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Questions tagged [orientation]

For question regarding the notion of orientation both in topology and in global analysis.

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1answer
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Metric to describe the “distance” between and orientation and position pair.

I have two points and would like a metric that tells me how close those points are to each other. Each point is described by both a 3D position and a 3D orientation. I can determine the distance ...
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2answers
669 views

Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
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If $S \subset X$, with $\partial S = S \cap \partial X$. Must $T(S \cap \partial X) = T(S) \cap T(\partial X)$?

Some background. This came about in the proof about boundary orientation $$\partial f^{-1}Z = (-1)^{\operatorname{codim}Z} (\partial f)^{-1}Z.$$ A reference would be Guillemin-Pollack page 101. ...
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1answer
57 views

Orientation on manifolds

I am trying to understand the definitions here. In many books (say Tu or even Guillemin-Pollack) an orientation on a manifold is an assignment to affix $+1$ and $-1$ to classes of (tangent) basis. It ...
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2answers
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Classification of surfaces

The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $...
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2answers
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A topological group which is also a (not necessarily smooth) manifold is orientable

I am trying to show that a topological group which is also a (not necessarily smooth) manifold is automatically orientable. I know of a proof involving transition functions for smooth manifolds, in ...
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1answer
52 views

If $\phi$ is a orientation preserving linear automorphism, do we really need to choose the same orientation for the domain and the codomain?

In the book of Linear Algebra by Werner Greub, at pages $131-132$, it is given that and However, when $\phi$ is a linear automorphism, it says that $\Delta_F = \Delta_E$, and derives the following ...
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Question related to orientation on a arbitrary oriented manifold

This is a section from Loring Tu's book Introduction to Manifolds page 244 Second Edition. My question is as follows: Towards the end of the text in the image he says that an oriented manifold can be ...
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On the unit sphere $S^2$, show the antipodal map $A:S^2\to S^2$ is orientation reversing using definitions.

I want to show that on the unit sphere $S^2$, the antipodal map $A:S^2\to S^2$ given by $(x,y,z) \mapsto (-x,-y,-z)$ is orientation reversing. I know that $S^2$ is a regular connected orientable ...
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1answer
26 views

Pairings of Simplex Edges

This is an open-ended question. If you'll allow, I'd like to keep its origins vague for the moment, so as not to bias responses. I am interested in any and all thoughts. There are three pairings of ...
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Angular error (in Euler angles) through quaternions

I found this formula in some notes but I would like to have a reference (book, paper, etc.) to understand where it comes from. I know that it works only for small angles. $ \begin{bmatrix} \phi_e\\ \...
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1answer
110 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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0answers
103 views

Continuity of point wise orientation

Prove that $[(X_1,...,X_n)]$ on a manifold $M$ is continuous if and only if every point $p$ in $M$ has a coordinate neighborhood $(U,\phi) = (U,x^1,...,x^n)$ such that for all $ q \in U$, the ...
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0answers
18 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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25 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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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
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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
17 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
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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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What is the difference between the orientation and the direction of a vector?

I came across this recently, and was confused regarding the difference between the orientation and the direction of a vector. Does the orientation refer to the relative coordinate which the vector is ...
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2answers
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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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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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1answer
979 views

How to rotate an orientation (Euler angles)

If I have an orientation defined by Euler angles and I want to simulate a rotation of the coordinate system about the origin (doesn't matter to me how the rotation is specified), how would I get the ...
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1answer
1k views

How to calculate rotation quaternion between two orientation quaternions?

I have some device (3D pointer) connected to my computer which returns it's position (in cartesian XYZ system) and orientation (in quaternions). I receive this values about 30 times/sec. Now I need ...
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0answers
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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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3answers
745 views

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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0answers
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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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0answers
46 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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0answers
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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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1answer
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Help to correct answer: Product of orientable manifolds is orientable

In the first reply to this post, I made a comment, but nobody answered me yet. Link: Orientability of a product of smooth manifolds implies orientability of each factor My problem is in the return of ...
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0answers
49 views

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
59 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

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
33 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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2answers
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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
43 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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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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1answer
31 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
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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
34 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
47 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
77 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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0answers
30 views

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
93 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
35 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
44 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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54 views

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
49 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
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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 ...