Questions tagged [orientation]
For question regarding the notion of orientation both in topology and in global analysis.
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Robotics, Sensors and Physics: quaternions, velocity, acceleration [on hold]
Problem: Feature engineering
Context: A robot with many sensors is moving around.
I have this features:
orientation (quaternions X,Y,Z,W)
angular velocity (X,Y,X)
linear acceleration (X,Y,Z)
<...
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1answer
76 views
Every connected orientable smooth manifold has exactly two orientations, Lee Proposition 15.9
The proof of Proposition 15.9 from John Lee's book "Introduction to Smooth Manifolds" is left as an exercise.
Here is the statement:
Let $M$ be a connected, orientable, smooth manifold with or ...
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30 views
Why should one think of orientation as a homology class?
Let $\pi:E\rightarrow B$ be a smooth vector bundle. I call this vector bundle to be an oriented vector bundle, if I can choose an orientation on $\pi^{-1}(b)$ for each $b\in B$ and a trivialization ...
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12 views
Rotation with roll pitch and yaw in different coordinate system
Say I am given a point in an x1,y1,z1 coordinate system. I have a different coordinate system, x2,y2,z2 that has the same origin as the x1,y1,z1 system, but the axis are not aligned. I have roll, ...
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20 views
Torque required to achieve a desired quaternion
I was hoping someone can either explain or direct me towards a source that can help me with the following problem (not for homework, more of a hobby).
Given an object with a current quaternion $q_c$ ...
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36 views
Diffeomorphism Between Surfaces Preserves Orientability
From Do Carmo (Exercise 2.6.2).
Let $S_2$ be an orientable regular surface and $\varphi:S_1\rightarrow S_2$ be a local diffeomorphism at every $p\in S_1$. Prove $S_1$ is orientable.
Up until this ...
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1answer
29 views
understanding orientable manifolds
I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." p. 138. I don't get the statement in the definition of orientable manifolds.
4.1 Definitions $\;$ (the preface omitted) ...
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1answer
15 views
Determine orientation of standard orthonormal bases
Let $\vec{e_1},\vec{e_2},\vec{e_3}$ be the standard orthonormal base $(1,0,0),(0,1,0),(0,0,1)$ which is positively oriented. Determine the orientation of each of the following bases:
$\vec{e_1},\vec{...
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2answers
53 views
Orientation of a Manifold with Trivial Tangential Bundle
Let $M$ be a smooth (eg $C^{\infty}$) manifold. Let assume that $M$ has trivial, oriented tangent bundle $TM$, so $TM \cong M \times \mathbb{R}^n$ for appropriate $n$ and orientable.
How to conclude ...
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23 views
Finding and Comparing 2 Sensors Rotations, with same reference frame but different initial Orientation
Let's say we want to Compare two different Arm (Humerus) Rotations (series of quaternions) and we do not care about space translation but only for rotation. To measure each rotation we use the same ...
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The orientation induced on the boundary of a manifold.
I just learned about the notion of orientability of a manifold which is difficult and abstract for me. If we consider all basis of a vector space, the matrix that transforms one basis in another basis ...
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Id $f,g$ orientation preserving cricle-diffeomorphisms, then $\rho(g^{-1}\circ f \circ g) = \rho(f)$.
Let $F, G : \mathbb{R} \rightarrow \mathbb{R}$ be a lift of $f$ and $G$ of $g$. That is, $ \pi \circ F = f \circ \pi$ with $\pi(x) = e^{2\pi i}$. We define $$\rho_{0}(F) = \lim_{n\rightarrow \infty}\...
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36 views
Smallest rotation angle between quaternions accounting for symmetry
I am trying to compute a similarity measure between the orientation of 3D objects accounting for symmetry invariance. I have a set of 3D objects, which are defined by a center of mass, and 3 unit ...
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35 views
Prove that an integration is left invariant.
$G$: Lie group of dimension $n$.
$\tilde{\Omega}$: Orientation on $G$.
$\Omega=\epsilon^1\wedge \epsilon^2\wedge \cdots \wedge \epsilon^n$ where $\epsilon^1\, \epsilon^2, \cdots ,\epsilon^n$ is ...
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Sard's theorem for orientation preserving diffeomorphism of the circle
thanks in advance for helping me. First I'll introduce some definitions:
(1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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1answer
27 views
If $(e_1, \ldots, e_n)$ and $(f_1, \ldots, f_n)$ are co-oriented, then so are $(e_1, \ldots, e_n, u)$ and $(f_1, \ldots, f_n, u)$
Suppose that $S$ is a smooth $n$-submanifold of $M$ where $\dim(M) = n+1$. Suppose also that $S$ and $M$ are both Riemannian and oriented.
Suppose that $(e_1, \ldots, e_n)$ and $(f_1, \ldots, f_n)$ ...
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1answer
58 views
Show that $\mathbb{S}^{n+m}$ is not homeomorphic to a product of orientable manifolds
I want to prove that the sphere $\mathbb{S}^{n+m}$ is not homeomorphic to the product of N and M, orientable manifolfs with $\textit{dim}\;N=n$ and $\textit{dim}\;M=m$. I know that I have to use the ...
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1answer
29 views
How/why does the contraction of standard volume form give the canonical form.
$M \subset \mathbb{R}^{N}$ is a (oriented) $n-1$ dimensional submanifold. Suppose $\nu \in T_{p}M^{\bot}$, of length one (a normal unit vector on $M$).
How and why does the contraction $\nu_{\neg}(...
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55 views
Submanifold of codimension 1 orientable iff there exists unit normal vector field.
Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Where a unit normal vector field is a section $\nu$ of the normal bundle $ TM^{\bot} \to M$. So the fibers are all the ...
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2answers
43 views
Every submanifold is orientable (co-dimension 1)?
Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Apparently it's orientable if and only if there exists a unit normal vector field on $M$. Where a unit normal vector field ...
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1answer
27 views
If $TM$ is trivial, then $\Lambda^n(M)$ is also trivial and $M$ is orientable
Suppose that $M$ is a smooth $n-$manifold. Suppose $$ TM=\coprod_{p\in M}T_pM $$ is the tangent bundle of $M$. And let $$ \Lambda^n(M)=\coprod_{p\in M}\Lambda^n(T_pM) $$ where $\Lambda^n(T_pM)$ is ...
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1answer
30 views
What is the direction of a rotation?
We usually talk about rotations in the clockwise or counterclockwise direction. But if a rotation is just a function defined on the space, then it is all about points and their images, and there is no ...
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0answers
19 views
Prove that the following function is an orientation of a curve
Assume that $f^{(j)}:\mathbb{R}^3\rightarrow \mathbb{R}$ for $j=1,2$ are some $C^1$ functions such that $(\nabla f^{(1)}, \nabla f^{(2)})$ are independent over the curve $\gamma = \{ x \in \mathbb{R}^...
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0answers
36 views
Orientation of 4 + 1 lines in $\mathbb{R}^3$.
I'm working on a 3D algorithm that at some point establishes orientation of two lines - the same way one would do using the triple product. The way those lines are described, however, makes the ...
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1answer
29 views
Create Wall 3D math oriented away from camera
I have 2 Points which has x,y,z let's say from and to I am drawing wall between them using ARkit ios
To draw wall I use static ...
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0answers
44 views
Visualisation of an orientable surface bounded by the Möbius curve
I'm learning multivariable calculus on MIT OpenCourseWare. When the teacher explained Stokes theorem he mentioned the Möbius strip. He showed it was non-orientable. Then he showed a somehow twisted ...
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1answer
64 views
Orientable manifold $M$ ,then $\partial M$ is orientable
Let $M$ a topological manifold of dimension $n$ with boundary $\partial M$.
We define $M$ to be orientable if $M- \partial M$ is orientable. Here when I say orientable, I mean there is a locally ...
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preserves or reverse orientation of sphere surface
Let $\varphi: (0, \infty) \times (0, \pi) \times (0, 2 \pi) \to \mathbb{R}^3 \setminus \{(x,y,z) \in \mathbb{R}^3| y=0, x \geq 0 \}$
$$(p,\phi,\theta) \mapsto (p \sin \phi \cos \theta, p \sin \phi \...
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1answer
32 views
Find the angles between two solids
I have 2 solids (A and B) and I need to find the three angles between their x, y, and z axes. If I calculate the geometrical center of the two solids (Ax, Ay, Az and Bx, By, Bz), is it correct to ...
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1answer
74 views
Orientation in the proof of Stokes Theorem
I'm reading the proof of Stokes theorem at page 83 of "Godinho, Natàrio, An introduction to Riemannian geometry" and I can't understand a passage in it, probably because the definition of ...
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1answer
37 views
Prove there exists a outward unit normal field on the boundary this manifold
Let $M$ be a compact subset of $\mathbb{R}^3 $ with the standard orientation $\mu =[e_1,e_2,e_3] $ and let $S = \partial{M}$ is its smooth boundary with the induced orientation from $M$. Prove there ...
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Null homologous loop and orientable surface
I am reading Algebraic Topology: A First Course written by Greenberg and Harper. On page 67 of this book it is stated that
Let $\gamma$ be a loop in $X$ regarded as a map $f:S^1\to X$. For $\chi[\...
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1answer
38 views
Smooth isotopy preserves orientation
Let $N$ a $n$-dimensional connected manifold and let $h: N \rightarrow N$ a diffeomorphism such that $h$ is smoothly isotopic to the identity map $\text{id}_N : N \rightarrow N$.
It's clair that the ...
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2answers
64 views
What does it mean for a (non-smooth) homeomorphism between oriented smooth manifolds to be orientation preserving?
The definitions I know of orientability of manifolds are in terms of tangent spaces. However, for example in this answer there is mention of orientation preserving homeomophisms (between orientable ...
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2answers
84 views
Fundamental group of $M$ has no subgroup of index $2\Rightarrow M$ is orientable
Let $M$ be a connected smooth manifold such that, for every $p\in M$, the fundamental group $\pi_1(M,p)$ has no subgroup of index $2$. Prove that $M$ is orientable.
Here's what I know: there is a ...
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2answers
28 views
Simple stokes - why is this the correct orientation?
We want to calculate $\iint_{S}\text{curl}(\vec F)dS$ where $\vec{F}(x,y,z)=(y^2z, xz,x^2y^2)$ and $S$ is the part of the paraboloid $z=x^2+y^2$ that's inside the cylinder $x^2+y^2= 1$ with an outward ...
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3answers
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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?
I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this ...
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1answer
32 views
$df_x$ maps $N(S.X)$ isomorphically
This is actually just a linear algebra problem, but this text is taken from Guillemin and Pollack, Differential Topology on page 100.
Real Problem: Let $f: X \to Y$ be smooth, $S = f^{-1}(Z)$ where ...
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2answers
74 views
Conditions for Matrix to be Product of Near-Identity Matrices
For $\epsilon > 0$, let $M_{\epsilon}$ be the family of $n$ x $n$ real matrices A such that $||$A$ - $I$_n|| < \epsilon$, where $|| \cdot ||$ is the standard operator norm. If $\epsilon$ is ...
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1answer
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If an orientation of a tree graph has no source vertices, must the in-degree of each vertex in said orientation be equal to one?
Given any polytree $T$ (any orientation of a tree graph) such that $\forall v\in V(T)(\text{indeg}(v)\neq 0)$ does this imply that $\forall v\in V(T)(\text{indeg}(v)=1)$? I'm pretty sure its true, but ...
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Maps between equivalence classes of vector bundles with and without orientation fixed at a point
I'm reading Hatcher's "Vector Bundles and K-Theory" and he says something that seems to be false.
Here is a screenshot of the part of the book in question.
He says that if we compare $Vect^n_0(S^k)$ ...
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36 views
Exponential map is orientation preserving
Let us define the map $f:\mathbb{R}\to \mathbb{S}^1$ by $x\to e^{2\pi i x}$. I want to prove that this map is an orientation-preserving map and also it's restriction on the unit interval has an ...
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0answers
51 views
How Euler angles change when we reverse direction of some axes
I looked at the other questions but couldn't find an answer for this particular question:
I have measured Euler angles of an object in one coordinate system and I need to use these data in another ...
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1answer
35 views
Relative orientation of satellite to a rotating plane.
We know the GPS position of a Geostationry satellite and a sensor. We also know the sensor orientation relative to the magnetic North of Earth. How do we find the orientation of Satellite relative to ...
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0answers
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$M$ is orientable if and only if $M \setminus \{p\}$ is orientable.
Let $M$ a manifold of class $C^{\infty}$. Show that $M$ is orientable if and only if $M \setminus \{p\}$ is orientable.
Comments:
($\Rightarrow$) Let $\omega: M \longrightarrow \Lambda^n(M)$ a ...
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1answer
75 views
Direction of path depends on sign of determinant of Jacobian
Let $f: K_1(0) \rightarrow \mathbb{R}^2$ be continuously differentiable, $\{z_1,..,z_m\}=f^{-1}(a)$ with a regular $a \in \mathbb{R}^2$.
We choose $\epsilon$ small enough so that $f\vert_{\overline{U}...
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1answer
64 views
(Negative) Gradient and Orientability of its flow.
Before asking my question, I put the necessary definitions and some context. If you are used with Morse Theory, you can skip the text within [[[...]]].
[[[Let me first define what I mean by gradient ...
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1answer
47 views
Prove a DAG can be obtained by an undirected graph's longest cycle
Let G(V, E) be a finite undirected graph and let κ be its longest undirected cycle.
Prove that it is always possible to obtain an orientation ω(κ) in which κ is topologically sorted and hence its ...
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1answer
31 views
Compatibility of orientation classes
Let $U$ be an oriented topological manifold of pure dimension $n$ and $K$ a compact of $U$. There is an orientation class $$or_{U,K} \in H_n(U, U \backslash K).$$ Let $V$ an open subset of $U$ such ...
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2answers
37 views
Let $M$ is not orientable and $p \in M$, then $M - \{p\}$ is orientable.
Is it true or false? Let $M$ a manifold of class $C^{\infty}$ not orientable and $p \in M$, then $M - \{p\}$ is orientable.
I believe this to be false, but I do not know a counterexample
