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Questions tagged [surfaces]

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

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When to parameterize to find equation of tangent plane and normal line?

I'm working through some problems asking to find the equation of a tangent plane and the normal line to the surface. I notice that some example questions parameterize the curve before solving and ...
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The sphere with three ends?

The preceding image take form Matthias Weber's Classical Minimal Surfaces in Euclidean Space by Examples notes is called the sphere with three ends. But what does it have to do with a sphere and why ...
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Cross Product in 3D Cylindrical or Spherical coordinates

When i take a cross product of two vectors (Cylinder base in x,y plane) for example d(phi)cross d(z), how do i know if the resultant Vector protrudes out of the page or goes inside? I know the right ...
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37 views

Question on what constitutes surfaces in R3

Our lecturer gave us the following definition of a surface in $\mathbb{R}^3$: $\Gamma$ Is a surface in $\mathbb{R}^3$ if for all $y\in \Gamma$ there exists a coordinate patch $\sigma : D\subset\mathbb{...
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Surface area and volumes, please solve [on hold]

The length of the diagonal of a cuboid is $5\sqrt{5}$ cm and the sum of its length, width and height is $19$ cm. Find its surface area.
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39 views

How to calculate the average projected area of a circle?

As a follow up to Average projected area in higher dimension, I started to think about a variation of the original post. Let's visualize a circle with radius $1$ in $3D$ space and cast its shadow on ...
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39 views

Can two spheres be thought of as a single surface?

Let's stick to $\mathbb{R}^3$ for the sake of simplicity. Say I've got a sphere of radius $1$ centered at the origin and another with the same radius centered at $(5,6,7)$. Can one think about them as ...
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Average projected area in higher dimension

I read the article The Average Projected Area Theorem – Generalization to Higher Dimensions and learnt that Augustine Cauchy proposed in the 19th century that the average projected area of any convex ...
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67 views

Am i correct in the way I am deciding whether or not subsets of $\Bbb R^3$ are surfaces or not?

I just want to make sure that there isn't any gaps in my reasoning ( or flat out mistakes!) before I try to learn anymore about classifying subsets of $\Bbb R^3$ as surfaces, so to that end ..... ...
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How to identify a face is circle using it's vertices [closed]

Is there a way to identify whether a face is of circular shape and it's center?. All I have is the face and it's vertices.
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39 views

Show there can be no co-ordinate patch at this point

I am attempting to prove that the subset of $\mathbf{R}^3$ satisfying $x^2 + z^2 = y^2$ is not a surface, where a surface is a subset of $\mathbf{R}^3$ for every point in which there is a co-...
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Helicoid is developable surface

Please help me to prove that helicoid whose parametric equation is given by $$x=u\cos v, y= u\sin v, z=pu$$ is developable, where $p$ is a constant and $u,v$ are the curvelinear coordinates of the ...
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Proving that a portion of a surface is developable

Suppose that a coordinate chart can be chosen for a portion of a surface such that the coordinate lines $u$=constant or $v$=constant are geodesics. Prove that the line element be brought into the ...
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Gaussian curvature of ruled surfaces

Let $c: I \rightarrow \mathbb{R}^3$ be a regular curve, $V: I \rightarrow \mathbb{S}^2$ a vector field and $a < b$. Then we call $$ f: (a,b)\times I \rightarrow \mathbb{R}^3, \quad f(s,t):= c(t) +...
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Closed 3D curve with linearly independent gradient

A curve $C$ in 3D can be described by the intersection of two surfaces, that is, $C=\{p \in \mathbb{R}^3: f_1(p)=0, f_2(p)=0\}$. $f_1(p)=0$ and $f_2(p)=0$ are the implicit representations of two ...
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What is the equation of the helicoidal staircase-like surface

I don't really have much to add to this question, except the picture I would think there has to be an analytical form for this surface. Also, if one could help build a vector, normal to it as a ...
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$I = \int_{- \infty}^{\infty} \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x} $ should not diverge

I'm having trouble evaluating this integral: $$I = \int_{-\infty}^\infty \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x} $$ where $\delta(x)$ is Dirac delta function, $\mathbf x$ is the real $n$-...
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prove ratio between hypervolume and surface volume of n-sphere is optimal

The ratio of the surface volume to hypervolume of an $n$-sphere is $\frac{r}{(n+1)}$ . Is there a way to prove that this is the highest ratio possible without appealing to the fact that $n$-spheres ...
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1answer
28 views

Calculate the flow of $\vec{F}(x,y,z) = 3x\vec{i} + 3y\vec{j} + z^5\vec{k}$ over the surface $x^2 + y^2 = 25$ for $0 \leq z \leq 1$

Calculate the flow of $\vec{F}(x,y,z) = 3x\vec{i} + 3y\vec{j} + z^5\vec{k}$ over the surface $x^2 + y^2 = 25$ for $0 \leq z \leq 1$. The normal considered points inside. The book uses cylindrical ...
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Boundary of polygonal presentation homeomorphic to bouqet of circles

Suppose $P$ is a regular 2n-gon, with sides in pairs to give a surface. I want to show that the image under quotient topology of boundary of this polygon is homeomorphic to wedge of n circles. I want ...
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Is the “immersed proper” hypothesis necessary in Half-space Theorem?

I'm using the following version of Half-space Theorem: $\textbf{Theorem}$(Half-space) A connected, immersed proper, nonplanar minimal surface $M$ in $\mathbb{R}^3$is not contained in a ...
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What’s the name of this surface?

I’m trying to find the name of this surface: $X(u,v)=(u-1/3u^3+uv^2, v-1/3v^3+vu^2, u^2-v^2)$ Can anyone help? Thank you!
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Whether preserving inner products follows from preserving intrinsic distances

I know that if a map F between two surfaces preserves inner products of tangent vectors and is 1-1 and onto, then it must preserve intrinsic distances, but I’m not sure whether the inverse is true. ...
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Fundamental polygon of surface with boundary

I was playing with some shapes recently and came across a surface $M$ with the following fundamental polygon: link This has surface word $$a c b f_1 a^{-1} c^{-1} f_2 b^{-1} f_3,$$ where the $f_i$ ...
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Representing a Multiple of the Fundamental Class of a Surface

Let us denote a connected, oriented surface of genus $g$ by $\Sigma_g$. It is easy to see that if we have a map of degree $1$ from $\Sigma_g$ to $\Sigma_h$ then $g\geq h$. Suppose now that $g\geq 2$ ...
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54 views

show that the parameterization of a torus is ____________ [closed]

I need help on a homework problem. a) A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about the z-axis in space. See accompanying figure. If C has a radius r>0 ...
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1answer
37 views

Derivation of formula for surface integral

Let $S$ be a surface given by the parametrization $\vec{r}(u,v)= x(u,v)\vec{i} + y(u,v)\vec{j} + z(u,v)\vec{k}$ for $(u,v)\in D$. If $f(x,y,z)$ is a function defined on $S$, then the surface integral ...
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191 views

How to calculate the surface area of parametric surface?

Suppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2\leq 1$. My homework is forcing me to use the parameterization $$\textbf{r}_1(s,t)= <s\...
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Calculating torsion of an asymptotic curve with nonzero curvature

I came across this problem when studying Gauss Map: Show that $\tau^2=-K$ on an asymptotic curve. Here $\tau$ is torsion of the asymptotic curve $\pmb r(u(s),v(s))$(with curvature $\kappa$ nonzero) ...
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If two minimal surfaces equal to neighborhood then they are equal

Does anyone know if there is any result that says "if two minimal surfaces equal to neighborhood then they are equal"? My teacher said that you think the result is true, but I do not find this result ...
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81 views

Which surfaces does a compact orientable surface with boundary cover?

Suppose I have a bounded, orientable genus 5 surface with 4 boundary circles. Is there a way to determine what surfaces it covers? First, I know that there is a covering map from the closed ...
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Question on judging a regular surface in differential geometry

This is the proposition and proof saying that if we know that S is a regular surface and x is a candidate of parametrization which satisfies 1)differentiable 2)surjective differential map 3)continuous ...
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31 views

A surface geometry such that all lines only and always intersect at right angles.

I'm looking for any 3D surface such that given any two lines on the surface, they intersect and only intersect in right angles.
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Understanding the definitions of Embedded Surface and Locally Parametrised Embedded Surface

I recently learned about the definitions of Embedded Surface (ES) and Locally Parametrised Embedded Surface (LEPS) in a lecture of mine. We defined them as follows: ES: A regular parametrisation $f$ ...
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38 views

Show that a curve is a geodesic on a surface.

Let $\alpha:[0,1]\rightarrow S^2$ be the curve $\alpha(t)=\left(\cos(e^t),\sin(e^t),0 \right)$. Show that $\alpha$ is a geodesic on $S^2$, but in the latitude-longitude parametrization of $S^2$, $\...
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Show that $C_{c}\cap \left\{ z=0\right\} =\partial B$

Let $d_{0}>0$, $R>0$ and $p_{3}\in \left( 0,d_{0}\right) $ given. Consider $C_{c}$ a any catenoid of radius $c>0$ centered $p=\left( 0,0,p_{3}\right)$ in $% %TCIMACRO{\U{211d} }% %...
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43 views

Ruled surface with negative/zero gaussian curvature.

So a surface $S \in \mathbb{R}^3$ is ruled if through each point $p$ there is a line in $\mathbb{R}^3$ entirely contained in $S$. Show that the line through $p$ lies along an asymptotic direction. ...
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25 views

Finding the surface area of a solid of revolution

I'm given the function $x=\frac{1}{15}(y^2+10)^{3/2}$ and I need to find the area of the solid of revolution obtained by rotating the function from $y=2$ to $y=4$ about the $x-axis$. I've tried ...
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1answer
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10-gon as connected sum of tori or projective plane

We're supposed to use cutting and gluing to find out, whether a given surface is a connected sum of tori or a connected sum of $\mathbb{R}P^2$. Now, there's one surface I'm stuck with. It is a 10-gon ...
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Are there convex surfaces with negative Gaussian curvature?

I've just started reading about convex surfaces and there are a few things which are breaking my intuition. According to this page: "Minkowski proved the existence of a closed convex surface with ...
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1answer
25 views

Differential Geometry of Surfaces in Heigher Dimensions

I am looking for a reference that discusses "surfaces in higher dimensions." Specially I need a book or a paper about analogue of fundamental forma in $\mathbb{R}^4$ and other higher dimensions. ...
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35 views

How to tell if an implicit surface is connected

Let $S_c := \{(x,y,z) \in \mathbb{R}^3 \,| \,\, z(z+4) = 3xy + c \}$. Find all values of $c\in\mathbb{R}$ such that: $1) \,\, S_c$ is a regular surface in $\mathbb{R}^3$ $2) \,\, S_c$ is connected $...
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How to determine number of genus from given equation of a surface?

In "Topics in Differential Geometry" book of Peter W. Michor page 15, he gives an example of a compact surface of genus $g$ as follows: Let $f(x) := x(x − 1)^2(x − 2)^2\cdots(x − (g − 1))^2(x − g)$....
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1answer
26 views

Removing a closed subset from a surface gives a surface

In Ethan Bloch's "A First Course in Geometric Topology and Differential Geometry" the definition of a surface is the following: A subset $Q \subseteq \mathbb{R}^n$ is called a surface if each point ...
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35 views

Calculate integral of Gaussian Geometry

I have this excercise but I am having problems becouse I dont know how to use the Gauss-Bonnet theorem. If $r \in \mathbb{R}^{+}$ and $\Sigma_r$ is given by: $$\Sigma _ { r } = \left\{ ( x , y , z ) \...
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29 views

How to bisect a quarter of a cheese to get minimal surface on the two halves?

The cheese is kind of "trappista cheese" ( https://en.wikipedia.org/wiki/Trappista_cheese ), which has a shape of "circle sect based prism". For simplicity it has a height equal to the radius of the ...
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Topological definition of curve and surface/planar regions

Is it possible to define curve and surfaces (or as a special case of the latter planar regions) using only topology concepts? I would define a curve as any set whose neighborhoods are equivalent to ...
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Using fundamental polygons to prove $PR^2\# T^2 = PR^2\# PR^2 \# PR^2$.

Background Let $P^2$ denote the real projective plane, and $T^2$ the torus. These generate a monoid where the operation is the connected sum. This monoid is abelian, so for notational brevity, we ...
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42 views

What is the Euler characteristic of the hyperboloid of one sheet

I would like to know what's the Euler characteristic of the hyperboloid of one sheet. I know that $2-2g$ is the Euler characteristic where g is the number of "holes". Using this fact, Euler ...
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3answers
608 views

Count the number of shapes in a polyhedron.

So this is a question that was asked in the International Kangaroo Math Contest 2017. The question is: The faces of the following polyhedron are either triangles or squares. Each triangle is ...