For questions about surfaces.

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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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0answers
14 views

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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0answers
13 views

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 ...
4
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1answer
25 views

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
35 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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16 views

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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0answers
12 views

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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0answers
21 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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1answer
82 views
+50

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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1answer
15 views

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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0answers
11 views

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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22 views

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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1answer
37 views

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 ...
2
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1answer
32 views

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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1answer
24 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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0answers
19 views

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
25 views

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
17 views

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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1answer
27 views

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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2answers
12 views

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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1answer
36 views

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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1answer
33 views

Find the area of the spaceships. [on hold]

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 ...
2
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1answer
67 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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0answers
26 views

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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0answers
29 views

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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0answers
18 views

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
23 views

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 <= ...
2
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2answers
32 views

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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2answers
24 views

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
41 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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26 views

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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0answers
17 views

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 ...
2
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1answer
42 views

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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2answers
81 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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0answers
35 views

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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2answers
107 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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1answer
19 views

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 ...
0
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1answer
35 views

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 ...
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0answers
29 views

The normal curvature is bounded by the principal curvatures.

Let the inclusion $i:S\subset\mathbb R^3$ be an immersion of a surface $S$, and let $N:S\to \mathbb R^3$ be a local Gauss map. Let $a:I\to S$ be an arc length parametrized curve, with $a(0)=p$ and ...
2
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1answer
45 views

Normal of a coons patch at a given point

Disclamer: Rendering the Coons patch is part of 3D Graphics homework, but finding the normals at a given point isn't. Just curious. Here's what I got so far: It's a Coons patch defined by four ...
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1answer
32 views

Determinant of this coherent sheaf on a surface $S$

If $C$ is a curve on a surface $S$, i.e. $i:C\subset S$, and $G$ is a line bundle on $C$, then $G|_U\cong \mathcal{O}_C$ where $U$ is an open subset of $S$, that is, $G$ is trivial on the complement ...
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2answers
96 views

If $\dfrac{\mathrm {circumference}}{\mathrm {diameter}}$ is the same for all circles, does the surface have to be flat?

Given a two dimensional Riemannian manifold with the property that the ratio of the circumference and the diameter is the same for all circles. What can be said about it? Does it have to be the ...
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0answers
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Convexity of a subset of $R^3$ based on its bounding surface

Let D be a closed region in $R^3$ whose boundary S is a closed, smooth, orientable surface. The tangent plane at every point on S intersects it in a connected set $M$. Is D (the interior of S) a ...
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0answers
63 views

a subspace of $\mathbb R^3$ with $\pi_1=\mathbb Z_2$

I've been wondering about such problems. It is well known that $\mathbb{RP}^2$ cannot be realized as a subspace of $\mathbb R^3$. But does there exist a space $X\subset\mathbb R^3$ (maybe even ...
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1answer
60 views

A question about the Möbius Strip and the Projective Plane

I know that both the Möbius Strip and the Projective Plane are both 2-manifolds. I try to prove that they are locally homeomorphic to $\mathbb{R}^2$ and Hausdorff. It seems easy to see that the ...
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0answers
29 views

Two cylinders cutting

Find equation of non-circular cylinder $ f (y,z)=0 $ which cuts a circular cylinder $ x^2+y^2 = a^2 $ to produce an intersection curve of constant geodesic curvature on the circular cylinder. What is ...
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0answers
19 views

Constant K surfaces producing lines of constant geodesic curvature

When two spheres intersect or when a sphere intersects a plane the line of intersection has constant tangential/geodesic curvature $ k_g$ (small circle). How is this generalized? Or, Under what ...
3
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1answer
50 views

surfaces, curves and lines

Could someone please assist with the following questions: Consider $f(x,y) = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and take $C$ to be the curve of intersection of $z = f(x,y)$ with the plane $y=x$. Show ...
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1answer
41 views

Using Principal Directions and Curvatures to Find Point On Surface

Given the principal directions (max and min), principal curvatures, and normal of a surface at point n, how would you go about looking for a point on the surface at a given vector distance from n? ...
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3answers
44 views

Describe the graph of the function: Level Sets, Intersections

Describe the graph of the function $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x, y) \rightarrow |y|$$ computing some level sets and some intersections. I have done the following: The level curves ...