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18 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 ...
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1answer
34 views

Find point where radius of curvature is minimum

Find the point where radius of curvature is minimum for the curve $$x^2y=a\left(x^2+\frac{a^2}{\sqrt{5}}\right)$$
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1answer
51 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 ...
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0answers
47 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 ...
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0answers
34 views

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

I am having difficulty proving the converse in part B. I understand part A and have shown that t/k-t/k = 0. I found that n=-1/k, n'-bt= b'+tn = 0, so n' = bt and b'=-tn. However, I am unable to find ...
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1answer
53 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
85 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
45 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
28 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, ...
4
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0answers
35 views

Non-vanishing differential forms and flatness

Let $M$ be a differentiable manifold of dimension $n$. If the tangent bundle is trivial, then the cotangent bundle is trivial, and so are its exterior powers. In other words, on a parallelizable ...
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0answers
12 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
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1answer
35 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
40 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
27 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 ...
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0answers
54 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 ...
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1answer
52 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 ...
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1answer
24 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 ...
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0answers
37 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 ...
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0answers
26 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
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2answers
62 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
34 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
24 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
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1answer
52 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
45 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 ...
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0answers
37 views

Finding curve that minimizes an integral due to constraints

In the euclidean plane I want a smooth curve $\gamma (t)$ which satisfy:$$\gamma (0)=(0,0)\quad\quad \gamma '(0)=(1,0)\quad\quad n\in \mathbb{N}\land n\neq 1\Rightarrow \gamma ^{(n)}(0)=(0,0)\\\gamma ...
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1answer
47 views

maximum curvature of 2D Cubic Bezier

Given a 2D cubic Bezier segment defined by P0, P1, P2, P3, here's what I want: A function that takes the segment and outputs the maximum curvature without using an iterative approach. I have a ...
0
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1answer
33 views

Quasi-concavity of a function of two variables such as $z=(x^a + y^b)^2$

If I have a function such as $z=(x^a + y^b)^2$ with $a$ and $b$ both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative? The ...
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0answers
14 views

point of maximal curvature of a vector of discrete values

How can I approximate the point of maximal curvature given a vector of points rather than a function? I do find the inflection point(s) by comparing the slope within a small window before and after a ...
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2answers
82 views

Global and local coordinates on a manifold, and their relations to curvature

I would be pleased to have some information about coordinates in differential geometry. A) First I would like to check whether or not the definitions I use are correct. (Mainly for the sake of ...
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0answers
21 views

Kahler condition related to Ricci curvature formula of a Hermitian, holomorphic vector bundle over a complex manifold

I read a local formula like this: Under some sort of Kahler condition, $$Ric(h)=-i\partial\bar\partial \log \det(h_{\alpha\bar\beta})$$ where $h_{\alpha\bar\beta}$ is the matrix of the Hermitian ...
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1answer
53 views

Resample Bézier Curve with curvature and number of points constraints

I have an algorithm that implements an uniform resample process throughout a Bézier curve. This is done using a chord parametrization process. However, the results achieved do not accomplish my ...
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2answers
61 views

Angle between two vectors on manifold

I'm parallel transporting a vector along a curve and trying to calculate how much this vector rotates relative to the curve's tangent vector. So if the path is a geodesic then I will get an answer of ...
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1answer
42 views

Proving that a surface is isometric to the plane

A surface $S$ has first fundamental form $du^2 + G(u,v)dv^2$ and curvature $0$. Also the curve $u=0$ is a geodesic when parametrized by arclength. Prove that $G(u,v) = 1$ i.e. that $S$ is isometric ...
4
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1answer
40 views

What curvature conditions make a surface rigid?

Consider a compact surface $S$, possibly with boundary, embedded in $\mathbb{R}^3$, with the induced Riemannian metric. I believe that if $S$ has constant positive Gaussian curvature (that is, $S$ is ...
1
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1answer
52 views

On Constant mean curvature surfaces.

I have two involved questions, firstly, I know that the gauss map sends a surface to the unit sphere, so for a surface $\Sigma\subset\Bbb R^3$, parametrised by $u:U\subset\Bbb R^2\to \Bbb R^3$. Would ...
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1answer
45 views

Show that total curvature of ellipse is $2\pi$

I'm trying to show that the total curvature $$K=\int_C\kappa\,ds$$ is equal to $2\pi$ over the ellipse $C$ with axes $a,b$ (and $\kappa$ is curvature). I computed: $$x(t)=(a\cos t,b\sin t,0) \\ ...
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1answer
20 views

Estimating curvature of oscillatory curve based on global constraints

I have a heuristic question about using global constraints of a problem to make estimates of local features of a curve, such as its curvature. Consider a suitably well behaved function on ...
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1answer
49 views

Calculate the curvature of a space curve at the point M(-1,5,-4)

In order to calculate the curvature for this space curve, do I use this formula? And where does the point $M$ come into this? P.S. This is probably a silly question, but I'm new to differential ...
1
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1answer
77 views

Radius of curvature for the plane curve $x^3 + y^3 = 12xy$.

Could someone help me with this problem? : Determine the radius of curvature for the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$.
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0answers
17 views

Grab major changes along a line

I have a line where there are many points along it. I want to be able to find the major changes in the line and just grab those points rather than grabbing all the points the line has. Here's a quick ...
2
votes
1answer
94 views

Prove the Gaussian curvature $K=0$

If two families of a geodesics on a surface intersect at a constant angle $\theta$, prove that the Gaussian curvature of the surface is zero, i.e. $K=0$. Please explain how to show the question. ...
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1answer
65 views

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ? Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...
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2answers
113 views

Where does this expression of Gaussian curvature come from?

In my Differential Geometry course, we have seen a way to calculate the Gaussian curvature $K$ given a metric expressed as the sum of two Pfaff forms $Q = ω_1^2 + ω_2^2$: we find another Pfaff form ...
3
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1answer
39 views

Calculating curvature of a curve on a the surface $x^2+y^2=1$. [closed]

Find a curve on the cylinder surface $x^2+y^2=1$ in $\mathbb R^3$ such that its curvature is equal to $\frac1{100}$ at each point of this curve. Does this easily generalize to different surfaces?
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1answer
73 views

Maximum/Minimum of Curvature - Ellipse

Find the sum of the maximum and minimum of the curvature of the ellipse: $9(x-1)^2 + y^2 = 9$. Hint( Use the parametrization $x(t) = 1 + cos(t)$) Tried to use parametrization like that, but then ...
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1answer
40 views

Mean curvature flow - implementation fails for some meshes

I am working on piece of software to deal with 3D meshes and I need to smooth some meshes. I have implemented MCF by using this formula $\vec{H} = {{t}\over{2}} \sum_{q \in\ link\ p} \vec{Ne} |e| ...
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0answers
25 views

Weakest curvature assumption for existence of harmonic coordinates

Let (M, g) be a Riemannian manifold. What are the weakest curvature bounds for which one can construct harmonic coordinates on M (or at balls contained in M)? Does anybody maybe know if it is possible ...
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2answers
67 views

Is a tangent to a curve in a hyperbolic plane straight?

Consider a projective plane with an absolute quadric, so that it is a hyperbolic plane. Given a curve I wonder how the tangent to a curve is defined in a plane with constant positive curvature. I ...
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2answers
76 views

Geometry of Commuting Hermitian Matrices

I am a physicist working on a project dedicated to the quantisation of commuting matrix models. In the appropriate formalism this problem is reduced to a quantisation in a curved space -- the space of ...
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1answer
40 views

moebius transforms preserve sum of signed curvatures

Let $P$ be a point where three arcs of circle meet at equal angles (120 degrees). Suppose that the sum of the curvatures (with sign given by orientation) of the three arcs is zero. Is it true that ...