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2
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0answers
29 views

Parameterization of surface 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
votes
1answer
56 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
vote
0answers
15 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
21 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
42 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
23 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} = ...
1
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0answers
16 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 ...
1
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0answers
24 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
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; ...
0
votes
2answers
80 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 ...
1
vote
1answer
147 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
votes
1answer
39 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
vote
1answer
35 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 ...
0
votes
0answers
12 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
votes
0answers
44 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
vote
1answer
23 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
votes
1answer
108 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
21 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 ...
1
vote
2answers
119 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
votes
2answers
41 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
votes
1answer
51 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
vote
1answer
84 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
votes
1answer
53 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
votes
0answers
46 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 ...
1
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0answers
18 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
votes
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 $\int_0^t||r'(u)||dt$
1
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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 ...
0
votes
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
votes
1answer
42 views

Find point where radius of curvature is minimum

Find the point where radius of curvature is minimum for the curve $$x^2y=a\left(x^2+\frac{a^2}{\sqrt{5}}\right)$$
0
votes
1answer
319 views

Proof of curvature of a curve described by Polar Coordinates

I have been looking everywhere for a proof on the curvature of a plane curve that is represented in polar coordinates. I am close in proving it myself, however, I seem to be missing a particular part ...
1
vote
1answer
136 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
vote
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
votes
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 ...
1
vote
2answers
65 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 ...
0
votes
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, ...
0
votes
0answers
14 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
44 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
votes
1answer
53 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 ...
1
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0answers
32 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 ...
0
votes
0answers
94 views

Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...
1
vote
1answer
57 views

Which axiom makes a vector space flat?

First of all, I'm not sure if this question even makes sense, i.e. is there a notion of curvature on a vector space structure. However, when dealing with vector spaces (here I am mostly thinking of ...
1
vote
1answer
27 views

An assumption used to derive the curvature tensor for Riemannian submersions

I was reading the literature about Riemannian submersions, and I came across the result showing the relation between the curvature tensor $\bar{R}$ in a manifold $M$ and the curvature tensor $R$ in a ...
1
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0answers
50 views

curvature of a plane curve

I'm trying to prove the formula to calculate the curvature of a plane curve. But I end up with the wrong sign and can't figure out why: I want to proof $\kappa(t) = \frac{\dot c(t) \cdot \ddot ...
3
votes
0answers
30 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
2
votes
2answers
77 views

Ricci Tensor, Curvature and Scalar Curvature computation from definition

I am studying a little of Riemannian geometry and I am having some problem in making the connection between two expressions of Ricci tensor, curvature and scalar curvature. Well, in the book that I am ...
0
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1answer
41 views

Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
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2answers
35 views

How to proof curvature and torsion are independent

As we know that curvature describes the change of curve in tangent and normal plane, while torsion describes the change the curve in binomial and normal plane. Assume we have a trajectory with length ...
1
vote
1answer
60 views

Definition of CAT(0) metric space

I have a question regarding the definition of CAT(0) spaces. I am using the following definition: $X$ complete metric space is CAT(0) if $\forall z,y \in X$, $\exists m \in X$ such that $\forall x ...
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0answers
52 views

Shape operator and orthogonality of eigenvectors

When studying differential geometry (at a hobby level) I always run into problems when it comes to the varying notations and statements about the shape operator ...