Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [curvature]

In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. (Def: http://en.m.wikipedia.org/wiki/Curvature_tensor)

In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. Reference: Wikipedia.

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds. Reference: Wikipedia.

1913 questions
0
votes
0 answers

Do principal curvatures $\kappa_1, \kappa_2$ represent absolute maximum/minimum normal curvatures?

According to Wolfram MathWorld [0], when talking about principal curvatures, $\kappa_1$ is the maximum and $\kappa_2$ is the minimum normal curvature. However, I also noticed that Grasshopper documentation [1] defines $K^1$ as the principal…
jordi
  • 279
0
votes
2 answers

Interpretation of the curvature of an helix

Consider the helix $h(t)=(a\cdot cos(t),a\cdot sin(t),bt)$ and the circle $c(t)=(a\cdot cos(t),a\cdot sin(t),0)$. I know that the curvature of them are: $k(t)_h=\frac{a}{a^2+b^2}$ $k(t)_c=\frac{1}{a}$ respectively. The curvature measures the…
Marius Alexandru Marinescu
  • 1
0
votes
0 answers

radius of the curvature at the origin

Find the radius of curvature at the origin of the curve $$y^2-2xy-3x^2-4x^3-x^2y^2 = 0$$ In this math, $$\frac {dy}{dx} = \frac {y+3x+6x^2+xy^2}{y-x-yx^2}$$ and $$\frac {dx}{dy} = \frac {y-x-yx^2}{y+3x+6x^2+xy^2}$$ At $(0, 0)$ both $$ \frac…
Al-amin Chowdhury
  • 1
0
votes
0 answers

Arc length of locus of center of curvature

If $ S_1 $is the arc length of the locus of center of curvature of a curve, then show that $$\dfrac{dS_1}{ds} =\dfrac{ \sqrt{(\kappa^2\tau^2 + \kappa^{'2})}}{\kappa^2} =\sqrt{ (\dfrac{\rho}{\sigma})^2+\rho^{'2}} $$ $\kappa$= curvature ,$\,\rho = $…
Atul Singh
  • 29
0
votes
1 answer

3D positively curved space

If we consider 2D euclidean surface consists of infinite concentric circles and 3D euclidean surface consists of infinite concentric spheres. If 2D surface is positively curved the radius of the circles at a distance $r$ from the origin becomes $R…
Apashanka Das
  • 11
  • 2
0
votes
1 answer

Generalization of curvature to "mixed units"

Every discussion of curvature I have ever seen involves an independent variable $x$ and a dependent variable $y(x)$, and both $x$ and $y$ represent some kind of distance. What if this is not the case? What if, for instance $x$ is a time and $y$ is…
bob.sacamento
  • 4,028
0
votes
1 answer

Estimating Gaussian Curvature from 3x3 surface grid without parameterization

Given a 3x3 fixed grid of surface coordinates (the center points of the cells of an elevation raster), is it possible to estimate the Gaussian curvature for the center point of this grid without parameterizing all 9 points into an equation?
TimB
  • 11
0
votes
0 answers

Can Menger curvature be used to determine if 3 points are closer to an oval or circle in shape?

I'm trying to understand Menger curvature in terms of shape determination in isolated cases of 3 well-distributed (defining) coordinate points that could form part of a shape that may be either a circle or an oval. It seems that the menger…
ina
  • 455
0
votes
3 answers

Laymen’s terms explanation of curvature

I’m attempting to explain curvature in layman’s terms to my class before explaining the formula. I like to do this first to give my students an idea of what we are finding. Some people explain curvature as a “measure of how fast a curve is…
B flat
  • 792
  • 1
  • 6
  • 17
0
votes
1 answer

curvature of a bended piece of paper?

According to the Theorema Egregium, a surface that is transformed isometrically, retains its gaussian curvature at every point. This means that for example a piece of paper cannot be folded into a sphere. We could do it with pizza dough, but paper…
user56834
  • 12,925
0
votes
2 answers

Show That The Perimeter Of The Evolute Of The Ellipse Is $4(\frac{ a^2}{b} - \frac{b^2}{a})$.

According to the question we have to find the perimeter of evolute of an ellipse. For an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, the equation of its evolute is $$(ax)^\frac{2}{3}+(by)^\frac{2}{3} = (a^2 - b^2)^\frac{2}{3}$$ To get the…
Afreen
  • 1
0
votes
1 answer

How to calculate effect of elevation change on vehicle normal force?

My name is Ricky and I run a site called Race Optimal. http://www.raceoptimal.com/about/physics I'm working on converting the 2D physics model to 3D. I have an algorithm to calculate a continuous piecewise polynomial for the x, y, and z…
Ricky Vesel
0
votes
1 answer

Defining curvature based on unit normal vector

I have a basic question regarding the definition of a curvature. Most of my searches revealed the following: κ = -T⋅$dN/ds$, where T is the tangent vector, N is the normal vector, s is the arc length, k is the curvature. I use a CFD code where the…
Saideep
  • 11
0
votes
1 answer

Calculate curvature of wave

I am looking for a way to calculate curvature of this wave (pic attached) in matlab. Sinc Wave I have generated this wave form in Matlab. t2 = linspace(6,2); y2 = sinc(t2); subplot(212),plot(t2,y2); xlabel('Time…
Basit
  • 1
0
votes
1 answer

Math Rap fact check

I'm a high school math teacher and I write math raps every year for my students. I'm working on my lyrics and I need help making sure something is mathematically accurate. I'd like to make reference to a 3 dimensional space with zero curvature…
Mike Andrejkovics
  • 339
1 2
3
4