For questions about surfaces.

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Area of a bounded region $R$ of surface $z = f(x,y)$ is $A = \int\int_{Q} \sqrt{1 + (\partial_x f)^2 + (\partial_y f)^2}\; dxdy$

This is Exercise 5 from do Carmo section 2.5 (page 100). Area of a bounded region $R$ of surface $\left\{ z = f(x,y)\right\}$ is $A = \int\int_{Q} \sqrt{1 + (\partial_x f)^2 + (\partial_y f)^2}\; ...
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Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
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Finding all points on a surface where the tangent plane is parallel to the plane 5x+3y-z=0

Consider z=f(x,y)=x^3 + 2xy + y Find all points on the surface where the tangnent plane is parallel to the plane 5x+3y-z=0 So I took the gradient of f (x,y,z) and got (2x^2 i , 2x+1 j , -1 k) And ...
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Gauss-Bonnet Theorem, External Angles and Orientation

The Global Gauss-Bonnet Theorem states: Let $R\subset S$ be a regular region and $C_1,\ldots,C_r$ be closed, simple, piecewise regular curves forming the boundary of $R$. Suposse $C_i$ is positively ...
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Preservation of the cross product by parametrization

Let $S$ be a regular surface and $X:U \subset \mathbb{R}^2\longrightarrow X(U)\subset S\subset \mathbb{R}^3$ a local parametrization. Does the following hold? If $e_1, e_2$ are two linearly ...
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28 views

Intersection of two spherical caps in $(n+1)$-dimensional Euclidian space

I want to compute the intersection area of two spherical caps whose tops are placed on two orthogonal axes in $\mathbb{R}^{n+1}$. In spherical coordinates, the surface element is given by $$ \,dS = ...
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What is a smooth family of divisors?

Suppose that $S$ is a smooth complex projective surface ($\mathbb C$-scheme, reduced, irreducible...). What do algebraic geometers usually mean with the term a smooth family of divisors in $S$? ...
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The relationship between Ricci and Gaussian curvatures

Why do we have that for a surface (dimension $2$) that $$\text{Ric}(X, Y) = K \langle X, Y \rangle ,$$ where $K$ is the Gaussian curvature and $X, Y$ are vector fields?
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Disk - Surface Area [closed]

Can anyone tell me what is the formula to find out the surface area of a disk if it provide with thickness and diameter
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22 views

Giving parametrization of Hyperboloid

I'd like to give a parametrization of an two sheet hyperboloid, such that this parametrization covers both sheets (Without using $±$ symbols). Does that parametrization exist?
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Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
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Blow up of Hirzebruch surface

I would like to prove that the blow up of $n$-th Hirzebruch surface $F_n=\mathbb P(\mathcal O \oplus \mathcal O(n))$ at a point outside the exceptional section is isomorphic to the blow up of ...
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About rational curves on elliptic K3 surfaces

I am trying to read this paper http://arxiv.org/pdf/math/9902092.pdf. I have some trouble at the end in Corollary 3.27, and also Proposition 3.24. 0)Morally speaking my main trouble is that i don't ...
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Orthogonal surfaces

Prove that the three surfaces of the family $xy/z=u$ $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}=v$, $\sqrt{x^2+y^2}-\sqrt{y^2+z^2}=w$ that pass through just one point are orthogonal I´m assuming that first I ...
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Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
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Completing compact surfaces with boundary to closed surfaces in $\mathbb R^3$

My question is whether any compact smooth surface in $\mathbb R^3$ (with smooth boundary) can be completed to a closed smooth surface in $\mathbb R^3$ without boundary? It is easy to complete it to an ...
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1answer
36 views

Curvature of plane curves

What is the neatest way to derive the formula for the curvature $\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $?
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Signed unit normal

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
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Conical surface with negative curvature

I was reading some physics papers and I read about cones possessing negative curvature on the tip (and k = 0 everywhere else). Basically, to build these surfaces instead of removing a sector of the ...
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23 views

How can this be the surface area of an intersection of cone and cylinder?

I have an exercise that requires me to calculate the surface of the intersection between two curves: $Z_1:z^2=x^2+y^2$ and $Z_2:x^2+y^2=2y$. So what I did is the following: Parametrise the cylinder ...
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A regular surface with non zero mean curvature is orientable

How can I prove that any regular surface with non zero mean curvature is orientable? UPDATE: The surface is embedded in $\mathbb{R}^3$.
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Finding surface area S using area of projection of S??

I was going over my calculus textbook and came across a question about surface area. and question is as follows. Let S be a parallelogram not parallel to any of the coordinate planes. Let ...
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Re-initialize a 3D spline surface using different control points

I have a 3d spline surface is that is modeled with 65 points where the x,y,z position of each point is known. I want to keep the surface shape, but have different x,y positions for my control ...
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Is that a regular surface?

Let $\mathbf{r}:(a,b)\times (0,1)\to\mathbb{R}^2$ be a injective application, given by: $$\mathbf{r}(u,v)=A(u)+v\cdot B(u), \forall\ (u,v)\in (a,b)\times (0,1)$$ where $A,B:(a,b)\to\mathbb{R}^2$ are ...
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A torsion-free sheaf of rank 1 on a surface

Let $X$ be a surface and $E$ be coherent sheaf on $X$. Now there is always a natural map $\mu:E\longrightarrow E^{\vee\vee}$. The kernel of this map is precisely the torsion subsheaf of $E$. Now if ...
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The curvatures of a transformed surface under a similarity transformation

Setup: Let $f:\mathbb R^3\to\mathbb R^3$ be a similarity transformation. Then $f=rA+b$ for some fixed orthogonal matrix $A$, vector $b$ and nonzero real $r$. Suppose $S$ is a surface, and $S'=f(S)$. ...
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25 views

How to determine the area of the paraboloid enclosed by the cone?

Is it possible to determine the exact area of the paraboloid that falls inside the cone? I've been trying for days without success...
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How to construct a surface with a closed curve?

in 3-dimension, suppose that there is a smooth closed curve $C$. Can I say that there is a smooth simply connected(no holes) surface whose boundary is $C$? and is it unique?(I guess not) like ...
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1answer
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Triangulations of surface.

Let $R$ be e regolar region of a surface $\Sigma$ such that $R$ is the closure `of an open set whose bourdary $\partial R$ is the union of simple closed regular curves. Let $T$ be a trangulation of ...
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1answer
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Square surface with four fixed points

I'm looking for a function of two variables, $f(x, y)$, that satisfies the following constraints: $f(0, 0) = z_1$ $f(0, 1) = z_2$ $f(1, 0) = z_3$ $f(1, 1) = z_4$ and within the unit square, it ...
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Curve Orientation on a Surface

Let $S\subset\mathbb{R}^3$ be an orientable surface, but not oriented (yet). Let $X:(u_1,u_2)\in U\subset \mathbb{R}^2\longrightarrow X(U)\subset S$ be a local parametrization of the surface $S$. We ...
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finding a point on a surface? the surface is an ellipsoid

I have drawn the cross-sections of the surface $2(x-1)^2 + (y+2)^2 +z^2 = 2$ for the given planes, but am now asked to write down a point which is on the surface. I have no idea how to go about this, ...
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Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'?

For me, intuitively, integral $2\pi y~dx$ make more sense. I know intuition can not be proof, but by far, most part of math I've learned does match with my intuition. So, I think this one should 'make ...
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Find the area of the spaceships. [closed]

Some spaceships are $2$ x $10^4$ times larger than other spaceships used in other movies. Find the area of the spaceships in $m^2$ if the other spaceships had a surface area of $1378cm^2$. I don't ...
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1answer
68 views

Building a tube around torus knot

I am currently trying to parametrize a surface constructed by thickening a rather complicated curve, defining its normal, binormal and tangent vectors. Even using Mathematica simplification, the ...
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How to mathematically formulate the surface of a spring?

I would like to mathematically map the surface of a cylinder constructed like a coil pot (or compressed spring), where the surface area and height of the pot is a function of the length of the coil, ...
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Derivative of the Gauss map is zero

If the derivative of the Gauss map is zero in every point in the image of a given local chart, can I conclude that the normal vector is constant and such image is contained in a plane? Edit: The ...
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Please check my calculations for finding a tangent plane to a parametric surface

Here's the question: Find the cartesian equation of the tangent plane to the surface $S : xy^2 + 3yz^2 − 2xyz = 1$ at the point $P(0, 3, 1/3)$. Here's what I did: The normal to the tangent plane at ...
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1answer
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How to find percentage of one rectangles area based on another rectangles area

I know I might sound the dumbest person in the galaxy, but I just wanted to make sure I am doing this right. I have a rectangle say [R1] placed inside a bigger rectangle [R2]. R1 will always be <= ...
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2answers
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How to calculate radius of a spherical surface having four circles touching one another?

There are four circles having radii $r_1, r_2, r_3 $ and $r_4$ touching one another on a spherical surface of radius $R$ (as shown in the picture below, four colored circles touching one another at 6 ...
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Surfaces described using norms and dot products

Okay this seems like a simple question, I think I'm just missing something obvious... The question asks to identify surfaces from the following formulae: $\vert{\bf r}\vert = a$, ${\bf r}\cdot{\bf ...
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1answer
42 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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Pappus theorem and area of a revolution surface.

Let $y=f(z)$ be a function. How can I calculate the area of the surface obtained rotating the function along the $z$ axis, where y is the revolution torus? Is it possible to do it using Pappus formula ...
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Arithmetic picard rank of smooth cubic surfaces

Assume a smooth cubic surface is defined over a field $k$ characteristic $0$, that it has line defined over $k$ and that its arithmetic Picard rank over $k$ is maximal i.e. $7$. Does this imply that ...
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What is a parametrized surface? How is it different from a surface? (Multivariable Calculus)

My textbook defines it like this: Let F be a continuous function from a subset D(F) R2 into Rq. Suppose that D(F) is pathwise connected, and that every point in D(F) is either an interior point of ...
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82 views

What equation will create a 3D rose curve?

The parametric equation $x=a\cos(bt)\cos(t)$, $y=a\cos(bt)\sin(t)$ where $a$ & $b$ are constants and $t$ is parameter gives a rose curve which looks like, On a similar basis, is there a equation ...
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What is the probability of selecting a random point q within x sq units of a random point p?

I was using an online app called "GeoGuessr" where you guess the location on a world map of a given Google Street view image. On one certain attempt, I picked a location within 92.7 square miles of ...
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108 views

Tangential space to the rational normal curve

Exercise 15.5 (Harris, Algebraic Geometry: A First Course): Describe the tangential surface to the twisted cubic curve $C \subset \mathbb P^3$. In particular, show that it is a quartic surface. What ...
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surface area of 'cylinder' with the top cut at an angle

I don't know what the name for this shape is, so in essence it is a cylinder, radius at base $r$, which has had a wedge of the top cut off at an angle so that rather than a circle the upper face is an ...
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How do I incorporate into the definition of a function that its output is constant?

Say I have a surface, x-y+xy+yz+z^2 = 0 (There is nothing particulary special about this expression. I've just made it too complicated to be simplified so that the variables are independent of one ...