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13 views

Arc length and curvature for logistic curves [on hold]

How can arc length and curvature for logistic or sigmoid curves can be calculated? Consider the logistic curve given by $y = \frac{y_i-y_f}{1 + \left(\frac{x}{C}\right)^{1/B}} + y_f$ where, ...
2
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0answers
35 views

Continuity of normal curvature

I want to show that the normal curvature is a continuous function. At first, here is the definition of normal curvature at point $p \in M \subset \mathbb{R}^3$ in direction $\mathbf{u} \in T_{p}M$: ...
3
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1answer
24 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
73 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
36 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
60 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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0answers
17 views

Delaunay surfaces - plane as surface of revolution

According to Wikipedia (and other sources) "Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These ...
1
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1answer
14 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. ...
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0answers
16 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
11 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 ...
0
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0answers
25 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 ...
2
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0answers
33 views

Parameterization of surfaces of revolution with constant mean curvature

Let $x(u,v) = (g(u), h(u) \cos v, h(u) \sin v)$ be a parameterization of a surface of revolution $M$, arising from rotating the regular curve $\alpha(u) = (g(u),h(u),0)$ around the $x$-axis with ...
0
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1answer
65 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, ...
1
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0answers
16 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
24 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 ...
0
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0answers
50 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} ...
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1answer
24 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
20 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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0answers
25 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
vote
1answer
26 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; ...
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2answers
83 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
150 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
185 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
36 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
13 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
47 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' ...
1
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1answer
26 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 $\gamma$ ...
4
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1answer
112 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 ...
1
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0answers
24 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
122 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
42 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
52 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 ...
1
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1answer
114 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
57 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
50 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
20 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
54 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
23 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
20 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
19 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 ...
0
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1answer
45 views

Find point where radius of curvature is minimum

Find the point where radius of curvature is minimum for the curve $$x^2y=a\left(x^2+\frac{a^2}{\sqrt{5}}\right)$$
1
vote
1answer
138 views

Prove that the line is tangent to the curve at the point.

Hello can someone please walk me through part a and b of the below question? I really want to understand it but am having a hard time figuring out the solution. I know how to calculate curvature for a ...
1
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1answer
57 views

A proof that involves Torsion, curvature, and differentiation that equates to 0

I know that I am supposed to write alpha = lambda*T + mu*N + v*B then differentiate then use the fact that {T,N,B} is a basis in R^3. I am just unsure how to write alpha as a combination using the ...
0
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2answers
96 views

A curve internally tangent to a sphere of radius $R$ has curvature at least $1/R$ at the point of tangency

Suppose $a$ is an arc length-parametrized space curve with the property that $\|a(s)\| \leq \|a(s_0)\| = R$ for all $s$ sufficiently close to $s_0$. Prove that $k(s_0) \geq 1/R$. So, I was going ...
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2answers
83 views

Tractricoid as a pseudosphere (surface with constant negative curvature)

How to motivate/calculate/prove see that the tractricoid, i.e. a tractrix rotated about its asymptote, has a constant negative curvature? What are the hyperbolic lines on a tractricoid and how to see ...
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0answers
33 views

clear statement about the relation between curvature and rotating vectors along a loop

in my research, I need to understand the relation between the formal definition of Riemann curvature tensor ($R_{jkl}^i$)and the ``vector rotation" approach: parallel transport a vector along a loop, ...
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0answers
15 views

How to draw a border at a specific distance from a cylinder outline

I have a small cylinder (Cylinder A) with its minor radius of A. Minor radius measures the minor radius of the cylinder ellipse. I need to draw a border at a distance of X from the border of ...
2
votes
1answer
45 views

Monotonic curvature and self intersections.

I'm trying to prove that if $\alpha: I \to \Bbb R^2$ is a differentiable curve, where $I$ is an interval, has strictly monotonic curvature, then $\alpha$ has no self-intersections. My attempt: We ...
2
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1answer
57 views

The curvature of a Cycloid at its cusps.

My lecturer proposed a question to particular result regarding the curvature of a Cycloid (generated by circle of radius 1) at its cusps. Having left it as an open problem, I thought it'd be ...
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0answers
36 views

Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...