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

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Finding a generating curve for a regular valued surface

Given a regular surface $X=\{(x,y,z):x^2+y^4+z^3=1\}$ and a point $p=(1,1,-1)\in X$ and a tangent vector $u = (2,-1,0)\in T_pX$ define a generating curve $\alpha (t):(-i,i)\rightarrow X$ for $u$ such ...
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Orientability of $ x(u,v)= \bigg(\bigg(1+v\cos\frac{u}{2}\bigg)\cos(u), \bigg(1+v\cos\frac{u}{2}\bigg)\sin u, v\sin\frac{u}{2}\bigg) $

Consider the map: \begin{equation} x: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}: (u,v) \rightarrow \bigg(\bigg(1+v\cos\frac{u}{2}\bigg)\cos(u), \bigg(1+v\cos\frac{u}{2}\bigg)\sin u, v\sin\frac{u}{2}\...
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Tangent plane of a surface at points with given gradient

So I'm stuck on the next problem: I need to find the tangent plane of the surface $$u=\ln\left( x+\frac{1}{y} \right)$$ at all the points where the gradient is equal to $$\nabla u=\hat i-\frac{16}{9}\...
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Proper name of a curve from “Vanishing Surfaces”

My question is maybe more about linguistics than maths... So, if you have a 3D surface that vanishes as a curve when projected on a 2D-plane (e.g. an axisymmetric surface projected to the r-Z plane). ...
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Is there a name for generalized ellipsoids?

In two dimensions, we have the following series of generalizations: circle $\rightarrow$ ellipse $\rightarrow$ smooth, convex, closed curve $\rightarrow$ smooth, simple, closed curve And in three ...
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50 views

Show that $(x\land y)z + (y\land z)x + (z\land x)y=0.$ where $x\land y=(x \times y)\cdot N$.

Let $P\subset \mathbb{R}^3$ be a plane through the origin and $N$ be a unit normal to $P$. For $x,y \in P$, set $x\land y=(x \times y)\cdot N$. Then for any three vectors $x,y,z \in P$, we have $$(...
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Given second fundamental form what is the geometric /topological invariant?

The Gauss Bonnet integrates the first and second forms into an elegant structure. But before that... If first fundamental form alone is given, a series of mutually bendables with isometric /intrinsic ...
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Given a first fundamental form, showing a particular second form cannot exist

If I have a first fundamental form $ \mathrm{d}u^2+\cos^2 u \mathrm{d}v^2$, I am trying to show that the second fundamental form cannot equal $f(u,v)\mathrm{d}v^2$ for a smooth function $f(u,v)$. I ...
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Circular cylinder $S=\{ (x,y,z) : x^2+y^2=1 \}$ can be covered with a single surface patch.

I somewhere found that we can take $U$ an annulus instead of a disc where $U=\{ (u,v): 0 < u^2+v^2 < π \}$. Can anyone please explain me that how a cylinder can be covered with a single surface ...
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25 views

Prove the boundary is a compact 1 manifold

A closed surface with boundary is a compact connected topological space $B$ with the property that each point $p \in B$ has an open neighborhood $U$ homeomorphic to either: $\{(x, y) \in \mathbb{R^2}|...
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28 views

Sketching the surface $z=\frac{x^2y}{3}$

I am trying to sketch the part of $x^2+y^2=9$ which lies in the first octant between the surfaces $z=0$ and $z=\frac{x^2y}{3}$. I understand that $x^2+y^2=9$ is a cylinder with radius three, ...
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19 views

Orthogonality of ruling and directrix of a ruled surface

Let $M$ be a ruled surface of $\mathbb{R}^3$ with a regular parametrization given by: $$x(u,v)= \alpha(u) + v\beta(u)$$ where $\alpha' \neq 0$ and $ ||\beta || = 1$. I want to show that $<\alpha'...
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Confusion regarding terminology in Pressley's E.D.G

Here are two definitions taken from page $77$ of Pressley's Elementary Differential Geometry - $2$nd edition. Definition $4.2.1$ A surface patch $\sigma: U \to \Bbb R^3$ is called ...
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46 views

Is the quotient space obtained by identifying the poles of a sphere homeomorphic to a closed surface?

I'm interested in the quotient space of $S^2$ obtained by identifying the poles, and in particular whether it is homeomorphic to a closed surface. I'm pretty sure its homotopic to one, just by ...
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72 views

Parametric Surface

A surface is given by $$r(u,v) = \langle u, v^2, uv\rangle$$ (a) Evaluate the unit normal vector, $\vec n$, to the surface at the point corresponding to $u=2$ and $v=1$. I've done this by ...
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How to derive 2D equation representing minimums of constrained 3d equation?

I have a 3D (multivariate) function f(x,y) which can be represented as a surface with constraints. When the surface is viewed from the side (as below), such that the Y axis is not visible, there is ...
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23 views

Finding the surface area of $S={(r\cos\theta,r\sin\theta,3−r):0\leq r \leq 3, 0\leq \theta\leq2 \pi }$

So we've been given this set: $$S={(r\cos\theta,r\sin\theta,3−r):0\leq r \leq 3, 0\leq \theta\leq2 \pi }$$ and I can see that this is part of a cone but I'm not too sure how to find the surface area....
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Regular parametrization of a surface is conformal iff it preserves angles.

Can anyone give me some hints of how to start the proof, because I have no idea where to start. I know if a parametrization is conformal, then $E=G$ and $F=0$, where E,F,G are values in the first ...
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27 views

Difference between a Möbius Strip and a Simple Surface

I am trying to distinguish between a Möbius strip and a surface that has no separations, holes and a connected boundary (homeomorphic to a disk or a half-sphere). Since a Möbius strip also has all the ...
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38 views

Parallel surface

For a regular surface $\mathbf{x} = \mathbf{x}(u,v)$ Define $\mathbf{y}(u,v) = \mathbf{x}(u,v) + t \mathbf{N} (u,v)$ where $\mathbf{N}$ is the unit normal of $\mathbf{x}$ How could I show the ...
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What is the equation for the walls of a 3D cylinder?

If for example I have the circle $x^2 + y^2 = 4$ in the $x$-$y$ plane, and I want to extend it upwards into the $z$ dimension, how would I write the equation for the circular walls in terms of $z$?
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identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the ...
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25 views

Euler's formula about graphs embedded in $\mathbb{R^2}$

State and prove Euler's formula about graphs embedded into $\mathbb{R^2}$ I know that if we suppose $ G $ is a finite connected graph drawn on the surface of a sphere $ S^2 $. Then the complement ...
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Prove Euler characteristic satisfies $\chi(X \times Y)=\chi(X)\chi(Y)$ for polyhedra $X$ and $Y$

Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$ I know that for polyhedron $P$ which is homemorphic to a ...
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58 views

Finding the derivative of an equation

I am currently doing an investigation in which I am required to design the dimensions of a juice box (can be cube/cuboid) which has the least possible surface area that can hold 200 ml of juice. I ...
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40 views

Optimisation of a juice box: finding the least possible surface area that can hold the most volume

I have an investigation which requires me to design the dimensions of a juice box (cuboid) which has the least possible surface area that can hold the most volume. I am not sure as to how I should ...
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43 views

Definition of a regular surface

Here is the definition of a regular surface from Differential Geometry of Curves and Surfaces by Manfredo do Carmo: A subset $S ⊂ \mathbb R^3$ is a regular surface if, for each $p ∈ S$, there ...
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Proper application of Surface Area of Revolution formulae?

I'm a little confused about how to properly apply the integrals used to calculate the area of a surface of revolution. Find the exact area of the surface obtained by rotating the curve about the x-...
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Ray intersecting a quad mesh

I am trying to solve the math behind rendering a quad-mesh surface. MatLab for instance can take a regularly spaced (x,y) grid with arbitrary third-dimension (z) values, treat each four neighbouring ...
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How can we find a normal to this plane? [closed]

A vector is said to be normal to a surface (plane) if it is perpendicular to that surface. Consider a plane P, and let points K(2,1,1), L(3,-1,2) and M(1,1,2) be on this plane. How can we find a ...
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Need help understanding what the curve made by two or three intersecting surfaces looks like

I have trouble visualizing what curves are traced out by the intersection of multiple surfaces in $R^3$. for example take the parametric equations $ <cos(t),sin(t),sin(t)$ > Clearly this would ...
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Suppose a surface contains a straight line. How can I prove that all the points on this line have non-positive Gaussian curvature?

Suppose a surface contains a straight line. How can I prove that all the points on this line have Gaussian curvature $K_P\leq0$?
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3D Analogue of a Catenary

When a cable is supported at its ends and droops due to its own weight, the resulting curve is called a catenary. However, is there a three-dimensional analogue of this shape? For example, let's say ...
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Representing results of CSG operations with spline-based surfaces

I've been playing with a few different CAD programs and have become interested in the math involed with CSG and spline-based surfaces. During my research, I found that the curve representing the ...
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Total Gaussian curvature

For a compact surface, $S$, in $\mathbb{R}^3$, how would I go about showing that the total Gaussian curvature $\int_S K da \leq 4 \pi$? I feel like Hopf's Umlaufsatz and the Gauss-Bonnet Theorem are ...
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Fundamental form of explicit surface

I'm trying to derive the formulae for the fundamental forms of an explicitly given surface $f(x,y)=z$ however I don't see how to set up my initial parametrisation. My intuition is that perhaps $\...
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How many parametrisations are needed to cover a sphere?

I have seen that a sphere can be covered with 6 parametrisations, but is it possible to totally cover a sphere with less parametrisations/charts?
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Homeomorphism $\phi : T^2/A \to X/B$. What are $ T^2/A$ and $X/B$?

The question I am working on asks me to construct a homeomorphism $\phi : T^2/A \to X/B$ where $T^2$, $A$, $X$ and $B$ are given as follows: $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by $A=S^...
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How to find the matrix for $dN_p$, the differential of the Gauss map?

Suppose that $x:U\rightarrow \mathbb{R}^3$ is a chart for a regular surface $S$. Using the notation (from Shifrin P.39, 46) that $N_p$ is the Gauss map at point $p$, whereas the matrix with $E$,$F$,$G$...
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Why this two surfaces have one end?

I want to prove that the infinite-holed torus and the infinite-jail cell window have one end but the doubly infinite-holed torus doesn't, my definition of one end is the following: A locally ...
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Fast Rational Bézier Surface Evaluation Problem

I am currently writing a NURBS ray tracer. What I do is convert the NURBS into rational Bézier patches and then perform the intersection test using Newton's method. To do this fast (the ray tracer ...
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How to tell whether a combinatorial surface is orientable.

I am learning about combinatorial surfaces, and I've encountered this question: What surface is represented by $a_1a_2\cdots a_na_1^{-1}a_2^{-1}\cdots a_n^{-1}$. I know the Euler characteristic is $...
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Parameterization of tubular surface and analysis of its geometry

Consider the tube defined by $$ α(s,v) = c(s) + r\big( \cos(v)\,b(s) + \sin(v)\,n(s)\big) , \quad r > 0. $$ Here $c$ is a Frenet curve with curvature $k>0$, torsion $\tau$ and $(t ,n, b) $ ...
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Finding the equation of the affine tangent plane

I understand the first part but for the second part of the question how do I find the equation of the affine tangent plane at the given point? In the solution I can see that they are working out the ...
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For surfaces in $\mathbb R^3$, does $ ds^2 = du^2 + G(u)^2 dv^2$ implies $|\frac{\partial G(u)}{\partial u}| ≤1 $?

Let we have a regular surface in $\mathbb R^3$, parameterized by $ \vec r(u,v)$. Suppose that the first fundamental form (metric) of this surface is given by: $ ds^2 = du^2 + G(u)^2 dv^2$ (G(u) only ...
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What is an equations “surface”?

Given an equation $$f(\vec x) = 0$$ in $n$ variables (and some constraint $\vec x\in X\subseteq\mathbb R^n$), what is the hypersurface of the $n-1$ dimensional submanifold $\{\vec x\in X: f(\vec x)=...
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60 views

What's the higher dimensional generalization of arc length?

Given a scalar field $f: \mathbb R^n \supseteq V \to S \subseteq \mathbb R, \vec x\mapsto f(\vec x)$, what is the $n$ dimensional hypersurface (or volume, however you want to call this submanifold of $...
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Counterexample of the fundamental theorem for hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$ in global sense

Are there any example of a closed Riemannian $n$-manifold $(M,g)$ and a symmetric bilinear form $A=h_{ij}dx^i\otimes dx^j\in\Gamma(T^\ast M\otimes T^\ast M)$ satisfying Gauss' equation \begin{eqnarray}...
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Prove that normal vector to the surface does not depend on parameterization

A given surface can be parameterize in many different ways. How to prove that a change in parameters, given a smooth, invertible map between the two parameter domains, does not change a normal vector ...