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

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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 ...
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57 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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2answers
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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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41 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 ...
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35 views

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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2answers
31 views

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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1answer
16 views

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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51 views

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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1answer
15 views

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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2answers
25 views

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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271 views

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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1answer
55 views

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 ...
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1answer
30 views

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 ...
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1answer
45 views

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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1answer
27 views

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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1answer
25 views

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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1answer
41 views

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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34 views

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 ...
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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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14 views

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 ...
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2answers
58 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 ...
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2answers
56 views

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 ...
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2answers
28 views

Axis of a cylinder

Could you explain to me what the axis of a cylinder is? Is it a line that passes through the center?
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1answer
34 views

local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...
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1answer
25 views

Show curve is tangent to a surface using gradient

The question is this: 'Show that the sphere $h(x,y,z) = x^2+y^2+z^2-8x-8y-6z+24=0$ is tangent to $f(x,y,z)=x^2+3y^2+2z^2=9$ at the point (2,1,1).' My approach was that grad(f) at P should give a ...
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Create a smooth surface based on 4 points?

To understand this question, please first understand the question and answer here: Create a formula that creates a curve between two points We are essentially transcending a 2d problem into a 3d ...
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Fundamental group of complement of image of diagonal embedding from $S_g$ to $S_g \times S_g$?

Let $S_g$ be a surface of genus $g \ge 0$. Let $\Delta \subset S_g \times S_g$ be the image of the diagonal embedding $x \mapsto (x, x)$. Let $X$ be the complement of $\Delta$ in $S_g \times S_g$. My ...
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Definition of Gauss curvature

In GR class a definition of Gauss curvature was introduced which I cannot understand. It says: In a curved surface in $R^3(x,y,z)$. For any small area (circle?) $\Delta A_1$ of the surface. ...
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38 views

I would like to find the inverse of $X$

Let $X(u,v)=(v−u,u^2−v^2,u+v)$, $(u,v)\in U=\mathbb R^2$ and $S=X(U)$. what is the inverse of $X$ where $X$ is the function which maps a $2$D object into a $3$D object $X^{-1}(x,y,z)$
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Polar forms of algebraic curves & surfaces

A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}) = 0$ be the equation of a surface of degree $n$. The first polar form of ...
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1answer
49 views

Surface of a polynomial

How can I find the surface represented by the polynomial $$x^2-y^2-2xz=0$$ any clue please?? I have tried to plot it using Maple
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Parametrisation of a surface and a cylinder

I have been asked to find C which is the curve I need to integrate over and C is the intersection of the cylinder $x^2+y^2=2y$ and the plane $y=z$. I assume you have to find a parametrization that ...
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63 views

Coverings of connected sum of four copies of $\mathbb{R}P^2$

G.Baumslag in one of his papers asserts that a group $G = \langle a,b,c,d | a^2b^2c^2d^2 = 1 \rangle$ contains all fundamental groups of closed compact orientable surfaces of genus $g\geq 2$? I think ...
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Exponential map on the ellipsoid.

Consider the ellipsoid $M \subseteq \mathbb{R}^3$ defined by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{x^2}{c^2} = 1,$$ where $0 < a < b < c$, equipped with the usual Riemannian metric ...
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1answer
30 views

Find the equation of the following curve

Suppose normal lines are drawn at all the points on the surface $z = ax^2 + by^2$, where $a$ and $b$ are some positive constants, that are at a given height $h$ above the $xy$-plane. Find an equation ...
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1answer
29 views

Finding velocity vector field over 3D surfaces

I have a surface, $S$, defined in three-dimensional space. For the sake of this question, let's assume it is a sphere with unit radius although the surface in my problem can be any shape. I have a ...
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5answers
45 views

Describing the motion of a particle (sphere)

If I have the following position at time t : $\hat{r}(t) = 3\cos(t)\hat{i} + 4\cos(t)\hat{j} + 5\sin(t)\hat{k}$ , then how can I tell if the particle's path lies on a sphere or not? If e.g. the second ...
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Why don't you add the top and bottom circle when finding the surfaces of revolution

I am studying Calculus II and I had a question about finding the surfaces of revolution. Why don't you add the top and bottom circle when you find the surface? If one is trying to find the surface ...
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How to prove a differentiable function from a surface to a surface

$S$ is the unit sphere without its north and south poles. $H$ is the hyperboloid $x^2+y^2-z^2=1$. For $p\in S$, $l_p$ is the ray perpendicular to the $z$ axis that starts on the $z$ axis and passes ...
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Finding the Surface that a Vector Function Lies On

I've been studying vector functions and one thing that I don't quite understand is finding the surface that some vector function r(t) lies on. For instance, in my textbook (Calculus Early ...
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26 views

Calculating surface integral (with gauss's theorem (?))

I would like to solve the following problem: Let $B_1$ be the unit ball in $R^3$ and $A := \delta B_1 \cap(\{x>0, z=0\}\cup\{x=0, z>0\})$. Let $F(x,y,z) := (-y+e^{x+z}, 0, e^{x+z})$. ...
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Surfaces in dimensions higher than $3$, how to compute the normal vector?

In my book surfaces are defined as maps $\sigma\ \colon A \subseteq \mathbb R^2 \to \mathbb R^n$ ($n \geq 3$). Then the book goes on to define regular surfaces and the normal vector. The latter is ...
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Degree of a ruled surface

If $X = C\times C' \subset \mathbb{P}^3$ for $C$ of genus $g$ and $C'$ of genus $g'$ (both smooth), then we know from Hartshorne exercise V.1.5 that $$ 8(g-1)(g'-1) = d(d-4)^2 $$ where $d = ...
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Does there exists any $x$ such that $x\geq AM\geq GM?$ If not,how do I prove it is unbounded?

We know $AM\geq GM$.Or,in words,$AM$ has a minimum value when it is equal to $GM$?But,by any chance is there any way to find $x$ such that it is the upper limit of $AM$? My inspiration for this ...
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50 views

Where does the following equation for the area of an ellipsoid come from?

I have been reading an old draft that I made a few years ago where I used the following expression for the area of an ellipsoid: $$A=2 \pi r^2 + \frac{2 \pi \tanh^{-1}(\sqrt{1-r^{-6}})}{r^4 \sqrt{1- ...