Questions tagged [surfaces]
For questions about two-dimensional manifolds.
3,148
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Novel Approach to Normal Estimation in Surface Reconstruction
Not sure where to post this exactly but I am hoping to find out if my approach is in any way a novel method for estimating normals in a point cloud when performing surface reconstruction (in e.g. ball-...
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A regular, connected, compact surface with curvature on $[0,1]$
today was my final differential geometry exam and there was a problem that I partially solved, but I have some doubts.
The problem asked to prove that there exists a regular, connected, compact ...
- 31
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When does the magnitude of the gradient equal the surface area of the $dxdy$ patch?
Given a surface $S$ in $\mathbb R^3$, what is the relationship between the gradient (when $S$ is defined as a level curve of function $F: \mathbb R^3 \to \mathbb R$) and surface area?
I noticed such a ...
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Can you explain to me the relationship between those two definitions of branched covering on surfaces?
I'm studying branched (or ramified) coverings between surfaces and the definition that was given in the book I'm reading is the following:
"Lets consider two closed and connected surfaces $M$ and ...
- 11
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1
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Basic question on Dehn twists
Let $\Sigma$ be a topological surface (compact and orientable, if necessary). Let $C$ be a subspace of $\Sigma$ homeomorphic to the annulus $A := [0,1] \times S^1$, together with a homeomorphism $h: A ...
- 1,227
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Problem with understanding Morse's Lemma / Function.
https://math.stackexchange.com/a/398282/1257548
In this answer it is said that $f:S→\mathbb{R}, (x,y,z)↦y$ is Morse function but I don't see why.
As far as I understand, because function is defined ...
- 33
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Laplacian version of surface tension gradient
I have read that the surface tension gradient operator is $
\nabla_{s} = (I - nn). \nabla$ , where n is the unit normal to the surface given by :
$$ n = (-\frac{\partial h}{\partial r},1) \frac{1}{\...
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Suppose for a region on a surface I can draw a "handle" can I cut the surface to reduce it's genus while leaving the region intact?
Suppose I have a smooth orientable surface $Q$ and a compact region $R$ of $Q$. Suppose there is a closed curve $C$ that divides R into two connected components $R_1,R_2$ but does not divide Q into ...
- 278
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How to explain strips on loss function when fitting a rice distribution
While investigating a Rice distribution fit, I have found a behaviour that I would like to investigate a bit deeper, hence my question to the community.
This is not about convergence issue but having ...
- 500
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Total curvature is area of image of Gauss map
I am trying to solve the following exercise from the book Differential Geometry by Loring W. Tu:
5.4 Total curvature
The total curvature of a smooth oriented surface $M$ in $\mathbb{R}^3$
is defined ...
- 31
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Help understanding the use of tetrahedrally arranged vectors to compute the gradient of a function
As a long-time user of the free, open-source raytracer POV-Ray, I'm trying to understand some of the source code used to compute and perturb surface normals. The method uses the evaluation of 4 ...
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38
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Why does a sphere not have a boundary
Let S be a geometric surface.
Definition: [Boundary of a surface] A surface with a boundary is a surface along with boundary points. A boundary point $P$ is a point such that there exists an open set $...
- 639
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How saddle points for surfaces in arbitrary semi-Riemannian manifolds can be defined?
I'm interested in how the concept of saddle point, easily defined for graphs of functions, can be generalized for two-dimensional surfaces embedded in arbitrary semi-Riemannian manifolds.
I think the ...
- 26k
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Geodesics on a Saddle Surface
Does anyone know of any free and relatively easy-to-use math apps like Geogebra that can be programmed (or were explicitly designed) to draw the geodesic between any two selected points on a saddle ...
- 70
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Finding absolute maximum and minimum in 3d function over a surface
I need do find global max and min of
$$
z=x+y^2
$$
inside (or on the border of) the volume described by:
$$
x^2+y^2-25=0
$$
I've already found maximum values in $(\frac{1}{2}, +\frac{3\sqrt{11}}{2}, \...
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Finding the embedded surface that induces a given 2x2 metric
I've been scratching my head for quite some time on how one could find the surface embedded in 3D flat space that induces a given 2x2 metric. Usually one asks the opposite question: given a surface, ...
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If length of gradient is constant along level curves then for level curves $\beta$, $\gamma$, exists $c \in \mathbb{R}$ with $\gamma = \beta + cn$
I'm trying to prove some results about geodesics and surfaces given by graphs. Im stuck with the following problem:
Let $f: \mathbb{R}^2 \to \mathbb{R}$ such that $\|\nabla f\|^2$ is constant along ...
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Embedding/immersion of $\mathbb RP^2$ into Euclidean space so lines are geodesics?
The real projective plane $\mathbb RP^2$ is introduced in algebraic geometry contexts as basically $\mathbb R^2$ but where all lines intersect at one point. One construction is the one that projects ...
- 8,363
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Immersion of $\mathbb RP^2$: explanation of Kirby's article on the Boy's surface
I am trying to understand Rob Kirby's AMS notice https://www.ams.org/notices/200710/tx071001306p.pdf on Boy's surface. From this blogpost https://divisbyzero.com/2020/04/08/make-a-real-projective-...
- 8,363
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What does it mean that the helicoid can 'glide' over itself?
In the Wolfram definition it says
The helicoid is the only non-rotary surface which can glide along itself.1
1Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 231-232, 1999.
I ...
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Where is the flaw in my reasoning for finding the surface area of a right circular cone using a definite integral? [duplicate]
So I've taken Calculus 1-3 and am taking 4 when I came across a problem that required me to know the surface area of a right circular cone. It is obviously trivial to search this up so I thought it ...
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Area of the rotation surface generated by a parametric curve around the Z axis
I know it might seem like a trivial question but I honestly couldn't find this formula anywhere.
If I have
$$\Gamma(t)=(x(t),y(t),z(t))\qquad t\in[a,b]$$
I want to calculate the area of the rotation ...
- 3,152
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65
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Geodesics and Laplace-Beltrami eigenfunctions
For a smooth, closed Riemannian 2-manifold $M \subset \mathbb{R}^3$, is there a relationship between the geodesics of $M$ and smooth functions $k : M \times M \to \mathbb{R}$ which can be expressed in ...
- 21
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Precisely defining the overlap depth, or deepest point of overlap, for ellipsoids and spheroids
I was wondering if there is a robust mathematical definition for the 'deepest point of overlap' of ellipsoid (or, equally as good, spheroid) 1 that has overlapped with ellipsoid 2. For non-overlapping ...
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2
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Tangent plane to 3 spheres
Given 3 spheres of radius 9 with center at the points $P = (2,1,0)$, $Q = (5,4,0)$ and $R = (3, 1, 2)$. Find the equation, $ax + by + cz = d$, of a plane tangent to the 3 spheres.
I calculated the ...
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How to map a plane onto a Octahedron without polar distortions?
https://www.math3d.org/ZfulehTVK
this is a link to where all my visualizations are.
While researching seamless procedural texturing I noticed people creating a 4d hypersphere to loop noise back on ...
- 11
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How to convert the formula for Gaussian curvature of a surface
In my previous post I was asking for some clarification on the derivation of gaussian curvature.
$$
\begin{align}
K=&\frac{1}{\left(E G-F^2\right)^2}
\left\{\left|\begin{array}{ccc}
\frac12E_{vv}+...
- 399
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Find the angle of the plane makes with the semimeridian $\phi=\phi_0$
(doCarmo 2.5-2)Let 𝑋(𝜃,𝜙)=(sin𝜃cos𝜙,sin𝜃sin𝜙,cos𝜃)
be parametrization of the sphere $𝑆^2$
. Let 𝑃
be the plane 𝑥=𝑧cot𝛼
, 0<𝛼<𝜋
and 𝛽
be the acute angle which the curve $𝑃∩𝑆^2$
...
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Parameterization cylinder under a plane
I have to evaluate the surface area of the cylinder
$$x^2+y^2=4$$
between the planes $z=0$ and $x+y+z=2$. I was trying to use the following formula to area
$$\iint_{D}|\sigma_{\theta}\times\sigma_{z}|...
- 1,831
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Different surfaces with the same Gaussian and mean curvature everywhere
I'm trying to teach myself the classical differential geometry of 2D surfaces in 3D Euclidean space but I'm struggling to understand exactly how much information the Gaussian and mean curvature ...
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For translation of axes, is there a definite equation for any of "the 27 lines" on the Clebsch Diagonal Cubic?
For the Clebsch Diagonal Cubic (related to a Quanta mag article on Hilbert's 13th Problem), I want to generate a point at will on any of these lines along the surface. Wolfram calls these "...
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answer
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Another cell complex structure of closed orientable genus $g$ surface?
Suppose we have a wedge sum of $2g$ circles, labeled by $a_1,b_1,\dots, a_g,b_g$. It is well known that if we attach a 2-cell with attaching map given by $a_1b_1a_1^{-1}b_1^{-1}\cdots a_gb_ga_g^{-1}...
- 2,114
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Curl of normal unit vector of a smooth and closed surface?
Let's say we have a curvilinear coordinate system $(\rho,\theta,\zeta)$. Also, let's say we have a smooth and closed surface $\Gamma$ parameterized as $\Gamma: \mathbf{S}(\rho(\theta,\zeta),\theta,\...
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Find the surface equation from the fundamental forms
If $I=(du)^2+\sin^2u(dv)^2=II$ then show that the surface is given by $$\underline{x}=\sin u\cos v\:e_1+\sin u\sin v\:e_2+\cos u\:e_3+\underline{c}$$
This exercise question is really interesting like ...
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1
answer
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Gaussian curvature in term of coefficient of first fundamental form and it's derivative
Show that the gaussian curvature $K$ can be written as,
$$
\begin{align}
K=\frac{1}{\left(E G-F^2\right)^2}
\left\{\left|\begin{array}{ccc}
\frac12E_{vv}+F_{uv}-\frac12G_{uu} & \frac{1}{2} E_u &...
- 399
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set of points where the Gaussian curvature is zero be a regular curve
I have a surface of the form $ X = (u,v,f(u,v)) $ around the non-planar parabolic point $p=(0,0)$, I take $f(u,v)$ as its Taylor expansion, then $X$ is of the form $$ X = (u,v,q_{00}+q_{10}u+q_{01}v+...
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Normal derivative of normal unit vector
Let's say we have a curvilinear coordinate system $(\rho,\theta,\zeta)$. Also, let's say we have a surface $\Gamma$ parameterized as $\Gamma: \mathbf{S}(\rho(\theta,\zeta),\theta,\zeta) = \mathbf{S}(\...
1
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1
answer
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Existence of a curve of finite length on the image of a Sobolev embedding
Suppose that we have an embedding $f:\mathbb{R}^2\to\mathbb{R}^3$ which belongs in the Sobolev space $W^{1,p}_{loc}(\mathbb{R^2},\mathbb{R}^3)$ for some $p>2$. Is it true then that for any two ...
- 363
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Subdividing any piecewisely linear surface in the Euclidean $R^3$ space by sets of planes and their signs.
Let us assume I have a set of planes
$$f_i(a_i,b_i,c_i,d_i) = a_ix+b_iy+c_iz + d_i = 0,\hspace{1cm} i \in \{0,\cdots,N\}$$
Then for each such plane $i$ I define a subset defined by selecting one of ...
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Two problems relating principal curvatures and circles contained in surface
Let $k_1=k_2=1$ be the principal curvatures of a regular surface $S$ at point $p\in S$ and assume that there is a circle $c$ of radius $1/2$ passing through $p.$ Prove that
the geodesic curvature of $...
- 5,275
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show that mean curvature equal zero or $v_p$ $w_p$ are are principal vectors
Let $M$ be a surface in $\mathbb R^3$ , and $p$ be a point in $M$ i.e. $p\in M$ .
Suppose that $v_p$ and $w_p$ are two tangent vectors to $M$ ate the point $p$ , such that $v_p.w_p=0$ and $S(v_p).S(...
- 21
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1
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Question on a stable geodesic lamination on a closed hyperbolic surface
Let me first state a theorem in Casson-Bleiler Automorphisms of Surfaces after Nielsen and Thurston.
Theorem 5.5: Let $h:F\to F$ be a non-periodic irreducible automorphism of a closed orientable ...
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How to calculate intersection of two surfaces in parametric form
I am given the two surfaces
$$S_1(s,r) = \left( 2\sin r \cdot (e^{s/2}-1)+r, s, e^{s/2}\cdot\sin r \right )$$
and
$$S_2(s,r) = (r, s+r, 0)$$
and am asked to find the intersection $S_1\cap S_2$.
I can ...
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1
answer
34
views
Which surface do these conditions determine (and how to visualize it)?
I'm interested in visualising the following surface of points $(x,y,z)\in \mathbb{R}^3$ which satisfy the following equation and inequalities:
$$ \begin{cases}
x + y + z = 0\\
x \leq y \leq z
\end{...
- 225
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Find $\epsilon$ such that $\varphi^{\epsilon} = \sigma + \epsilon(\cos t \mathbf{n} + \sin t \mathbf{b})$ is injective ($\sigma$ biregular curve) .
I’ve been struggling for an entire day trying to approach the third point of this exercise (Ex. 4.23 from ‘Curves and Surfaces’ by M. Abate and F. Tovena).
Let $\sigma:I \to \mathbb{R}^3$ be a ...
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31
views
Orientable surface $M\subset\mathbb{R^{m+n}}$ such that has no normal linearly independent vector fields
We know the following results:
Theorem 1: If a suface $M^m \subset \mathbb{R}^{m+n}$ of co-dimension $n$ admits $n$ linearly independent continuous normal vector fields $v_1, \dots, v_n:M \rightarrow \...
- 2,284
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Classification of good foliations of a pair of pants
The following is a proposition from FLP (Thurston's work on surfaces).
Proposition 6.7 (Classification of good foliations of a pair of pants) The function $\mathcal{MF}_0(P^2)\to\Bbb R^3_+$, which to ...
- 4,839
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54
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Divergence theorem with normal component of a curl to a surface
Let $\mathbf{A}$ be a vector function in $\mathbf{R}^3$ and we want to find the normal and tangent components of $\nabla \times \mathbf{A}$ on a smooth and closed surface $\Gamma$. $\mathbf{n}$ is the ...
2
votes
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answers
29
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Estimating mean curvature of a surface by two perpendicular curves of the surface
Let S be an oriented smooth surface containing a circle of radius 1 and a straight line, which intersect perpendicularly at a point $p\in S$. Show that if the Gauss curvature K of
S satisfies
K(p)=0, ...
- 21
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10
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Linear fitting of values on non-uniformly parametrized B-spline surface
I have a 2d (u,v) surface in 3d space (x,y,z) that is defined as a b-spline surface. The surface is not arc-length parametrized.
Additionally, I have scalar values defined on the surface at given u,v ...
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