# Tagged Questions

Surface is a two-dimensional submanifold of three-dimensional Euclidean space.

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### Rulings of One Sheet Hyperboloid

Let $M$ be a hyperboloid of one sheet satisfying $x^2+y^2-z^2=1$. Show that $x(u,v)=(\frac{uv+1}{uv-1},\frac{u-v}{uv-1},\frac{u+v}{uv-1})$ gives a parametrization of $M$ where both sets of parameter ...
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### Help with understanding a proof of compact surface having an elliptic point

In my studies of differential geometry from do Carmo's book, I have come across a very nice claim which states that a regular compact surface has an elliptic point that is a point with positive ...
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### determining equation of a surface

I was wondering if there is a way to determine the equation of a surface if three R2 linear equations are known. I work in a research lab that produces a lot of correlation equations (mx+b), and we ...
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### Does this base change yield another dominant morphism?

Here's something that seems to be true, or at least I hope it to be true, but I'm unable to prove it: Let $S$ be a $k$-rational surface and $B$ a curve, both projective, smooth and geometrically ...
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### Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that $X$...
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### Geodesics on surfaces of revolution about z axis with negative curvature

This is a question in differential geometry of surfaces that I could not do We are given S a surface of revolution about the z axis with everywhere negative Gaussian curvature. We are to show that the ...
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### Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $\gamma (t) = (sin(t)+2,0,t)$ on XZ plane for $t \in R$, ...
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### Proving subset of regular surface - hyperboloid - is a regular surface

I have stumbled upon this in differential geometry dealing with regular surfaces: We define the following surface (a hyperboloid) as $K = \{ (x,y,z) \in R^3 | x^2+y^2-z^2 = 1 \}$ and ...
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### Can a surface of revolution be built from a self-intersected curve?

I'm reading "Differential Geometry of Curves And Surfaces" of Manfredo Do Carmo. There's a point in his book about Surfaces of Revolution which confuses me a lot. Here is the part: The part ...
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### Understanding the first fundamental form of a surface, how the parametrization doesn't matter.

The following is an excerpt from Pressley's Elementary Differential Geometry on the definition of the first fundamental form. However, there are some parts of this concept that I'm unclear about. It ...
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### Minimal surface with radially symmetrical function

The following image is from the book "Regularity Theory for Mean Curvature Flow", by Ecker. I consider the plateau problem, whose goal is to solve minimal surface given fixed boundary values. In ...
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### Area of ​​the surface of revolution of the ellipsoid

I need to find the surface area of an ellipsoid using the equation of an ellipse. I believe my calculations are correct but the formulas I meet on the Internet are complex and have arcsin or arctan in ...
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### How can we calculate the unit normal $\textbf{N}$ of the sphere?

I want to show that the normal curvature of any curve on a sphere of radius $r$ is $\pm \frac{1}{r}$.  The normal curvature is $\kappa_n=\gamma '' \cdot \textbf{N}$, where $\gamma$ is a unit-...
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### Compact surface

To check if the surface $x^2-y^2+z^4=1$ is compact, we have to check if the surface is closed and bounded. Could you give me some hints how exactly we check that? How can we check if it closed and ...
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### Revolution of fractal

How to find the volume and surface area of a shape which made from revolution of Koch Snowflake? (I think the surface area will be an infinity, because length of the Koch snowflake is infinity.) And ...
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### Open subset of a plane [duplicate]

Suppose that the second fundamental form of a surface patch $\sigma$ is zero everywhere. How can we prove that $\sigma$ is an open subset of a plane? The second fundamental form of a surface patch ...
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### Two twisted cubic curves in $\mathbb P^3$ intersect iff they lie in a common cubic surface

Let $C_1$ and $C_2$ be twisted cubic curves in $\mathbb P^3$. I want to prove that they intersect if and only if they lie in common cubic surface, perhaps singular. The second condition can be ...
Show that the ellipsoid $$\frac{x^2}{p^2}+\frac{y^2}{q^2}+\frac{z^2}{r^2}=1$$ where $p$, $q$ and $r$ are non-zero constants, is a smooth surface. To do this do we have to take a parametrization of ...
I want to find the equation of the tangent plane of the surface patch $\sigma (r, \theta)=(r\cosh \theta , r\sinh \theta , r^2)$ at the point $(1,0,1)$. I have done the following: The point \$(1,0,...