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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.

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Hypothetical City's Foundation vs Curvature of the Earth

If there is a City whose foundation is 1500 miles square, then how far down in our present earth's curvature would one have to excavate for its corners to remain ground level? Thanks, Kenneth Layne
KennethLane
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Gaussian curvature of the (unit) sphere

I am looking for the most straightforward way of showing that the Gaussian curvature of the unit sphere is 1. How could I go about this? I found a way for this which involves showing that $${R^{\theta}}_{\phi} = \sin(\theta) \; e^{\theta} \wedge…
John
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curvature and center of curvature of a quadratic parametric curve

A General Statement I read is: Given a parametric curve $$\vec r(s)=(x(s),y(s))$$ the tangent vector is represented by the first derivative, $$\vec r'(s)=(x'(s),y'(s))$$ Then the vector $\vec n=\vec r''(s)$ is perpendicular to the Tangent and…
ONe
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Sign of a curve

Concept: "signed" [positive/ negative] curvatures, Literature overview: I come to see some rules like curvature is positive/ negative based on the turn of the curve (right/ left) (or) based on concavity/ convexity. Can someone help me with the…
Saideep Pavuluri
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how can I show this equation?(curvature)

$$\dfrac{dT}{ds} = \kappa N$$ $\kappa$ = curvature $T$ = unit tangent vector $N$ =unit normal vector How can i show this equation? I don't know why direction of $\dfrac{dT}{ds}$ is direction of $N$.
Dobu
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Curvature, Torsion and Change of coordinates

Is there a general way so we can find curvature and torsion of a curve $$\gamma : \mathbb{R} \rightarrow \mathbb{R}^n$$ where $$n=2,3$$ from a coordinate system to a new coordinate system?
Stavros Cohen
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Curvature derivation for arbitrary parameterization

Note: ${x_p}$ and ${x_{pp}}$ are derivative of x wrt p once and twice, respectively. I am deriving the curvature for an arbitrary curve C=(x(p),y(p)). The formula for derivation satisfies $\frac{1}{|C_p|}$$\frac{\delta}{\delta…
user1696420
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connection between cubic interpolation and curvature of a function

I am reading "Numerical Optimization", second edition, written by Joerge Nocedal and Stephen Wright. On page 59 the authors claim: Cubic interpolation provides a good model for functions with significant changes of curvature. I don't understand…
user359583
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Prove a curvature identity

Show that N = (dT/dt)/(abs (dT/dt)) using dT/ds = k N
Math
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Invariance of curvature in Curvature Scale Space

Paper is here I have read this paper. But i don't know what the Invariance means in page 3 and 5. Calculation/estimation of κ is NOT invariant and Invariant to viewpoint. What's the meaning of these sayings? When i read "Invariant to viewpoint", i…
jack fukey
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How can I find the curvature of $r(t) = ( \sqrt{2}t, e^t, e^{-t} )$ using the unit tangent vector?

I've been given the function $r(t) = ( \sqrt{2}t, e^t, e^{-t} ) $ and asked to find: (a) the unit tangent and unit normal vectors, and (b) the curvature using the formula $|T'(t)| / |r'(t)|$ only (i.e. other formulae for K disallowed) My answers…
Katherine Rix
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Principal Curvatures of axisymmetric function

I have an axisymmetric function ( z = f(r) with r being the radius ) describing how a specific material is shaped under an applied force, a somewhat complicated function implemented in Mathematica. Now I need to calculate the two principal…
Carsten
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Find point where radius of curvature is minimum

Find the point where radius of curvature is minimum for the curve $$x^2y=a\left(x^2+\frac{a^2}{\sqrt{5}}\right)$$
harish
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In case of Folium of Descartes $x^3+y^3=3axy$ , prove that the radius of curvature at the point $(3a/2,3a/2)$ is numerically equal to $3\sqrt2 a/16$.

In case of Folium of Descartes $x^3+y^3=3axy$ , prove that the radius of curvature at the point $(3a/2,3a/2)$ is numerically equal to $3\sqrt2 a/16$. The solution given in the book is as follows: If $f(x,y)=x^3+y^3=3axy=0$ we may easily obtain…
Arthur
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Curve equation from curvature

I'm trying to get a curve equation from curve curvature (trying to design a bent air pipe with later boundary layer separation). I should also mention that my math is pretty rusty. My idea was to have a bend curve that would have a linear decreasing…
user2882635
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