1
vote
0answers
22 views

normalized mean curvature flow with convex initial hypersurface has finite velocity

I can't understand the prove in [Xi-Ping Zhu] Lectures on mean curvature flows. The statement as follow. Lemma 3.5 (page 32) There exists a positive constant $C$ such that ...
4
votes
0answers
67 views

Tangent developable of helix.

Let $T$ be union of tangent lines to helix $C=(\cos x, \sin x,x)$. 1) I want to prove that $T - C$ is a smooth manifold and find equation for $T$. 2) I want to find how many times a line can ...
1
vote
0answers
42 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
4
votes
0answers
61 views

Soft: Why does the existence of a singularity cause problems for deRham cohmology?

I've heard that if a variety has a singularity then the deRham theory has "problems". What exactly are these? Im guessing there is some sort of issue with the defintion of a differential form, but ...
0
votes
0answers
27 views

dimension of tangent space to a boundary point of a convex shape

I have a basic question regarding the dimension of the tangent space at a point $P\neq0$ that lies on the boundary of a pointed convex cone with its point centered at 0. For a 3D cone that is ...
2
votes
1answer
29 views

Given two closed curves, when is their minkowski sum differentiable?

Suppose you are given closed curves, $\gamma_1$ and $\gamma_2$, which define convex figures in the plane. If we take the minkowski sum of $\gamma_1$ and $\gamma_2$, when is the resulting curve ...
4
votes
1answer
111 views

Tensor notation (practicing)

I'm praticing tensor notation, and I want to prove this way that given vectors $A,B,C,D$ then $(A \times B) \times (C \times D) = \det(A,C,D)B - \det(B,C,D)A$, where $\det$ means the triple product. ...
0
votes
1answer
56 views

Tangent at a singular point

I'm looking at this question If the tangent at the point $P$ with coordinates $(h, k)$ on the curve $y^2 = 2x^3$ is perpendicular to the line $4x = 3y$, find $(h, k).$ This is how I attempted it ...
2
votes
3answers
81 views

Parametrization of $y^2 - x^2=1$

I have found parametrizations for the level curve $y^2-x^2=1$, however, I have a question regarding one of them. From the Pythagorean trigonometric identity $\cos^2 x + \sin^2 x =1$ we obtain ...
0
votes
1answer
167 views

3D Road - Rotate around 3d curve

First of all, I'm not sure whether to post this on stackoverflow or here, but since there's some mathematics needed here (especially at the end of this question) I posted it here. I'm given a ...
1
vote
1answer
71 views

Willmore energy of an ellipsoid

Given an ellipsoid of equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ How can I calculate the Willmore energy of this surface knowing that its definition is: ...
3
votes
1answer
157 views

The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)

In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it. Let $\gamma: ...
1
vote
1answer
243 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
4
votes
2answers
383 views

Length of curve in 3D spherical coordinate

let $r$ be the magnitude of a vector in 3D with Spherical co-ordinate $(r,\theta,\phi)$ and cartesian coordinates is $(x,y,z)$, whose angle with $z$ axis is $\phi$ and projection of the vector makes ...
1
vote
2answers
304 views

How can I calculate the Euclidian displacement of two places on a sphere (earth in this case ) and calculate the

I would like to get the formula on how to calculate the distance between two geographical co-ordinates on earth and heading angle relative to True North. Say from New York to New Dehli , I draw a ...
-1
votes
1answer
139 views

Length of a curve on $S^2$

$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$? $2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates? ...
5
votes
1answer
419 views

What are some isometries of $S^2$ without fixed points?

This spherical geometry question involves isometries. I am particularly looking for isometries with no fixed points.
2
votes
1answer
188 views

Tangent cone to a subset of $\mathbb{R}^3$

Well, I have the set $X=\{(x,y,z) \in \mathbb{R}^3 | 3x^2+2x^3+y^2+z^2=1\}$ How can I calculate the tangent cone at the point $(-1,0,0)$ ? What are the standard ways to calculate the tangent cone to ...
6
votes
7answers
862 views

Detect when a point belongs to a bounding box with distances

I have a box with known bounding coordinates (latitudes and longitudes): latN, latS, lonW, lonE. I have a mystery point P with ...
1
vote
1answer
139 views

Parametrization of a solid

Find a parametrization $\sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3$, with $I$ a parallelepiped, of $\lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace $.
2
votes
1answer
138 views

Showing: point of polytope which maximizes the minimum distance to a vertex is a barycentre?

Let $T_1$ and $T_2$ be two regular $(n-1)$-dimensional simplices with vertices $$(t,0,\ldots,0), (0,t,\ldots, 0),\ldots, (0, 0, \ldots, t),$$ and $$(t-n+1,1,\ldots, 1), (1, t-n+1, \ldots, 1), \ldots, ...
1
vote
1answer
120 views

Degree of Hessian surface invariant under linear transformations?

Given a surface $V(f) \subset \mathbb{P}^n$ for a homogeneous polynomial $f$ of degree $d$ on $\mathbb{P}^n$ and a linear transformation $g \in SL(n+1)$. Is the degree of the Hessian $H_f = V(\det ...
0
votes
1answer
740 views

Find point on sphere with directional tangent vector

Say a sphere equation like this: $x^2+y^2+z^2=5$. I want to find a point on the sphere whose tangent vector is perpendicular to the vector $\begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix}$. I go ...
0
votes
1answer
63 views

Can any smooth planar curve which is closed, be a base for a 3 dimensional cone?

A cone in 3 dimensions has a vertex and a base. The contour of the base is a circle which is a smooth closed planar curve. Can there be a more general cone which can have any smooth closed planar ...
3
votes
1answer
250 views

References for the basic theory of surfaces of revolution, cylinders and cones

I'm looking for references to books were the following types of problems about finding the equation defining a surface of revolution, a cylinder or a cone are treated. These are problems that are ...
1
vote
1answer
483 views

Locus of osculation of concentric ellipses (elliptic pond ripples)

If you dropped two rocks in a pond, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line. Now imagine that instead of ...