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

learn more… | top users | synonyms

6
votes
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 ...
2
votes
1answer
26 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 ...
0
votes
0answers
60 views

What is the cap body produced by the unit sphere?

For ecach number a > 1, let C(a) be the cap body produced by the unit sphere in E^(3) and the points (+-a,0,0). Calculate the volume, surface area and mean width of C(a). For this question, I don't ...
0
votes
0answers
15 views

What is the boundary component of the polygon?

A polygon has the edge equation, R(f)P(b^-1)Q(d)S(e^-1)R=1 where R, P, Q and S are the vertices and f, b, d and e are the edges. What would the boundary component be. I know the boundary edge is f, b, ...
0
votes
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 ...
2
votes
1answer
34 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) $ ...
0
votes
0answers
23 views

Solve for $y$ in terms of $x$ at $z=0$

Feeling like a complete idiot here (seems this should be easy)...the equation of a surface I fit is: $$ sf(x,y) = p_{00} + p_{10}x + p_{01}y + p_{20}x^2 + p_{11}xy + p_{02}y^2 $$ What is the ...
0
votes
0answers
30 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 ...
2
votes
0answers
29 views

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 ...
0
votes
0answers
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 ...
0
votes
2answers
57 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 ...
1
vote
0answers
34 views

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 ...
0
votes
2answers
51 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 ...
0
votes
2answers
27 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?
1
vote
1answer
32 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 $X$ ...
0
votes
1answer
24 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 ...
1
vote
2answers
36 views

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 ...
2
votes
1answer
31 views

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 ...
0
votes
0answers
19 views

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. ...
1
vote
2answers
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 $X^{-1}(x,y,z)$
3
votes
0answers
46 views

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 ...
1
vote
1answer
48 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
0
votes
1answer
26 views

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 ...
1
vote
1answer
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 ...
13
votes
0answers
288 views

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 ...
2
votes
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
1answer
42 views

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 ...
0
votes
2answers
30 views

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 ...
0
votes
1answer
27 views

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 ...
0
votes
1answer
25 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 $F(x,y,z) := (-y+e^{x+z}, 0, e^{x+z})$. ...
0
votes
1answer
39 views

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 ...
0
votes
0answers
9 views

Neighborhood of Surfaces created with NURBS

I created surfaces with NURBS. I want to check connectivity of these surfaces. However I cannot check if these surfaces' edges overlaps. Because even if the curves are completely same, their control ...
1
vote
0answers
44 views

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 = ...
1
vote
1answer
54 views

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 ...
1
vote
1answer
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- ...
0
votes
1answer
24 views

How can I prove that a point on a surface can be parametrized by a certain parametrization? [closed]

So the specific question is as follows: how can I show that the surface $y(x-a)+zx=0$ can be parametrized by $\alpha(u,t)=(au,ut,t-ut)$? Or equivalently, the set of points defined by the first ...
2
votes
1answer
52 views

Klein Bottle as the connected sum of two projective planes

Can someone explain intuitively why the Klein bottle is diffeomorphic to the connected sum of two projective planes? I can do this using origamis/fundamental graphs )w/e they are called. is it ...
1
vote
1answer
29 views

Normal and geodesic curvatures of intersection curve of two given surfaces

I was recently presented this in differential geometry class which I cannot seem to solve involving curve of intersection of two surfaces in $ R^3 $ which reads as Let us define two surfaces in $ ...
1
vote
1answer
33 views

A regular curve on regular surface in $ \mathbb R^3 $ being of constant tangent length

I was given this in diff. geometry on which I am stuck: Let $ S \subset \mathbb R^3 $ be a regular orientable surface in $ \mathbb R^3 $ such that on it we define the regular curve $ \gamma : I ...
0
votes
1answer
43 views

Can there be a closed geodesic on surface embedded in R3 with zero Gaussian curvature

I was asked this in differential geometry class and it.is bugging me as i do not know the answer I know a surface of constant Gaussian curvature zero is locally isometric to a plane on which no ...
2
votes
1answer
76 views

Determining if there can be smooth closed geodesic given its curvature

In my class on differential geometry I have been given the following question on which I am stuck: Let S be a regular orientable surface in $ R^3 $ with Gaussian curvature $K$ (not necessarily ...
0
votes
1answer
22 views

Proving That $\alpha^{\prime}(t)$ is orthogonal to $N(\gamma(t))$

Let $\Phi$ be a surface in $\mathbb{R}^3$ with parameter domain $K\subset\mathbb{R}^2$ and let $\gamma:[a,b]\to K$ be a $\mathcal{C}^1$-curve. Also let $\alpha=\Phi\circ\gamma$. Prove that ...
4
votes
0answers
85 views

How to calculate the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the ...
1
vote
0answers
38 views

Intersection of a surface and a line.

I have a surface ($H$) that passes every corner points(one coordinate gets its maximum $1$ while others $0$), such as $(1,0,...,0), (0,1,0,...,0),....,(0,...,1).$ H is characterized by ...
0
votes
1answer
59 views

Rulings of One Sheet Hyperboloid

Let $M$ be a hyperboloid of one sheet satisfying $x^2+y^2-z^2=1$. Show that $x(u,v)=(\frac{uv+1}{uv-1},\frac{u-v}{uv-1},\frac{u+v}{uv-1})$ gives a parametrization of $M$ where both sets of parameter ...
1
vote
1answer
18 views

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} ...
0
votes
1answer
21 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 ...
0
votes
0answers
37 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 ...