The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
2answers
21 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
34 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 ...
1
vote
0answers
44 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 ...
1
vote
0answers
30 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 ...
0
votes
1answer
23 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
votes
0answers
12 views

Confusion about quasi-concavity

If I have a function such as $y=(x^a + y^b)^2$ 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 ...
0
votes
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 ...
1
vote
2answers
50 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 ...
1
vote
0answers
15 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 ...
1
vote
1answer
31 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 ...
1
vote
2answers
49 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 ...
0
votes
1answer
40 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
votes
1answer
35 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
vote
1answer
45 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 ...
1
vote
1answer
40 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) \\ ...
0
votes
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 ...
-1
votes
1answer
45 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
vote
1answer
67 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 $(0, 0)$.
0
votes
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
90 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. ...
1
vote
1answer
55 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: ...
7
votes
2answers
100 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
votes
1answer
37 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?
1
vote
1answer
61 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 ...
2
votes
1answer
31 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| ...
1
vote
0answers
23 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 ...
1
vote
2answers
61 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 ...
5
votes
2answers
71 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 ...
1
vote
1answer
38 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 ...
1
vote
1answer
42 views

Gaussian curvature of a parallel surface

Question 11 section 3.5 in Do carmo part c.Let a surfae x have constant mean curvature equal to c does not equal 0 and consider the parallel surface to x at a distance 1/2c. Prove that this parallel ...
3
votes
2answers
63 views

Computing the Gaussian curvature of this surface $z=e^{(-1/2)(x^2+y^2)}$.

Compute the Gaussian curvature of $z=e^{(-1/2)(x^2+y^2)}$. Sketch this surface and show where $K=0 $, $K>0$, and $K<0$. So would the easiest way to do this question be to construct a ...
0
votes
1answer
18 views

Mean curvature an asymptotic directions

I'm a little confused by this question I've come across in do carmo while studying for my final. (Sect 3.2 #7) Show that if the mean curvature is zero at a nonplanar point, then this point has two ...
1
vote
2answers
51 views

Gauss Curvature of a Surface

Find the Gauss curvature of a surface with the first quadratic form: $$\mathrm{d}s^2 = \mathrm{d}u^2 + 2\cos a(u,v)\mathrm{d}u\,\mathrm{d}v + \mathrm{d}v^2.$$ I have found $E$, $F$, and $G$. $E = ...
2
votes
0answers
40 views

Curvatures of 4-Dimensional Parallel Curve

Let $\vec{\alpha}: I \rightarrow \mathbb{R}^4$ be an arc-length parametrized curve in $\mathbb{R}^4$ with curvatures $k_1, k_2, k_3$. The principal normal unit vector is $\vec{n} = ...
1
vote
1answer
52 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
1
vote
0answers
20 views

Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
2
votes
0answers
39 views

Frenet formulas for curves in arbitrary Riemann manifold

As far as I understand, to have Frenet formulas one would need a curve, embedded in $\mathbb{R^n}$ and, desirably, naturally parametized. But there are homonomic notions of curvature and torsion of ...
1
vote
1answer
34 views

Why use Gauss and mean curvature to characterize a surface's deviation from being “flat” at one point?

We know for a 2-dimensional surface there are two orthogonal principal directions at every point, where the principal curvatures $\kappa_1$ and $\kappa_2$ are the two ends of the curvature spectrum ...
1
vote
1answer
47 views

Binormal vector, $B(t)$, is independent of $t$?

What does it mean that the binormal vector, $B(t)$, is independent of $t$? Also, if the curvature, $k(t)$, of a curve equals $\frac{1}{t}$, where $t\ge 0$, does the curve posses any points in which ...
2
votes
3answers
85 views

Finding the Total Curvature of Plane Curves

I'm trying to find the total curvature (or equivalently, rotation index, winding number etc.) of a plane curve (closed plane curves) given by $$\gamma(t)=(\cos(t),\sin(nt)), 0\leq t\leq 2\pi$$for each ...
1
vote
0answers
48 views

Proof check for critical point definition with mean curvature

I'm currently trying to prove: "Definition: We say that a surface $S \in R^3$ is minimal if it is a critical point for the area functional" Starting with this: "If we consider a family of smooth ...
1
vote
1answer
55 views

Cartan formalism calculation

Just to test out the Cartan formalism, I decided to apply it to the sphere. So, it admits a metric, $$\mathrm{d}s^2 = \mathrm{d}r^2 + r^2 \sin^2 \phi \mathrm{d}\theta^2 + r^2 \mathrm{d}\phi^2$$ from ...
1
vote
2answers
84 views

Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why?

What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?
3
votes
1answer
54 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
1
vote
1answer
57 views

Find the Gauss Curvature of This Particular Metric:

Let D be an open disc centred at the origin in $ \Bbb R^2 $. Give D a Riemannian metric of the form $ (dx^2 + dy^2)/f(r)^2 $, where $ r = \sqrt{x^2 + y^2} $ and $ f(r) > 0 $. Show that the Gauss ...
3
votes
1answer
54 views

Curvature of saddle by definition

I'm trying to compute the principle curvatures of the saddle $M$ defined by $z= y^2 -x^2$ at the point $p = (0,0,0)$, but I know my computations are wrong. Maybe you can help to see where I went ...
3
votes
1answer
30 views

How to evaluate the curvature by using normal gradient of a function?

The gradient of the function $\phi$ is: $$ \nabla\phi =(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}) $$ and the unit normal is: $$ ...
2
votes
1answer
123 views

Curvature and the Arrow Pratt Absolute Risk Coefficient

So I'm in my first year of grad school, and I'm taking a decision analysis course. One of the topics we're covering is risk aversion, and with that comes discussion of the Arrow Pratt Absolute Risk ...
0
votes
1answer
43 views

Similarity in formulae for curvatures

At each point on a curve $\mathcal{C}$, the tangent vector is parallel to a non-vanishing vector field $\mathbf{F}$. Show that the curvature $\kappa$ of $\mathcal{C}$ is given by: ...
1
vote
0answers
50 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...