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

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Is there an efficient method to merge multiple ($2d$) areas into one?

Is there a mathematical method to merge multiple ($2d$) areas into the smallest possible total area? Example: ...
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Possibility of regular surface with specific first and second fundamental form matrices

I have met this in diff. geometry class which states: We are to determine if there exists a regular surface in $ R^3 $, $ S = f(u,v) $ with fundamental forms as follows: $ I = \begin{bmatrix} ...
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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 ...
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14 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 ...
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33 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
14 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 $, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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66 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 $, $ ...
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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 ...
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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 ...
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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 ...
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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 = ...
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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 ...
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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 ...
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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 ...
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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 ...
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53 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 ...
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3answers
60 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 ...
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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 ...
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1answer
27 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 ...
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1answer
57 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 ...
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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 = ...
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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 ...
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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 ...
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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 $\{\sigma_u,\sigma_v\}$ ...
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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 ...
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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" ...
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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 ...
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1answer
80 views

What's the name of the surface and Is it a $C^2$ smooth surface? [duplicate]

what's the name of the surface? Is it a $C^2$ smooth surface? Its implicit equation is: $(x−2)^2(x+2)^2+(y−2)^2(y+2)^2+(z−2)^2(z+2)^2+3(x^2y^2+x^2z^2+y^2z^2)+6xyz−10(x^2+y^2+z^2)+22=0$
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26 views

maximum number of faces with n lines

I was wondering a formula F(n) to guess the maximum number of faces made with n lines, for example: with 1 line, we cant create a face; F(n) = 0; with 2 lines, we also cant create a face F(n) = 0; ...
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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 ...
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When does there exist a isometric transform between the surfaces $S$ and $\widetilde{S}$?

Suppose there are two $E^3$ surfaces, $$S:\mathbf{r}(u,v)=(au,bv,\frac{au^2+bv^2}{2})$$ ...
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How can we calculate the unit normal $\textbf{N}$ of the sphere?

I want to show that the normal curvature of any curve on a sphere of radius $r$ is $\pm \frac{1}{r}$. $$$$ The normal curvature is $\kappa_n=\gamma '' \cdot \textbf{N}$, where $\gamma$ is a ...
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Compact surface

To check if the surface $x^2-y^2+z^4=1$ is compact, we have to check if the surface is closed and bounded. Could you give me some hints how exactly we check that? How can we check if it closed and ...
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81 views

Revolution of fractal

How to find the volume and surface area of a shape which made from revolution of Koch Snowflake? (I think the surface area will be an infinity, because length of the Koch snowflake is infinity.) And ...
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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 ...