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

learn more… | top users | synonyms

2
votes
2answers
32 views

Is there a way to parametrise general quadrics?

A general quadric is a surface of the form: $$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fxz + 2Gx + 2Hy + 2Iz + J = 0$$ It can be written as a matrix expression $$ [x, y, z, 1]\begin{bmatrix} A && ...
1
vote
1answer
84 views

What is the word associated with the connected sum of two surfaces with boundary?

I want to calculate the word associated with the connected sum of two surfaces with boundary but I don't know how to proceed. I know that the word associated with the connected sum of two surfaces ...
1
vote
1answer
75 views

Understand the corrispondence between rational maps and linear systems.

A key fact in "birational geometry" (on $\mathbb C$) is the following theorem: Let S be a surface. Then there is a bijection between the following sets: (i) {rational maps ...
2
votes
2answers
54 views

How to calculate Euler characteristic of surfaces $K$ and $P$?

The book Introduction to Topology by C. Adams and R. Franzosa says : From the triangulations in Figure 14.8, we see that $\chi(S^2) = 2$, $\chi(T^2) = 0$, $\chi(K) = 0$ and $\chi(P) = 1$. And ...
0
votes
0answers
29 views

RBF - triangle mesh interpolation - skinny triangles and incorrect results

I have triangle mesh, that I need to describe by RBF. I need to do this only locally on vertex neighborhood. All is working correctly if underlaying triangulation is reasonably regular. But if there ...
1
vote
1answer
36 views

Prove that $S$ is colorable if and only if it is orientable

I am taking a course on algebraic topology and I am trying to prove the following exercise: Let $S$ be a differentiable surface in $\mathbb{R}^3$. Prove that $S$ is colorable (you can paint one ...
3
votes
1answer
67 views

Can any surface be described by an equation?

There seems to be two definitions of a surface: The set $S$ of points $(x,y,z)$ satisfying the equation $f(x,y,z)=0$ for some smooth/differentiable function $f:E^{3} \to R$ with $\nabla f \neq 0$ on ...
0
votes
1answer
50 views

Is a point on a plane part of a face on the plane?

There is a line and a face in $\Bbb R^3$, does the line inersect the face? I have a plane (infinite area) in $\Bbb R^3$ defined by a point $(x_0,y_0,z_0)$ and its normal $n$. The plane contains a ...
1
vote
1answer
51 views

Relation between curves on a surface and divisors

Given a projective surface $S$, the irreducible curves contained in $S$ are exactly, by definition, the prime (Weil) divisors of $S$. I was wondering what are reducible curves on $S$ in terms of ...
0
votes
1answer
55 views

Is the principal curvature of a cylinder positive or negative according to the second fundamental form?

First off, what is the name of the tensor associated with the second fundamental form? For the first fundamental form, I believe we call the associated tensor, "the metric tensor." Principal ...
1
vote
2answers
114 views

Points on ellipsoid with maximum Gaussian curvature/mean curvature.

Find the points on the ellipsoid $$x^2/a^2+y^2/b^2+z^2/c^2=1$$ with maximum Gaussian curvature and mean curvature respectively. I parametrized it as $(a\sin u\cos v,b\sin u\sin v, c\cos v)$ and ...
0
votes
0answers
142 views

Surface area common to two perpendicularly intersecting cylinders

I need help to calculate the following surface area: the surface area common to the two cylinders $x^2 + y^2 = a^2$ and $x^2 + z^2 = a^2$ using surface integrals essentially. My attempt: Let ...
6
votes
1answer
154 views

Deriving the surface area of a sphere from the volume

I am a high school student, so I know how to derive the volume $V=\dfrac{4}{3}\pi r^3$ using calculus, but I am unable to derive its surface area. However, I notice that we can approximate the ...
0
votes
1answer
27 views

Verification of a proof regarding the connected sum of two surfaces

I am trying to solve the following exercise: Let $X_1, X_2$ be two surfaces. Lets consider charts $\varphi_j: U_j \to \mathbb{R}^2$ with $U_j \subset X_j$, $j= 1, 2$ and let $B_j = ...
5
votes
2answers
98 views

What does it mean to 'preserve the first fundamental form'?

I'm a bit confused about the phrase 'preserving the first fundamental form', or 'The Gaussian curvature is determined by the first fundamental form'. For example, let's say I have two surfaces $M$ ...
7
votes
1answer
125 views

Intersection of curves on surfaces and the sheaf $\mathcal O_{C\cap C'}$

Let $S$ be a (complex) projective surface and let $C,C'\subset S$ two closed irreducible curves (namely two prime Weil divisors). It is well defined the scheme $C\cap C'$ (as fibered product of ...
4
votes
0answers
127 views

How to classify this surface

I know that it should be either a sphere, torus, Klein bottle, real projective plane, or a connect sum of any combination of these, but I don't know the steps in identifying what kind of surface ...
3
votes
1answer
86 views

What's the name of these two surfaces?

I've plot two implicit surfaces which are shown in the above, I only know their expression, but I don't know how to call them.
1
vote
1answer
49 views

Effective Weil divisors on a Surface and local equations

Let $S$ be a non singular surface over an algebraically closed field $k$ ($S$ is a $k$-scheme integral, of finite type and separated). Suppose that $D\subset S$ is an effective Weil divisor; I don't ...
0
votes
0answers
34 views

Identify the direction of a loop of Edges on a Face

I have a geometrical face which is defined by a surface. The face is also defined by a few edges which are curves, they bound the face. These edges have start and end points. They flow in a loop ...
0
votes
1answer
74 views

Why $\sigma(u,v)=(\cos u\ \cos v,\cos u\ \sin v,u)$ is not a surface?

During evaluation of different special cases of Gaussian curvature for the surface of revolution in Ch. 7 (page 152) of Elementary Differential Geometry by Pressley, it comes to the case when ...
0
votes
0answers
50 views

Showing that the closed unit disk is not a regular surface

First of all, I realize that this question has been asked on this site before; but looking at the answers already given none of them go into the details which are causing me trouble (all the relative ...
3
votes
0answers
39 views

Is the given set an open subset of the $\mathcal{G}^r_d(|L|)$

Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be a very ample line bundle on $X$. We have a variety $\mathcal{G}^r_d(|L|_s)$ associated to the linear system of curves $|L|$. The ...
3
votes
1answer
93 views

Existence of a vector field with one singularity on a surface

In a paper I'm reading, it is stated that "It is known that on a compact, connected, oriented two dimensional manifold, there exists a vector field with only one singularity". Where can I find a ...
0
votes
0answers
23 views

Showing that the intersection of two particular vector spaces has codimension 1 in the smaller of the two spaces

Here is the setting: I have a compact symplectic manifold $(X^{2n},\omega)$ and a compact symplectically embedded submanifold $(M^{2d},\sigma)$; that is, $\iota^{\ast}\omega=\sigma$. The dimension 2d ...
0
votes
0answers
50 views

find resultant XYZ points in a laser 3D scanner

I am working on a 3D scanner using a laser and camera and some other stuff needed, the idea is that the camera captures an image to the target when the laser is on, then use the image to gain the ...
1
vote
0answers
20 views

The Walking Saddle Curve and other weirdly moving shapes

The link at the end of this sentence seems to be a video of a walking saddle curve. Where can we obtain these? What are more exact parameters of this curve? What is the relation between wind speed ...
4
votes
1answer
55 views

Measure on surfaces in $\mathbb{R}^3$

I am interested in the following three surfaces in $\mathbb{R}^3$: $$S_1=\{(x,y,x+y+x^3+\sqrt{y}): x \in [0,x_0], y\in [0,y_0]\},$$ $$S_2=\{(x,y,x+y+x^2 +y^2+1): x \in [0,x_0], y\in [0,y_0]\},$$ ...
0
votes
0answers
30 views

Developable surfaces in $\mathbb{R}^4$

It is known that there are developable surfaces in $\mathbb{R}^4$ which are not ruled: the famous example is of Hilbert and Cohn-Vossen in their book "Geometry and the Imagination" (p. 342). The ...
6
votes
0answers
290 views

Response Surface Methodology using Moving Least Squares Method

I would like to obtain the response surface of a mathematical function for reliability-based design optimization (RBDO). To obtain a reliably response surface, I learned that moving least squares ...
2
votes
1answer
87 views

converse of theorema egregium

Suppose $(M,g)$ is a $3$ dimensional Riemannian manifold and $N$ is any surface imbedded in $M$. If the theorema egregium holds for $N$ does it follow that $M$ is flat? The way I'm thinking of the ...
0
votes
0answers
27 views

Principal directions of surface given by data points

I am wondering how I can easily find the principal directions of a surface given by data points. The points are given in a fixed matrix. Assuming that the surface is differentiable, I can easily find ...
1
vote
0answers
43 views

Parametrization of helicoid like surface for Faraday's law of induction of a solenoid?

I want to visualize with mayavi a possible surface for Faraday's law of induction in the electrodynamics of a solenoid. I.e. something like a helicoid with a smooth transition to a rectangular area, ...
3
votes
0answers
102 views

Grothendieck and Segre's proof of the Hodge Index theorem

On a field of positive characteristic we have the following well-known and important result: (Hodge Index theorem) Let $H$ be an ample divisor on a surface $X$, and suppose that $D$ is a ...
4
votes
0answers
77 views

What is the mean curvature of a developable surface?

I am trying to calculate the mean curvature of a developable surface but having some difficulty. Any help would be greatly appreciated! I am very new to differential geometry so please bear with me, ...
2
votes
2answers
44 views

Surface with all tangent plane pass through one common point is a cone surface

I want to prove that a surface with all tangent plane pass through one common point is a cone surface (a surface like this $\mathbf{r}(u,v)=\mathbf{a}_0+v\mathbf{b}(u)$, here $\mathbf{a}_0$ is a ...
3
votes
1answer
75 views

Why $\dfrac{\partial \sigma}{\partial u}=\dfrac{\partial \sigma}{\partial \bar{u}}$?

According to Elementary Differential Geometry by A N Pressley: $\Large\textbf{5.3. Conformal Mappings of Surfaces}$ Now that we understand how to measure lengths of curves on surfaces, it is ...
0
votes
0answers
45 views

Is there affine surface of general type of the form $y^2=f(x) f(z)$ or $y^2=f(x) g(z)$?

Let $f,g$ be univariate polynomials with integer coefficients of degree $n$. Is there affine surface of general type of the form (1) $y^2=f(x) f(z)$ or (2) $y^2=f(x) g(z)$? I would expect for $n$ ...
3
votes
1answer
40 views

Surface of $(x^2 + y^2 + z^2)^2 = a^2 * (x^2 - y^2)$ using surface integrals

I have to find the surface of $$(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2)$$ using a surface integral and really have no idea what to do... I would really appreciate it if you could give me an idea.
0
votes
0answers
81 views

Find the maximum volume of the pyramid bounded by the plane and the coordinate planes?

Surface $\sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z}$ , $(c>0)$ I found that at $(x_{0},y_{0},z_{0})$ a tangent plane to the surface is : ...
3
votes
1answer
78 views

Differential geometry: restriction of differentiable map to regular surface is differentiable

From Do Carmo: Let $S_1$, $S_2$ be regular surfaces. Suppose $S_1\subset V\subset \mathbb{R}^3$ and $\varphi:V\rightarrow \mathbb{R}^3$ is a differentiable map such that $\varphi(S_1)\subset S_2$. ...
0
votes
0answers
44 views

Computing Gauss curvature using Meusnier theorem

I have troubles with finding Gauss curvature and mean curvature in a certain point of an oblique cylindrical surface. I know the way using the fundamental forms, but I am supposed to use the Meusnier ...
0
votes
0answers
45 views

Why can't the pseudosphere be completed in $R^3$?

Without appealing to Hilbert's theorem on the non-embeddability of complete hyperbolic surfaces in $R^3$, is there a way to "see" that one can't extend the pseudosphere / surface of revolution of a ...
2
votes
0answers
39 views

name this Romanesque surface

I happened to notice that the surface $$ x = \sin(u-v), y = \sin(v), z = \sin(-u) $$ or equivalently (if I haven't blundered) $$ x^4 + y^4 + z^4 - 2 x^2 y^2 - 2 x^2 z^2 - 2 y^2 z^2 + 4 x^2 y^2 z^2 = 0 ...
-1
votes
2answers
53 views

Vector parametrization of a surface intersection

How does one parametrize the following curve in 3-space to $\vec{g}(t): [a, b] \to \mathbb{R}^3$: the intersection of $x^2+y^2+z^2=a^2$ and $x+y+z=0$ ? What I could come up with is as follows: ...
5
votes
0answers
52 views

Solving a 2nd-order elliptic PDE with non-constant coefficients

I wonder how I can solve the following 2nd-order PDE on the positive semiplane $\{x>0\}$: $$(\partial_x^2+\frac{1}{x}\partial_y^2)\phi=\delta(x-x_0)\delta(y).$$ I notice that the l.h.s. is the ...
2
votes
1answer
80 views

What line bundle pulls back to the trivial line bundle

Let $X$ be an abelian surface. $C$ be a curve in $X$. Consider the projective bundle $\pi:\mathbb{P}^1_C\longrightarrow C$. This is a projective morphism. I have two questions : 1) Can we find an ...
0
votes
0answers
27 views

Volume and surface of knock out drum

I have to calculate the volume and the surface of some KO drums (knock out drum). To avoid ambiguous understandings here's a picture of one: http://www.zamilsteel.com/ped/images/projects/11.jpg I ...
2
votes
0answers
61 views

What is the type of the surfaces $x^5 - y^5 + z^2 + x=0$ and $x^5 - y^5 + z^2 + x+1=0$?

I am interested what is the type of the surfaces over the rationals $$ x^5 - y^5 + z^2 + x=0$$ and $$ x^5 - y^5 + z^2 + x+1=0$$ Magma's ...
2
votes
0answers
154 views

What is the push forward of the canonical class?

Let $X$ be an abelian surface over $\mathbb{C}$. And let $i:X\longrightarrow X$ be the inverse map. $i$ is a degree 2 morphism. We consider $Y$ the quotient of $X$ by the action of $i$, that is, ...