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20 views

Sectional curvature in 3-dimensions

I wonder how to compute the sectional curvature of 3-dimensional objects eg. unit ball, $H=\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb{R}^{4}:x_{0}^{2}-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})=1$ and ...
0
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2answers
51 views

Prove that $\frac{dN}{ds}=-\kappa T$

Prove that $\frac{dN}{ds}=-\kappa T$, where $N$ is the oriented normal, and $T$ is the unit tangent vector, and $s$ is arc-length parameter. Here's what I've got so far from my note and I don't ...
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0answers
8 views

compute next frame curvature torsion based on existing tangent, normal, binormal, position, curvature and torsion

I have a question that is making me headache. Suppose I have $r_{s}$, $T_{s}$, $N_{s}$, $B_{s}$, $\kappa_{s}$ and $\tau_{s}$ for position of the sample, tangent, normal, binormal, curvature and ...
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2answers
26 views

Using eigenvalues of a hessian matrix vs D operation to classify critical points.

Having recently covered using the discriminant, $D(x_0,y_0)$, for classifying critical points of equations of two variables. For example: $$R(x,y)=-x^2+4x+2xy+8y-2y^2$$ to find that $(6,8)$ is the ...
1
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1answer
29 views

Discrete Gauß and geodesic curvature

Imagine that you have an n-polygon $S$ and you wanted to calculated the discrete Gaussian or gedoesic curvature. How are they defined? If $p$ is a vertex of $S$ then Gauß-Bonnet suggests that the ...
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1answer
32 views

How to find the sum of maximum and minimum curvature in an ellipse?

I am having difficulty finding the sum of maximum and minimum curvature of the ellipse $9(x-1)^2 + y^2 = 9$. I know that I am supposed to parametrize the ellipse as $f(x(t), y(t))$, with $x(t) = 1 + ...
0
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1answer
23 views

Negative Gauss Curvature

Let S be a manifold of dimension 2, compact and orientable. Suppose its border is made of k geodesic circumferences, with $k \geq 3$. Show that there exists a point in S with negative Gauss ...
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1answer
48 views

Christoffel symbols of a surface of revolution

I am looking for a way to write down the Christoffel symbols for a surface of revolution. They are given by ...
1
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0answers
46 views

Order of Riemann tensor indexes and the Ricci Identity

I have seen the Ricci identity written variously as $R_{ijk}{}^l x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R_{ij}{}^l{}_k x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R^l{}_{kij} x^k = ...
4
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1answer
36 views

What happens to geodesic curvature under the Gauss map?

$\def\RR{\mathbb{R}}$Let $D$ be a closed disc, smoothly embedded in $\RR^3$. The Gauss-Bonnet theorem tells me that $\int \!\! \int_D K + \int_{\partial D} \kappa = 2 \pi$, where $K$ is the Gaussian ...
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0answers
21 views

sum of two curvatures

I have a question about sum of two relative curvature and torsion. The problem is as follow. Suppose we have three points, Ps, ...
0
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1answer
28 views

Curvature and torsion of a helix

I would like to calculate the curvature and torsion for a helix knowing the radius $r$, pitch $2pπ$, center $s$, and direction axis $d$. Can anyone help? I know how to compute curvature and torsion ...
0
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0answers
19 views

higher order schemes or curvature to be smoothed

I want to calculate the radius of curvature of a meandering river. I have done a first estimated using a tool in arcgis. These are the results. The values of -99999 are recorded for points of no ...
3
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1answer
29 views

Curvature of plane curve, formula disagrees with Mathematica?

I have the following equation $$y=\pi\ln(2x)$$ And when I ask Mathematica/WolframAlpha for the curvature I get $$K=\frac{\pi x}{(x^2+\pi^2)^{3/2}}$$ However the formula for curvature of a plane ...
2
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0answers
83 views

Showing a property of a curvature tensor in $S^2$

Consider $S^2 \subset \mathbb{R}^3$. I need to show that if $$R_{ijkl} = -g(R(\partial_i,\partial_j)\partial_k,\partial_l)$$ is a curvature tensor in $S^2$ and $g$ is a metric also in $S^2$, then ...
1
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1answer
40 views

Does this surface exist

Does anybody of you know if there is a surface with first fundamental form $(g_{ij}) = \operatorname{diag}(1, \cos^2(u))$ and second fundamental form $(h_{ij}) = \operatorname{diag}(1, \sin^2(u))$? ...
2
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1answer
72 views

Sectional curvature of product metric?

If $M$ and $N$ are Riemannian manifolds, can we relate the sectional curvature of the product Riemannian manifold $M \times N$ to those of $M$ and $N$? If both $M$ and $N$ have non-negative (or ...
1
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1answer
16 views

Where Am I Going Wrong on this Curvature Problem?

Massively deviating from the given answer on this one, have no idea what's going on. I'm sure that I'm using the correct formulas, but the answer provided is very different from what I come up with. ...
0
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0answers
19 views

Parameterized Curvature Problem

Massively deviating from the given answer on this one, have no idea what's going on. Find the curvature of the following function: x = 5cos(t) ; y = 4sin(t) ; t = $pi$/4 Formula for curvature is k ...
0
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1answer
19 views

Defining Principal Radius of Curvature for Zero Principal Curvature

In differential geometry, the principal radius of curvature $\rho$ is defined as the reciprocal of the principal curvature $\kappa$. But suppose one of my principal curvatures is 0. How can the ...
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0answers
32 views

Natural curvature tightening of parametric curve

I'm looking to compute the "tightening" of curvature for a curve (mine is mainly 2D but could be of any dimension). In particular, since I am mainly 2D, I'm staying away from the cross-product based ...
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1answer
14 views

What is the proper name of a point a long a smooth curve where the radius changes but not direction of curvature?

What you call a point a long a smooth curve where the radius changes? When it reverses curvature, it’s an “inflection point”. What if it doesn’t change direction, just radius? I seem to remember ...
0
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1answer
70 views

Gaussian Curvature

I am able to show that if a curve lies in a plane then it's curvature at a point $p$ is $$\kappa=\lim_{\mu\to 0}\frac{\sigma}{\mu}$$ where $\mu$ is the length of a segment of the curve containing p, ...
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0answers
20 views

Is the standard embedding of the torus the tight embedding?

Definition of tight: A mapping of a surface into $\mathbb{R}^3$ is called tight if its image, equipped with the induced metric, has minimal total absolute curvature. (A definition of this kind is ...
2
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0answers
27 views

“Projection of metric” vs. “projection of curvatures”

Suppose we have a submanifold $M^n$ which is embedded in manifold $M^{n+2}$ and $g_{\mu \nu}$ denotes the metric of $M^{n+2}$. We know that the induced metric on the submanifold is defined by ...
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0answers
55 views

Total curvature of a piecewise smooth planar curve

I am trying to show that the total curvature of a piecewise smooth planar curve $\alpha$ has the property that $\displaystyle \int_{C}\kappa $ $\text{d}s + \displaystyle \sum_{j=1}^{l} \epsilon_{j} ...
0
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1answer
26 views

Definition of Curvature

Derivation of Definition of Curvature $$ \phi = tan^{-1}(y')$$ Differentiate with respect to s. $$ \frac {d \phi} {ds} = \frac d {ds} (tan^{-1} y')$$ Chain rule. $$ \frac {d \phi} {ds} = ...
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0answers
22 views

Torsion of an asymptotic curve with nonzero curvature

I wish to solve the following problem using the matrix of the shape operator $S_{P}$. Suppose $K= \text{det} S_{P} <0$, and $C$ is an asymptotic curve with curvature $\kappa (P)$ nonzero. I want to ...
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1answer
51 views

Interpolation between two points

I am looking for an interpolation between two points $P$ and $Q$. I need the curve to have derivative (direction) $\vec{v_1}$ at point P and $\vec{v_2}$ at point Q. In addition, there is a maximum ...
1
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1answer
31 views

Sectional Curvature of Paraboloid

I seem to have made a mistake while doing the simple exercises of calculating 2D sectional curvature of paraboloid $z=\frac a2 (x^2+y^2)$. I used polar coordinates to do this; ...
0
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2answers
87 views

Intersection and Curvature of Surfaces

a) Describe the intersection $(C)$ of sphere $x^2 + y^2 + z^2 = 1$ and the elliptic cylinder $x^2 + 2z^2 = 1$, and find out the total arc-length of this intersection. b) Determine the points on the ...
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1answer
154 views

Curvature of a Planar Graph

a) Show that the curvature of the planar graph r = f($\theta$) at a general point is $\kappa(\theta)$ = $\frac{[2f'^2(\theta) + f^2(\theta) - f(\theta)f''(\theta)]}{[(f'^2(\theta) + ...
2
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1answer
188 views

Curvature of function given by points

I have function surface given by f(x,y) values (plus x and y). I know normal at each point and neighbourhood points as well. Is it possible to use this information to calculate Gaussian and Mean ...
1
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1answer
40 views

Do closed submanifolds have nonzero curvature on a non-zero-measure set

If $M\subset \mathbb{R}^n$ is a $m$-dimensional submanifold, does the set with non-zero gaussian curvature always have $m$-dimensional Hausdorff measure greater than zero? For a manifold homeomorphic ...
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0answers
16 views

How to design own the parametric vector?

I try to design the parametric vector that looks like a roller coaster I know that my equation will like $r(t) = A\sin(t)i + Btj+ C\cos(t)k$, but i want ...
2
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0answers
50 views

Is there a surface S$\subset R^3$ whose Gaussian curvature is -1 at each point S?

Is there a surface $S\subset \Bbb R^3$ whose Gaussian curvature is $-1$ at each point $S$? At first I think this does not make a sense. But googling and googling.. I found a 'final exam problem' ...
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1answer
27 views

Proving for differentiable curve $\kappa_1=1+\kappa_2^2$

Let $\gamma$ be a differentiable and regular curve in $\mathbb{R}^3$ which satisfies $|\gamma|=1$. Prove that for every point $$\kappa_1=1+\kappa_2^2$$ where $\kappa_1$ is the curvature of $\gamma$ ...
5
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1answer
118 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
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0answers
26 views

Embedding of manifolds of constant negative curvature

Consider the manifold of constant negative curvature $G=SL(2, R)/\Gamma$ where $\Gamma$ is such that $G$ is compact (I have no special constraint on $\Gamma$). I know that by the Whitney embedding ...
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2answers
129 views

Curvature of curve not parametrized by arclength

If I have a curve that is not parametrized by arclength, is the curvature still $||\gamma''(t)||$? I am not so sure about this, cause then we don't know that $\gamma'' \perp \gamma'$ holds, so the ...
3
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2answers
46 views

Number of Curvature Maxima of a 2D Cubic Bezier curve

I am trying to prove that a standard cubic Bezier curve can only have at most 2 curvature maxima over $t \in [0,1]$. Assuming that no 3 adjacent control points are colinear, the curvature will either ...
3
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1answer
54 views

Reparametrize by arc length and find the signed curvature

Let $$g(t)=(e^t \cos(t),e^t \sin(t))$$ Reparametrize $g(t)$ by arc length starting at $0$, then find the signed curvature of the unit-speed reparametization. I found that the ...
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1answer
128 views

What are some interesting topics for a differential geometry essay?

As an assignment I have been asked to write an essay on an aspect related to differential geometry. My lecturer has proposed topics such as architecture, navigation and computer graphics. I personally ...
2
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1answer
70 views

Curvature kappa with known acceleration, unit normal and unit tangent vectors

The acceleration of a particle is $a(t) = (4\sin t \cos t)T(t) + (4e^t \sin^2 (t/6))N(t)$, where $T(t)$ is the unit tangent vector and $N(t)$ is the unit normal vector. At $t = \pi/2$, the speed of ...
3
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0answers
57 views

Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature ...
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0answers
24 views

Principle Lines of curvature

In the text by Manfredo P. Do Carmo entitled Differential Geometry of Curves and Surfaces, an analysis of the principle directions is made near a non umbilic point on pp 160-161. I have followed his ...
2
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1answer
56 views

Curvature in complex analysis

Apparently, the curvature of the image contour of the unit circle $|z| = 1$ under the conformal mapping $w = f(z)$ is $$\frac{1}{|zf'(z)|} \text{Re}\bigg(1 + \frac{zf''(z)}{f'(z)}\bigg).$$ How does ...
0
votes
1answer
25 views

Curvature of curve

$r(t) = (-3sint)i + (-3sint)j + (cost)k$ I got as far as:$$||r'(u)|| = sqrt{(18cos^2u + sin^2u)}$$ But I cannot evaluate $\int_0^t||r'(u)||dt$
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0answers
22 views

A non-rigorous explanation of curvature of curves in 3-space

In a second year calculus class there is usually the definition of curvature as $$\kappa=\left|\frac{d\bf{T}}{ds}\right|$$where $\bf{T}\rm(t)=\frac{r^\prime(t)}{|r^\prime(t)|}$ is the unit tangent ...
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0answers
21 views

Point of maximum curvature

I'm trying to find the point where the change in y becomes more or less independent of the change in x in this equation: $$y=7\times 10^{-5}\exp\left(\dfrac{-x}{3.0453\times 10^{-7}}\right)+9.1\times ...