Tagged Questions

1
vote
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: ...
3
votes
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: ...
1
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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 ...
4
votes
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 ...
1
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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 ...
-1
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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? ...
4
votes
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.
2
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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 ...
6
votes
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 ...
1
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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 $.
2
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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, ...
1
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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 ...
0
votes
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 ...
0
votes
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 ...
3
votes
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 ...
1
vote
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 ...