# Tagged Questions

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### 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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### 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 ...
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### 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 $k_2$, ...
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### 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 ...
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### 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?
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### compute principal directions of a cylinder

I calculated the parametric equation of a cylinder, $$x(u,v)=a\cos(u)$$ $$y(u,v)=a\sin(u)$$ $$z(u,v)=v$$ I do not know how to calculate principal directions ? I am not sure what it means neither ...
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### Numerical computation of surface curvature

In 2 dimensions, the definition of curvature of a curve $y = y(x)$ is $$C = \frac{y''}{(1+y'^{2})^{3/2}}$$ and it is easy to estimate the curvature numerically for given ...
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### Prove that Gauss map on M is surjective

Let $M$ be a closed, orientable, and bounded surface in $\mathbb{R}^3$. (a) Prove that the Gauss map on $M$ is surjective. (b) Let $K_+(p) = \max \{0, K(p)\}$. Show that $$\int K_+dA \ge 4\pi.$$ ...
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### Curvature of surface

So lets say I have a mesh and for each face I have the position of its $3$ vertices and the area of the face. So let's say I have a point $p$ on this face and a vector $v$ that goes from $p$ to the ...
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