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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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Orientation-preserving local difeomorphism

I was studying some notes of classes and I am stucked at the following proposition: Let $S_1,S_2,S_3$ be orientated surfaces. If $S_1\stackrel{f}{\to}S_2\stackrel{g}{\to}S_3$ are local ...
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Two connex components with different orientations - The surface cannot be orientable!

I need to prove that if $S $ is a surface covered by two coordinate neighborhood $V,U $ s.t. $V\cap U $ has two connex and the jacobian of coordinate change is positive in once and negative in other, ...
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If exists two-form not null in $S$, so $S$ is orientable

I need to prove that, being $S$ a regular surface, If exists a two-form not null in $S$, so $S$ is orientable. Here, I need to use estricly the basic definition for orientability, ie, any two ...
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Orientation of a curve in $\mathbb{R^3}$

Let $\gamma: \mathbb{R} \to \mathbb{R^3}, \gamma(t) := (0,0,1)+\cos(t)v_1+\sin(t)v_2$, where $v_1 =(\frac{2}{3},\frac{1}{3},-\frac{2}{3}),v_2=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$. I know this ...
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Rigorous definition of left hand coordinate system.

It seems that for a 3D vector space over $\mathbb{R}$, we cannot define whether a coordinate system if left-handed or right-handed, since we cannot compare an abstract vector with our fingures. So how ...
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47 views

Cross product definition with confusion around handedness

I want to check if my understanding about the cross product is correct. Wikipedia page on cross product says the definition of cross product depends on the orientation of the vector space. ...
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1answer
21 views

Sign independency when rotating an inertia tensor with a rotation matrix

I have a set of points that represent a rigid cylinder for which I calculate its rotation in space (x,y,z) using the formula $ \mathbf{J} = \mathbf{R^T} \mathbf{J_0} \mathbf{R} $ with R being the ...
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Definition of orientation of curve using implicit function theorem

I am reading Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables. On p.64, sec. II.1.9, he tried to show that if $\gamma=(\gamma_1,\gamma_2):[a,b]\to\mathbb R^2$ ...
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Connected intersection of a manifold and orientation

From Do Carmo's book (Riemannian Geometry, P. 19) If M can be covered by two coordinate neighborhoods $V_1$ and $V_2$ in such a way that the intersection $V_1\cap V_2$ is connected, then $M$ is ...
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XYZ position with orientation and altitude?

I am working on a project with sensors: I am able to have the orientation output of an object as quaternion, rotation matrix or euler angle. A second sensor is measuring the change is altitude of ...
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Orientation for a vector spaces determines a canonical orientation for the dual vector space

Suppose $V$ is an $n$-dimensional real vector space, with $n > 0$. Show that an orientation for $V$ determines a canonical orientation for $V^*$, the dual of $V$. The idea I had in mind to show ...
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Non orientable normal bundle gives a non simply connected manifold [closed]

Let $X$ be a compact connected manifold and $M\subset X$ be a compact connected hypersurface. If the normal bundle $NM$ is not orientable, then $\pi_1(X)\not= 0$.
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Chain rule for differential calculus of 3D rotations

In the paper "A Primer on the Differential Calculus of 3D orientations" by Bloesch et al. they give an example on how to apply the identities derived for the derivatives of various rotation ...
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1answer
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Volume element of Haar measure on SO(3) with Euler angle parametrization

I have a real-valued function $f(\alpha, \beta, \gamma)$ which takes Euler angles $\alpha, \beta, \gamma$ as input, that I would like to average over the uniform distribution on orientations of 3-...
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Seifert surfaces for knots $6_1, 6_2, 6_3$.

I have been trying to calculate the genera of these knots, but the first step in doing so is to convert them into orientable knots by constructing Seifert surfaces for those knots. I started to do ...
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Confusion about the top homology group of a compact manifold.

I know that if the manifold is compact, then all of its homology groups are finitely generated. But on the other hand, we know (for example Hatcher 3.26) that if the manifold is closed and orientable, ...
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Is orientation of Stokes theorem a convention?

Is orientation of Stokes theorem, that is, right hand rule, a convention? Can we also choose the left hand rule? But will not it create problems in Physics where the sign of our physical quantity (...
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1answer
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Position and orientation of local coordinate system relative to another local coordinate system (both described in world coord. system)

This question might have been answered before, but I couldn't find one that will best describe my problem, or before I got confused. The following image depicts three coordinate systems: a world ...
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I don't understand this theorem about covering spaces or it is a typo?

Isn't it a trivial conclusion the first part of the following theorem from Theodore Frankel book (The Geometry of Physics: An Introduction) or it is a simple typo? Theorem: The orientable cover of $...
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Existence of volume form implies orientability

I'm trying to prove the following: Let $M$ be a smooth manifold, $\omega\in \Gamma(\bigwedge^n TM^*)$ non-vanishing. Then $M$ is orientable. My approach is the following. I want to define an atlas of ...
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Isomorphism in all homology and cohomology groups

Let $M^n$ be a closed, connected and orientable topological manifold of dimension $n \geq 2$ and let $f : M \to M$ be a continuous map. Assume that $f_* : H_n(M) \to H_n(M)$ is an isomorphism. The ...
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Euler angles, Quaternion and mobile device rotation

I've written a JS SDK that listens to mobile device rotation, providing $3$ inputs: $\alpha$ : An angle can range between $0$ and $360$ degrees $\beta$ : An Angle between $-180$ and $180$ degrees $\...
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1answer
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Is a smooth immersion $c: [a,b] \to M$ injective if its image is a 1-manifold with non-empty boundary?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, it is confirmed that the example is an error. ...
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What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under these maps?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, Prof Jack Lee says that the example is ...
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Getting the orientation for a torus from the parametrisation

Say I have the torus $T$ in $\Bbb R^4$ given by $x^2+y^2=z^2+w^2=1$ parametrised (almost everywhere) by $$\Phi:(0,2\pi)\times (0,2\pi)\to \Bbb R^4$$$$\Phi(u,v)=(\cos u,\sin u,\cos v,\sin v)$$ Then I ...
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How is a $k$-form integrated over an oriented smooth $n$-manifold in the case it is connected?

I have seen in several answers to questions on this page stating that there is no way to integrate a $k$-form over an oriented smooth $n$-manifold if $k \neq n$. However I cite Tu in his book on ...
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Orientation for compact manifolds

Page 253, -8 to -4 line Hatcher defines orientation for compact manifolds with boundary. A compact manifold is $R$_orientable if $M-\partial M$ is $R$-orientable. If $\partial M \times [0,1)$...
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$R$ orientation in Hatchers

The precise source is Chapter 3, page 235, - line 9. So what Hatcher has stated is: $M$ is an $n$ manifold, and $R$ is a ring, and $x \in M$. Then $H_n(M,M-x;R) \simeq H_n(M,M-x; \Bbb Z) \...
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Problems with 3D object orientation during rotation (Euler Angles)

I'm currently working on adding a multi-object-selection feature for a game world editor. I want to add a function that allows the user to rotate an object around the middle of all selected objects. ...
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How to get RPY(Roll, Pitch, Yaw) from directional cosines from a 3D vector?

I'm trying to find the pose of an 3D vector in terms of RPY(Roll, Pitch, Yaw). Let's say the two end points of the vector is $P_0(x_0, y_0, z_0)$ and $P_1(x_1, y_1, z_1)$. So the centered vector I get ...
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Question of deck transformation on double cover $\tilde{M}$ of non-orientable manifold $M$.

Suppose $(M,g)$ is a non-orientable, compact, connected Riemannian manifold with positive sectional curvature, $\tilde{M}$ is its orientable double cover. $\varphi$ is deck transformation of $\tilde{M}...
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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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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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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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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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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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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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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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216 views

What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
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112 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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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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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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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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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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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}\...