# Tagged Questions

1answer
42 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: ...
1answer
61 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: ...
1answer
133 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 ...
2answers
71 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 ...
2answers
67 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 ...
1answer
91 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? ...
1answer
147 views

### what are some isometries of S^2 without fixed points?

Spherical geometry question involving isometries. Particularly looking for isometries with no fixed points.
1answer
94 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 ...
7answers
289 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 ...
1answer
73 views

### Parametrization of a solid

Find a parametrization $\sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3$, with $I$ a parallelepiped, of $\lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace$.
1answer
92 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, ...
1answer
81 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 ...
1answer
412 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 ...
1answer
54 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 ...
1answer
184 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 ...
1answer
369 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 ...
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