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

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What equation will create a 3D rose curve?

The parametric equation $x=a\cos(bt)\cos(t)$, $y=a\cos(bt)\sin(t)$ where $a$ & $b$ are constants and $t$ is parameter gives a rose curve which looks like, On a similar basis, is there a equation ...
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55 views

What is the probability of selecting a random point q within x sq units of a random point p?

I was using an online app called "GeoGuessr" where you guess the location on a world map of a given Google Street view image. On one certain attempt, I picked a location within 92.7 square miles of ...
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2answers
177 views

Tangential space to the rational normal curve

Exercise 15.5 (Harris, Algebraic Geometry: A First Course): Describe the tangential surface to the twisted cubic curve $C \subset \mathbb P^3$. In particular, show that it is a quartic surface. What ...
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1answer
281 views

surface area of 'cylinder' with the top cut at an angle

I don't know what the name for this shape is, so in essence it is a cylinder, radius at base $r$, which has had a wedge of the top cut off at an angle so that rather than a circle the upper face is an ...
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1answer
39 views

How do I incorporate into the definition of a function that its output is constant?

Say I have a surface, x-y+xy+yz+z^2 = 0 (There is nothing particulary special about this expression. I've just made it too complicated to be simplified so that the variables are independent of one ...
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1answer
44 views

The normal curvature is bounded by the principal curvatures.

Let the inclusion $i:S\subset\mathbb R^3$ be an immersion of a surface $S$, and let $N:S\to \mathbb R^3$ be a local Gauss map. Let $a:I\to S$ be an arc length parametrized curve, with $a(0)=p$ and $a'(...
2
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1answer
157 views

Normal of a coons patch at a given point

Disclamer: Rendering the Coons patch is part of 3D Graphics homework, but finding the normals at a given point isn't. Just curious. Here's what I got so far: It's a Coons patch defined by four ...
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1answer
41 views

Determinant of this coherent sheaf on a surface $S$

If $C$ is a curve on a surface $S$, i.e. $i:C\subset S$, and $G$ is a line bundle on $C$, then $G|_U\cong \mathcal{O}_C$ where $U$ is an open subset of $S$, that is, $G$ is trivial on the complement ...
6
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2answers
127 views

If $\dfrac{\mathrm {circumference}}{\mathrm {diameter}}$ is the same for all circles, does the surface have to be flat?

Given a two dimensional Riemannian manifold with the property that the ratio of the circumference and the diameter is the same for all circles. What can be said about it? Does it have to be the ...
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0answers
21 views

Convexity of a subset of $R^3$ based on its bounding surface

Let D be a closed region in $R^3$ whose boundary S is a closed, smooth, orientable surface. The tangent plane at every point on S intersects it in a connected set $M$. Is D (the interior of S) a ...
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81 views

a subspace of $\mathbb R^3$ with $\pi_1=\mathbb Z_2$

I've been wondering about such problems. It is well known that $\mathbb{RP}^2$ cannot be realized as a subspace of $\mathbb R^3$. But does there exist a space $X\subset\mathbb R^3$ (maybe even $...
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1answer
141 views

A question about the Möbius Strip and the Projective Plane

I know that both the Möbius Strip and the Projective Plane are both 2-manifolds. I try to prove that they are locally homeomorphic to $\mathbb{R}^2$ and Hausdorff. It seems easy to see that the ...
3
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1answer
53 views

surfaces, curves and lines

Could someone please assist with the following questions: Consider $f(x,y) = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and take $C$ to be the curve of intersection of $z = f(x,y)$ with the plane $y=x$. Show ...
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1answer
138 views

Using Principal Directions and Curvatures to Find Point On Surface

Given the principal directions (max and min), principal curvatures, and normal of a surface at point n, how would you go about looking for a point on the surface at a given vector distance from n? I'...
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3answers
114 views

Describe the graph of the function: Level Sets, Intersections

Describe the graph of the function $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x, y) \rightarrow |y|$$ computing some level sets and some intersections. I have done the following: The level curves ...
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1answer
66 views

Level set for the function

Draw or describe the level surface and an intersection of the graph for the function $$f: \mathbb{R}^3 \rightarrow \mathbb{R}, (x, y, z) \rightarrow x^2+y^2$$ I have done the following: The level ...
3
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1answer
37 views

Compute surface area about the $x$-axis

Compute the surface area swept out if the graph $y = e^{-x}$, for $0 \leq x \leq 1$, is revolved about the $x$-axis. The formula for the surface area is $$S = 2\pi\int_{a}^{b} f(x) \sqrt{1 + f'(...
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128 views

What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$. Suppose there is a surjection: $E\longrightarrow i_*A\...
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1answer
112 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points (...
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29 views

How to prove that dim$|L|=D^/2+1$ on a $K3$ surface?

Let $D$ be an irreducible curve on a $K3$ surface $S$, and let $L=\mathcal{O}_S(D)$. The Riemann Roch formula on the $K3$ surface is given by : $\qquad\qquad\qquad\qquad\qquad h^0(S,L)+h^0(S,L^{-1})=...
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0answers
83 views

Existence of periodic orbits (non-linear systems)

I'm trying to solve the following problem: Use the Poincaré-Bendixson's criterion to show that the system has a periodic orbit $$ \dot{x}_1 =x_2 \\ \dot{x}_2=-x_1+x_2-2(x_1+2x_2)x_2^2 $$ The unique ...
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2answers
44 views

Why is $h^1(S,\mathcal{O}_S(-D))=h^0(D,\mathcal{O}_D)-1$ on a $K3$ surface?

Let $D$ be a divisor on a $K3$ surface $S$. We have an exact sequence : $\qquad\qquad\qquad\qquad\qquad0\longrightarrow\mathcal{O}_S(-D)\longrightarrow\mathcal{O}_S\longrightarrow\mathcal{O}_D\...
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2answers
63 views

Why is this divisor on a K3-surface not effective?

Let $D$ be a divisor on a $K3$ surface and set $L:=\mathcal{O}_S(D)$. Riemann Roch theorem : $\chi(L)=\chi(\mathcal{O}_S)+\frac{1}{2}D.(D-K)$ reduces to $h^0(S,L)+h^0(S,L^{-1})=2+\frac{1}{2}D^2+h^1(S,...
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1answer
22 views

Pyramid Surface Area

Square based pyramid has a side length of $220$ (b) and a height of $105.$ Find the surface area. I tried by "doing" Pythagorean theorem $110^2+105^2=s$ then i did the equation for surface area (...
2
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1answer
91 views

Connected Sum of Surfaces

I am trying to prove that the connected sum of surfaces is a surface. My definition of surface is: A topological space locally homeomorphic to $\mathbb{R}^2$, second countable, Hausdorff and ...
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1answer
35 views

Cylindrical coordinates - Surfaces

I found the following: Cylindrical coordinates $(\rho , \theta , z)$. This system consists of the following coordinate surfaces: Cylinders with common $z-$axis: $\rho=\sqrt{x^2+y^2}=\text{ ...
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1answer
164 views

Linking surface integral of a gradient field to a contour integral [duplicate]

I have a vector field $F$ deriving from a scalar potential $f$, i.e. $F=\text{grad}(f)$. I want to compute the integral of $F$ over a surface (To evaluate the flux of $F$). I think there exists a ...
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1answer
25 views

From a family of projective curves to a surface

Suppose that $$\mathcal F:\;(X^3+Y^3+Z^3)\lambda+Z^2X\mu=0$$ is a family of projective plane curves parameterized by $(\lambda:\mu)\in\mathbb P^1(\mathbb C)$. This family of curves forms a surface $...
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1answer
399 views

Parametrization where coordinates lines are lines of curvature

I am asked to prove that given a surface $S$ and a point $p\in S$ non-umbilical, then there exists $U$ open in $\mathbb{R}^2$, there exists $Y:U\subset \mathbb{R}^2\longrightarrow \mathbb{R}^3$ a ...
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1answer
228 views

Find the surface area generated when the curve is revolved around the x-axis

Find the surface area generated when the curve is revolved around the x-axis $y=\frac{x^3}{10}$ on $[0,\sqrt{10}]$ This is what I have so far: $$f'(x)=\frac{3x^2}{10}$$ $$f'(x)^2=\frac{9x^4}{100}$$ ...
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1answer
142 views

Get 4 points lying on the plane by given normal

I would like to create plane using 4 points (which I need to find out), when I know the intersection point of the 2 diagonals in the plane. Next thing I know, that the Y coord of 2 bottom points will ...
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1answer
48 views

Evaluate a double integral over a domain

I'm asked to evaluate $$\iint(x^3+y)$$ over the ellipse on the xy plane such that $2x^2+y^2<2y$ I figured that the ellipse can be parametrized by $$\vec r(t)=\left(\frac{\cos t}{\sqrt2};1-\sin ...
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1answer
50 views

Surfaces of constant curvature using the conformal method

I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around ...
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1answer
42 views

How to find the side for the biggest area?

Let $x$ be one side of a rectangle and $a$ its perimeter. We know that it's area is given by: $$ S = x\cdot\left(\frac{a}{2}-x\right). $$ $$ S=-x^2+ax/2$$ where a=-1, b= a/2 and c= 0 $$D=a^2/4$$ $$n=-...
2
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2answers
218 views

Can someone clarify the definition of flux?

I am confused by the concept of flux as used in vector calculus. Suppose I have a sphere. On the inside of this sphere is a spherically symmetric electric charge distribution. Now I want to find the ...
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1answer
43 views

Canonical map from fundamental group to Fuchsian group?

Suppose we have a Riemann Surface $S$ of constant negative curvature $-1$. What is the canonical map from the fundamental group $\pi_1(S)$ to the discrete subgroup $\Delta \subset PSL_2(\mathbb{R})$ ...
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1answer
84 views

Rational map on $\mathbb P^1$ and its fibers

Consider a non-singular complex projective surface $S$ and a rational map $\psi:S\longrightarrow \mathbb P^1$; moreover suppose that $\psi$ is not defined on $\Delta=\{x_1,\ldots,x_m\}\subset S$. Now ...
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1answer
48 views

Minkowski metric on a surface

Do closed surfaces admit a metric with lorentzian signature? Any reference?
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1answer
108 views

Length of geodesic representative on hyperbolic surfaces

Let $S$ be a closed oriented hyperbolic surface. Let $x,y \in S$ and let $\alpha,\beta$ be two geodesic arcs with endpoints $x$ and $y$. Let $\alpha \beta$ be the closed piecewise geodesic curve ...
0
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1answer
36 views

Multivariable Calculus: Intersection of surface and planes

I have this math problem: Let $S$ be the surface that consists of all points $(x,y,z)$ that satisfy the equation $x^2+y^2=z^2$. 1) What are the intersections of $S$ with horizontal planes $...
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1answer
83 views

Principal directions bissect the asymptotic directions

How can one prove that at a hyperbolic point, the principal directions bissect the asymptotic directions?
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1answer
211 views

Blow-up and base change

Consider a complex smooth (projective) surface $X$ and a blow-up $\epsilon:S\longrightarrow X$ at a point $x\in X$. Let $\sigma\in\text{Aut}(\mathbb C)$ be a field automorphism and moreover let $$\...
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3answers
89 views

Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = x^2+...
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1answer
201 views

Can anyone solve a stochastic differential equation - related to neuroscience research?

I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ...
2
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1answer
92 views

Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
2
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0answers
72 views

Cubic surface as P$^2$ blowing up 6 points

This question is from the chapter 1 of Reid's note: Chapters on algebraic surfaces Suppose that L:(x=y=0), M:(z=t=0), and L$_5$:(y=t=0) lie on a nonsingular cubic surface X in P$^3$, define a ...
2
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94 views

An algorithm to find isometry between surfaces in $\mathbb{R}^3$?

Given two surfaces in $R^3$, i would like to find isometry between these two. Usually, in class, we did some examples, like bending the plane into a cylinder, or cone, and they were not hard, quite ...
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65 views

Area of a region on the surface of a prolate spheroid

Is there a general expression for the area of a region bounded by 3 great ellipses on the surface of a prolate spheroid (where a great ellipse is the intersection of the spheroid with a plane passing ...
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1answer
62 views

Calculating surface area

I have the following surface in $$R^3:{(x,y,z),(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2) \ ,\ x,y >=0}.$$ I want to find it's surface area. I've tried using spherical coordinates but calculating the ...
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1answer
95 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 \...