb{R}^3$ surface given by $(\sin u, \sin 2u, v)$.

For a domain $U=\{\, (u,v) \in \mathbb{R}^2 \mid -\pi<u<\pi,\ 0<v<1 \,\}$ we have a mapping $X \colon U \to \mathbb{R}^3$ defined by $X(u,v) = (\sin u, \sin 2u, v)$. The resulting surface ...
6
votes
0answers
123 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
0
votes
0answers
29 views

Two unit disks spliced together?

$$ x^2 + (y-z)^2 = 2 x^2 z/y $$ The surface represented by above equation is formed by radial cuts on two separate unit diameter disks spliced together forming a "continuous" surface around ...
0
votes
0answers
29 views

“Circle” on pseudosphere

How should parametrization of the 2 parameter surface of a pseudosphere ("latitude" u and longitude v) change to result in a 1 parameter curve of constant geodesic curvature? EDIT: In other ...
0
votes
2answers
49 views

Viviani on Sphere parametrization

How should parametrization of the 2 parameter surface of a sphere (latitude u, longitude v) be changed to result in 1 parameter curve of Viviani?
2
votes
1answer
72 views

The projections are differentiable $1$-forms.

Suppose $M$ is a surface and suppose $X: U \subset \mathbb R^2\rightarrow M$ is a coordinate patch. Then for every $p \in X(U)$, the pair of vectors $(X_u(X^{-1}(p),X_v(X^{-1}(p) )$ is a basis of the ...
1
vote
1answer
66 views

Curvature proof of a convex plane curve

Having a little trouble with part b. Is there a way to show that this curve would be arc length paramaterized? I am assuming that we cannot say this. If it is not we can take alpha', alpha'' and ...
-1
votes
1answer
87 views

Give a closed plane curve C with k (curvature) > 0 that is not convex, Draw closed plane curves with rotation indices 0, 2, -2, and 3

1.) Give a closed plane curve C with k (curvature) > 0 that is not convex. can someone please explain these concepts to me. How can you have a closed plane curve like this? Do you used signed ...
0
votes
0answers
28 views

sphere parametrization

We have the standard spherical surface parametrization in which one set describes geodesics (longitudes) and another ( latitudes/parallels). What parametrization may be possible for both sets to be ...
0
votes
2answers
96 views

A curve internally tangent to a sphere of radius $R$ has curvature at least $1/R$ at the point of tangency

Suppose $a$ is an arc length-parametrized space curve with the property that $\|a(s)\| \leq \|a(s_0)\| = R$ for all $s$ sufficiently close to $s_0$. Prove that $k(s_0) \geq 1/R$. So, I was going ...
1
vote
2answers
46 views

surface presentation

Given the following group and presentation, how could I go about showing if there exists a compact surface with that fundamental group? The group is $\big \langle a, b, c, d, e $ $\mid$ ...
1
vote
1answer
40 views

Total variation as surface area smooth functions of two variables.

I learnt we have different definitions for the total variation for functions of the form $f:\mathbb{R}^2\to\mathbb{R}$ which are in some way analogous to the total variation of functions of one ...
1
vote
0answers
46 views

What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$
0
votes
1answer
13 views

Equation for multiple peak surface to test particle swarm optimising algorithm

i have developed a particle swarm optimisation algorithm that i am running some tests on. It is able to solve simple equations like this: $x^2 + y^2 + 300y - 254x + 3$ with only one optimum but ...
1
vote
3answers
55 views

Proof that this surface is of revolution

I have a surface with parametric equation $$\mathbf{x}(u,v)=(u\cos(v),u\sin(v),u^2),$$ $u$ is any real number, $v$ is between $0$ and $2\pi$. I don't know how to show that this is surface of ...
0
votes
0answers
31 views

lifting loops on surface with abelian fundamental group for decrease their self-intersection number

I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian: Lemma: Let $M$ be a compact orientable ...
0
votes
1answer
48 views

Classical presentation of fundamental group of surface with boundary

It is well known fact about fundamental group of orientable compact surface: Letting $g$ be the genus and $b$ the number of boundary components of surface $M$. There is a generating set ...
2
votes
1answer
946 views

Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
2
votes
1answer
24 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
2
votes
0answers
64 views

Finding the leftmost, rightmost, top, and bottom, points, on a surface, of a sphere.

So I'm making a 3D game, and the player is inside a glass sphere. I'm projecting a bunch of points onto the sphere, and I need to find the leftmost, rightmost, topmost, and bottommost points, so I can ...
6
votes
2answers
56 views

1-dimensional foliation on a surface

Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary?
3
votes
1answer
47 views

Why does every noncompact orientable surface have a complex structure?

There is a high-powered proof of the fact that orientable noncompact surfaces have free fundamental group here that invokes the ability to put a complex structure on any such surface. But why should ...
0
votes
1answer
21 views

How do I find the ridges and valleys given a surface elevation function

Given a surface with a single elevation value for every x and y how can I find the places where the isoelevation contours have the tightest bends? And how can I differentiate between bends that are ...
1
vote
1answer
61 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
3
votes
1answer
68 views

Relationship between Surface Area and Volume

Question: Is there a general relationship between surface area and volume analogous to the below examples? Example 1. Consider a ball $B$ centered at the origin of a spherical coordinate system. The ...
0
votes
1answer
56 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
0
votes
1answer
24 views

Check if a point is within a quadratic surface (with arbitrary rotation)

Is there a general way to check whether a point is on a quadratic surface given that the principal axes do not need to coincide with the coordinate axes and that the quadric's centroid does not need ...
0
votes
0answers
45 views

Strain mapping sphere to plane

I need general indications or guidance. I do not know how to map a surface $ z = \sin (\pi x) \sin (\pi y), (x,0,\pi), (y,0,\pi)$ to a unit square. Nor do I know how to map a quadrant of a unit ...
1
vote
0answers
46 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
4
votes
0answers
41 views

Property that defines Quadric Surface

The book < Geometry and the Imagination > (written by David Hilbert) introduces a property of a Quadric Surface without a proof. Property : The cone consisting of all the tangents from a ...
2
votes
1answer
66 views

What is the meaning of $d\vec S$ in a surface integral?

Can someone explain if I have a surface $z= 9-x^2-y^2$ What would $\vec{n}$ be? What would $d\vec{S}$ be? Why is $d\vec{S}$ $(2x,2y,1)$ and not $(2x,2y,1)/\sqrt{4x^2+4y^2+1}$? Thanks!
0
votes
0answers
32 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
1
vote
1answer
38 views

Describing Bézier surfaces

I'm having some trouble with Bézier surfaces and I was hoping someone could help me. Question is rather simple: lets say we have 2 Bézier curves with control points: P00,P10,P20,P30 and second ...
0
votes
1answer
24 views

Optics surface equation to quadric form

This should be straightforward, but honestly I forgot even the names Google for... I've got a surface description in this form (what is it called?): $$z=\dfrac{cr^2}{1+\sqrt{1-(1+k)c^2r^2}},$$ ...
21
votes
2answers
605 views

Convex surface on which any two points $a,b$ can be joined by a curve of length $(\pi/2-\epsilon)|a-b|$

I am trying to solve an exercise on page 13 of the book Metric structures on Riemannian and non-Riemannian spaces by Gromov. Construct a closed, convex surface $X$ in $\mathbb R^3$ such that any ...
0
votes
0answers
110 views

Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...
2
votes
1answer
40 views

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface of degree greater or equal to 4?

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface in $\mathbb{CP}^3$ of degree greater or equal to 4? Someone told me that the answer is ...
0
votes
0answers
24 views

Is there an algorithm that, given a point cloud, infers an optimal wireframe (surface) structure?

I have a point cloud that I would like to convert to a surface, in the form of a wireframe lattice structure. This means, from a sequence of 3D points (x,y,z), obtaining three 2D matrices X,Y,Z of ...
1
vote
0answers
34 views

Geometric definition of the stable commutator length

In his book, D.Calegari proves the equivalence of the algebraic and geometric definitions of stable commutator length (Proposition 2.10, p. 15). I actually have some difficulties in understanding the ...
0
votes
0answers
33 views

Area of the portion of the cylinder $x^2+y^2 = 9$ for which $-1 \leq z \leq 2$ and $ 0 \leq \theta \leq \pi/2$

Problem: Find the area of the portion of the cylinder $x^2+y^2 = 9$, for which $-1 \leq z \leq 2$ and $ 0 \leq \theta \leq \pi/2$ I first solved this by parametrizing the surface. $x = 3\cos(u)$ , ...
3
votes
1answer
81 views

simple closed curve is nullhomologous iff is separable

A simple closed curve $\gamma$ in an orientable genus $g$ surface $M$ is nullhomologous if and only if $M \setminus \gamma$ consists of two connected components, one of which is a surface $N$ with ...
1
vote
2answers
53 views

Verification of the Stokes theorem for the surface that is a part of a cone

Let $S$ consist of the part of the cone $z=(x^2+y^2)^{1/2}$ for $x^2+y^2\leq9$ and suppose $${\bf A}=(-y,x,-xyz).$$ Verify that Stokes theorem is satisfied for this choice of $\bf A$ and $S$. In ...
3
votes
1answer
55 views

Sketching a surface

If $${\bf F}=2y{\bf i}-z{\bf j}+x^2{\bf k},$$ and $s$ is the surface of the parabolic cylinder $y^2=8x$ in the first octant, bounded by the planes $y=4$ and $z=6$, evaluate $$\int_S{\bf ...
-1
votes
3answers
80 views

How many feet of rope to wrap a column

A heating pipe in my bathroom measures 105" in height. It is 8" in circumference (so about 2.55" diameter). I want to wrap it with a 1/4" thick rope. How many feet should I buy? (All measurements in ...
0
votes
1answer
49 views

Areas of tetrahedron surfaces - how to calculate?

Reading up on Cauchy's stress theorem, I have stumbled over the so-called Cauchy tetrahedron, which is an important part of the theorem's proof. The following is cited straight from Wikipedia, but a ...
4
votes
1answer
52 views

How do I find the area of a triangle, in 3D, that lies between two planes, z = A and z = B

Very simple problem to conceptualize, but I don't have a good mathematical solution. I have a triangle with P0 = (x0, y0, z0), P1 = (x1, y1, z1), and P2 = (x2, y2, z2). The triangle represents part ...
2
votes
1answer
32 views

Parametrizing to Calculate Flux

Evaluate the flux of $\mathbf{f}$ across the oriented surface $\Sigma$ by computing the surface integral $\iint_{\Sigma} \mathbf{f} \cdot d\sigma$, where $\Sigma$ is the surface $z=xe^y$ for $0 \leq x ...
1
vote
1answer
25 views

Show a smooth map from a compact, connected, orientable surface to a cyllinder has singular derivative at 2 points.

Let $M$ be a compact, connected, orientable surface in $\mathbb{R}^3$. Let $N$ be the cyllinder in $\mathbb{R}^3$ defined by $x^2+y^2=1$. Suppose $f:M\to N$ is $C^{\infty}$. Show that $f_*:TM\to ...
1
vote
2answers
71 views

Notation for Surface Integral in $\mathbb{R}^3$

Recently, a paper of mine got accepted, but the reviewers are struggling with the (in my view) standard notation for surface integrals in $\mathbb{R}^3$: Let $\Gamma \subset \mathbb{R}^3$ be a ...