Surface is a two-dimensional submanifold of three-dimensional Euclidean space.

learn more… | top users | synonyms

0
votes
0answers
12 views

Computing the approximate area of an iso surface

I'm searching for an approximation to the surface area of isosurfaces. They are defined by a constant value v in a scalar field. The scalar field is defined by placing n vectors in k-space such that ...
0
votes
0answers
14 views

Determining the unit normal field of a paraboloid $P$, and integrating a vector field over $P$

Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are ...
3
votes
1answer
360 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
0
votes
1answer
28 views

Computing the approximate or exact area of an isosurface

The isosurfaces I'm reading about are defined by a constant value v in a scalar field. The scalar field is defined by placing n vectors in k-space such that ...
0
votes
0answers
30 views

Proof of relation between normal of a surface and principle curvatures of surface.

If F(x,y,z) is a scalar function. Then how to prove that, $$\nabla . n = K_1 +K_2$$ where n is normal to surface of constant $F$ given as $$n=\frac{\nabla F}{|\nabla F|}$$ $K_1$ and $K_2$ are ...
1
vote
2answers
16 views

Geodesics on surfaces of revolution about z axis with negative curvature

This is a question in differential geometry of surfaces that I could not do We are given S a surface of revolution about the z axis with everywhere negative Gaussian curvature. We are to show that the ...
1
vote
0answers
24 views

Does this base change yield another dominant morphism?

Here's something that seems to be true, or at least I hope it to be true, but I'm unable to prove it: Let $S$ be a $k$-rational surface and $B$ a curve, both projective, smooth and geometrically ...
1
vote
1answer
13 views

Help with understanding a proof of compact surface having an elliptic point

In my studies of differential geometry from do Carmo's book, I have come across a very nice claim which states that a regular compact surface has an elliptic point that is a point with positive ...
1
vote
0answers
22 views

determining equation of a surface

I was wondering if there is a way to determine the equation of a surface if three R2 linear equations are known. I work in a research lab that produces a lot of correlation equations (mx+b), and we ...
1
vote
0answers
30 views

Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that ...
2
votes
1answer
13 views

Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $ \gamma (t) = (sin(t)+2,0,t) $ on XZ plane for $ t \in R $, ...
3
votes
2answers
46 views

How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...
2
votes
1answer
20 views

Proving subset of regular surface - hyperboloid - is a regular surface

I have stumbled upon this in differential geometry dealing with regular surfaces: We define the following surface (a hyperboloid) as $ K = \{ (x,y,z) \in R^3 | x^2+y^2-z^2 = 1 \} $ and ...
0
votes
2answers
32 views

Can a surface of revolution be built from a self-intersected curve?

I'm reading "Differential Geometry of Curves And Surfaces" of Manfredo Do Carmo. There's a point in his book about Surfaces of Revolution which confuses me a lot. Here is the part: The part ...
0
votes
1answer
30 views

Understanding the first fundamental form of a surface, how the parametrization doesn't matter.

The following is an excerpt from Pressley's Elementary Differential Geometry on the definition of the first fundamental form. However, there are some parts of this concept that I'm unclear about. It ...
0
votes
2answers
49 views

How to find the surface area of a spherical cap by integration?

I don't really understand how they derived the formula in the following picture. The aim is basically to find the formula for the surface area of a spherical cap. Why do you differentiate the ...
0
votes
0answers
35 views

Proof of Theorem 3.2 - Elementary Differential Geometry, O'Neil.

I am going through Elementary Differential Geometry by O'Neil, and I am at Theorem 3.2 on page 151. O'Neil comments that a rigorous proof of this theorem requires the methods of advanced calculus, and ...
1
vote
1answer
38 views

Gradient in terms of first fundamental form

In Do Carmo's Differential Geometry of Curves and Surfaces, I'm having a quite hard time trying to solve Excersise 14 on pages 101-102. He defines the gradient of a differentiable function $f:S\to ...
0
votes
0answers
14 views

Minimal surface with radially symmetrical function

The following image is from the book "Regularity Theory for Mean Curvature Flow", by Ecker. I consider the plateau problem, whose goal is to solve minimal surface given fixed boundary values. In ...
0
votes
0answers
16 views

Area of ​​the surface of revolution of the ellipsoid

I need to find the surface area of an ellipsoid using the equation of an ellipse. I believe my calculations are correct but the formulas I meet on the Internet are complex and have arcsin or arctan in ...
0
votes
0answers
64 views

Identification space of square. Net, triangulation and surface classification

Space Z is made as an identification space of unit square $Q=${$(x,y) | 0\leq x, y \leq 1$} by making the following identifications: $ (0,y)$~$(1,y) $ for all $0\leq y\leq 1 $, $ ...
0
votes
1answer
44 views

The normal curvature is bounded by the principal curvatures.

Let the inclusion $i:S\subset\mathbb R^3$ be an immersion of a surface $S$, and let $N:S\to \mathbb R^3$ be a local Gauss map. Let $a:I\to S$ be an arc length parametrized curve, with $a(0)=p$ and ...
1
vote
1answer
704 views

Homeomorphism between $\mathbb{C}/L$ and $S^1\times S^1$

We have the lattice $L = \{m_1w_1 + m_2w_2 \mid m_1, m_2 \in \mathbb{Z}, w_1, w_2 \in \mathbb{C}\}$. We want to construct a homeomorphism between $\mathbb{C}/L$ and $S^1\times S^1$. I've read that ...
0
votes
2answers
43 views

Examples of surfaces

I have to find an example of a surface of revolution excluding a sphere and a cone. Is $\sigma(x,y)=(\cos x, 5, x^2+y^2)$ such an example? $$$$ I also have to find an example of a surface the ...
0
votes
0answers
14 views

Surface Normal To Euler Angles

I am working on an application to extract positions from a point cloud. My point cloud has three axis X, Y, Z I am using PCA to generate a surface normal from a section of a surface so I end up with ...
0
votes
0answers
12 views

Surface Area for a Curve Rotated Around the Y-Axis

I tried finding the surface area of a function rotated about the y-axis but I don't trust my answer. If I am looking for the surface area of a function y=f(x) rotated about the y-axis. $$S= 2\pi ...
0
votes
1answer
42 views

Laplace-Beltrami of the Gauss map

I'm looking for the proof of very nice identity about the Laplace-Beltrami operator of the Gauss map $N$ of a regular surface in $\mathbb{R}^3$ given by a patch $X$. I want to show that $$\Delta N = ...
0
votes
1answer
41 views

The parameter curves are asymptotic curves

I am looking at the following exercise: Let $p$ be a hyperbolic point of a surface $S$. Show that there is a patch of $S$ containing $p$ whose parameter curves are asymptotic curves. Show that ...
1
vote
1answer
44 views

Condition to be conformal

I am looking at the following exercise: Let $\Phi : U \rightarrow V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$. Write $$\Phi (u, v)=(f(u, v), g(u, v))$$ where $f$ and $g$ are ...
5
votes
4answers
293 views

How can we find geodesics on a one sheet hyperboloid?

I am looking at the following exercise: Describe four different geodesics on the hyperboloid of one sheet $$x^2+y^2-z^2=1$$ passing through the point $(1, 0, 0)$. $$$$ We have that a curve ...
1
vote
2answers
65 views

Which is the intersection?

I am looking at the last question of the following exercise: $$$$ Which exactly is the intersection of any surface from one family of the triply orthogonal system with any surface from another ...
2
votes
1answer
67 views

Is this a misprint in Do Carmo's 'Curves and Surfaces'?

I'm reading the following section from the book 'Curves and Surfaces' by Do Carmo, but I'm stuck and after having gone over this like 10 times I'm starting to think it must be a misprint. The problem ...
0
votes
2answers
52 views

Equation of a cone

Find the equation of the cone whose vertex is at the origin and whose directing curve is given by the equations: $$\begin{cases} x^2-2z+1=0 \\ y-z+1=0\end{cases} $$ We know that an eliptic cone is ...
0
votes
3answers
59 views

Orientable surface

Suppose that two smooth surfaces $S$ and $\tilde{S}$ are diffeomorphic and that $S$ is orientable. I want to prove that $\tilde{S}$ is orientable. $$$$ Since $S$ and $\tilde{S}$ are ...
11
votes
0answers
118 views

If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$? [migrated]

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
1
vote
1answer
83 views

The surface is an open subset of a sphere

I am looking at the following exercise: $$$$ Could you give me some hints how we could show that? Do we use the matrix of the Weingarten map with respect to the basis $\{\sigma_u,\sigma_v\}$ ...
0
votes
1answer
26 views

How to minimize the surface area taken by a cylinder?

In my math class, we are working on Geometric Optimization problems. We have to create an equation, and then solve for one variable, in terms of another variable. Then, using an expression, we find ...
2
votes
1answer
68 views

The straight lines are contained in $S$

I am looking at the following exercise: The hyperboloid of one sheet is $$S=\{(x,y,z)\in \mathbb{R}^3 \mid x^2+y^2-z^2=1\}$$ Show that, for every $\theta$, the straight line $$(x − z) \cos \theta = ...
1
vote
1answer
56 views

Topological surface covered by hexagons and heptagons

I've found an interesting exercice that I don't know how to approach. It goes like this. We have a topological space which is Hausdorff, compact, connected and locally homeomorphic to ...
0
votes
1answer
27 views

How could we show that these are perpendicular?

I am looking at the following exercise: Suppose that the first fundamental form of a surface patch $\sigma (u, v)$ is of the form $E(du^2 + dv^2)$. Prove that $\sigma_{uu} + \sigma_{vv}$ is ...
0
votes
0answers
25 views

Is this (self intersecting) surface considered one sided?

I wanted to apply stokes theorem on a curve (in black) that possibly looks like the seam of a tennis ball. I make a surface by drawing a line from the origin to each point of this curve. I get a ...
0
votes
0answers
42 views

Unit normal of the surface $S$

We have that $$\sigma (u,v)=\gamma (u)+v\delta (u)$$ and $$K=\frac{-(\dot\delta \cdot \textbf{N})^2}{EG-F^2}$$ I want to show that if $\gamma$ is a curve on a surface $S$ and $\delta$ is the unit ...
0
votes
2answers
786 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
1
vote
0answers
42 views

Equation of a 3D curve shaped like a logarithmic spiral

I'm not exactly certain of the mathematical description of this surface (if I were I wouldn't have a question), but I basically want to make a "3D spiral" which is basically a sine wave "wrapped" ...
2
votes
1answer
175 views

Fundamental forms of constant mean curvature surfaces

For surfaces of constant mean curvature in $E^3$, prove that either they are all-umbilic-points surfaces or their fundamental forms can be represented as following: I $=\lambda(u,v)(dudu+dvdv)$ ...
3
votes
0answers
24 views

Reference to an atlas of curves and surfaces?

I remember at more than one university math department there being a set of glass cabinets with a number of physical models of surfaces. They were all algebraic varieties on the reals (of limited ...
3
votes
1answer
56 views

Common surface between two equations

What is common surface between: $(x+5)^2+z^2=y$ and $z^2+y^2=25$ ? I have found that at the $XY$ plane the common surface is a hyperbola, but it cannot be right because at the paraboloid there isn't ...
2
votes
1answer
32 views

When does there exist a isometric transform between the surfaces $S$ and $\widetilde{S}$?

Suppose there are two $E^3$ surfaces, $$S:\mathbf{r}(u,v)=(au,bv,\frac{au^2+bv^2}{2})$$ ...
1
vote
0answers
71 views

Open subset of a plane [duplicate]

Suppose that the second fundamental form of a surface patch $\sigma$ is zero everywhere. How can we prove that $\sigma$ is an open subset of a plane? The second fundamental form of a surface patch ...