Questions tagged [projection]
This tag is for questions relating to "Projection", which is nothing but the shadow cast by an object. An everyday example of a projection is the casting of shadows onto a plane. Projection has many application in various areas of Mathematics (such as Euclidean geometry, linear algebra, topology, category theory, set theory etc.) as well as Physics.
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Casting shadows of parametric convex surfaces to arbitrary planes
Im aware that, given a smooth function $f: \mathbb R^3 \rightarrow \mathbb R$ a surface $S = \{f=0\}$ casts a (orthogonally projected) shadow to a plane with unit normal $\mathbb n$ that, provided $S$ ...
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Is there anything to stop an image being projected onto the side walls in a pinhole camera/camera obscura?
(I was unsure whether this should be posted in the Physics Stack Exchange but decided to post it here instead. Apologies if this is the wrong place)
I always see diagrams of how a camera obscura ...
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Projection of finite dimensional space with ell infinity norm
Consider $\mathbb{R}^n$ with $\ell_{\infty}$ norm. Let $Y \subset \mathbb{R}^n$ be a linear subspace with dimension $r < n$.
I am considering the following projection: for each $x \in \mathbb{R}^n$,...
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Define projection transformation onto plane [duplicate]
Let $L: \Bbb R^3 \to \Bbb R^3$ be the projection onto the plane $x-y-z=0$.
I need to prove $L$ is linear which I have no problem with, but first I am need to define $L$. I came up with the following $...
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inverse of a matrix representation of a projected operator
Let $X$ be a separable Hilbert space, and let $X_n$ be the subspace spanned by orthonormal vectors $(e_i)_{i=1}^n$. Consider the operator $I_n:\mathbb{R}^n\to X$ be such that
$I_n u=\sum_{i=1}^n ...
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Projection of $W$ onto the subspace of $\triangle W$?
In the LoRA paper (https://arxiv.org/pdf/2106.09685.pdf), they stated in H.4 that:
One can naturally consider a feature amplification factor as the ratio $\frac{\|\Delta W\|_F}{\left\|U^{\top} W V^{\...
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Idea about the projections of the $C^*$-algebra $pAp$, where $p$ is a projection in $A$
Let $A$ be a unital $C^*$-algebra and $p$ be a projection in $A$. Now consider the $C^*$-algebra $pAp$. I want to classify the projections in $pAp.$ I know that, if $q \in A$ is a projection such that ...
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Filling functions on hypercubes given the function's integrals over each coordinate
$\newcommand\dif{\mathop{}\!\mathrm{d}}$
Suppose $f:[0,1]^2\rightarrow\mathbb{R}_{>0}$ and that:
$$\int_0^1{f(x,y)\dif x}=a(y),\hspace{1cm}\int_0^1{f(x,y)\dif y}=b(x),$$
for some known functions $a,...
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Mapping a hemisphere onto the unit circle
How can I approach attempting to map a hemisphere onto the unit circle such that the meridians become arcs of a circle through $(0,1)$ and $(0,-1)$ and are evenly spaced around the x-axis (which is ...
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How to find a quasi-distance-preserving mapping from $\mathbb{R}^3$ to $\mathbb{R}^2$
I have a finite set of points in $\mathbb{R}^3$ and would like to project them on a plane in such a way that distance between points is preserved as much as possible.
Is there a nice way to do it? I ...
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Question on showing that $\frac{1}{n}\sum_{k=0}^{n-1}U^kf$ converges in norm to the orthogonal projection $Pf$ to the space $\{f\in H: Uf = f\}$
Edit: The reference I am reading is Yves Coudène's Ergodic Theory and Dynamical Systems, chapter 1, proof of theorem 1.1 on page 5.
Let $H$ be a Hilbert space and $U:H\to H$ a bounded linear operator ...
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Prove that a collection of certain polynomials sum to unity
Consider a (finite) collection of $n$ distinct constants $\{f_1,\dots,f_n\}$ and define the polynomial in $x$
$$P_i(x):=\prod_{j\in\{1,\dots,n\}}^{j\neq i}\frac{(x-f_j)}{(f_i-f_j)}~,$$
for $1\leq i \...
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When is linear projection of a TVS not open?
For a general topological vector space, is it always true that linear projections are open maps, even the discontinuous ones? If not, are there well-known counterexamples? What about in normed spaces? ...
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Projecting Y axis values from one function to another to find X
I have two functions as shown in the graph below. I need to project the Y axis sigma values from the blue function onto the orange function and find the ...
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How to find the parameters necessary for drawing the lines of latitude and longitude of a spherical orthographic projection as partial ellipses?
I recently asked this question, which is a preface to this one. In that one I explain roughly what I'm attempting to achieve, but I realized that once I found that cutoff angle I still didn't have a ...
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Infinite decomposition of a Hilbert space
Let $H$ be a Hilbert space. Suppose we have $H_{n}\oplus G_{n}=H$ for all $%
n\geq 0$ so that $\left( H_{n}\right) _{n\geq 0}$ is an increasing sequence
of closed subspaces and $\left( G_{n}\right) _{...
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How to get the exact cutoff angle for a given latitude in a spherical orthographic projection?
I'm trying to compute a grid that's orthographically projected onto a sphere for the purposes of overlaying onto images of Sol (which is so far away that this is a good approximation), so that one can ...
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lines passing through two points in $\mathbb{P}^n$
The reference is the Example 7.5. b) from ag notes by Gathmann. The paragraph explains the idea of projection from a point. He claims that the unique line passing through $a=(1:0:\cdots:0)\in\mathbb{P}...
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Need help on Spherical sweep calculation based on animated 3D camera.
I need some maths help to crack this problem, so I can implement something in Maya (3D animation program.
If we have an animated 3D camera ( -z is aiming direction, y is up, x is right) and focusing ...
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inverse of projection tensor in continuum mechanics
There is a 4th-order tensor in continuum mechanics, called projection tensor, which gives the deviatoric part of any tenor on the initial configuration. If we ignore the difference between initial and ...
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Characterizing "almost-vector spaces" that don't assume the axiom $1\cdot v=v$ for scalar multiplication
Consider a structure satisfying all the axioms of a vector space except for $1\cdot v=v$. I don't know if there might might be a name for these, but lest call them "almost-vector spaces". ...
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Is the projection $\pi:R^3 \to R^2$ given by $ \pi(x,y,z)=(x,y)$ on a regular suface continuous?
$\pi:R^3 \to R^2$ given by $ \pi(x,y,z)=(x,y)$
In the textbook, do Carmo-Differential geometry of curves & surfaces, page 66, the proof of proposition 4, it says that the projection $\pi$ ...
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Define the image of the projection matrix $Q:=X(X^{T}X)^{-1}X^{T}$
I recently came across the following claim:
Let $X \in \mathbb{R}^{n \times p}$ ($p\leq n$) be a matrix with linearly independent columns. Then $Q:=X(X^{T}X)^{-1}X^{T}$ is a projection matrix onto $...
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Linear Algebra projecting vector into a plane defined by 2 unit vectors
Given 2 unit vectors l and v (for light and view directions). And given a third unit vector n which is the normal to surface and not coplanar with l and v.
It is required to prove that the projection ...
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Projection of a vector onto a non-orthonormal basis vector?
Suppost $v$ be a vector and I want to project onto a non-orthonormal basis vector $u$. There is no span just two of these vectors.
How do I do that? Is it correct that if I say there is no span?
There ...
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Extending a vector so its projection match another vector's length, there's a name for that?
My Problem
I'm a game developer, and I had a problem yesterday in which I needed to extend a vector $\vec{v}$ so its projection into $\vec{w}$ was equal to $||\vec{w}||$.
(Illustrating the Problem)
...
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Question about projections acting on dual space
Let $X$ be a complex Banach space, and let $P$ be a bounded linear operator acting on the dual $X^{*}$ such that that $P^2=P$.
I research for a bounded linear operator $Q$ acting on $X$ such that its ...
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How can the conditional expectation be the $L^2$ projection, when the $L^2$ projection depends on the choice of the norm?
one can define the conditional expectation of a random variable $X$ on $\mathbb{R}^n$ in an axiomatic way, without relying on any norm by being the $\mathcal{G}$-measurable random variable $Y$ which ...
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Is there a smooth global section from projective space to the sphere?
I'm having a hard time trying to find (if there is any) global section from $\mathbb{RP}^n$ to the sphere $\mathbb{S}^n$ or $\mathbb{R}^{n+1}$ (global sections of the the natural projection map $\pi$ ...
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Rewriting a linear transformation from formula to matrix
This is a question from my exam that I just cannot figure out how to do it. Thank you for helping me in advance. I will try to write it in here but it will be easier to understand using the image I ...
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Finding the projection of a vector onto another vector
I have a vector $x$, and another vector $x_7$, and I want to find the projection of $x_7$ on $x$.
I know that $x = \begin{pmatrix} \frac{1}{36} (\sqrt{1465} - 13) & 1\end{pmatrix}$
and I'm given $...
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Why does it suffice to define the homogeneous projection operator by only testing with continuous instead of measurable functions?
In Lenaic Chizat's "Sparse Optimization on Measures with Overparametrized Gradient Descent" one finds the following definition: for a measure $\mu \in \mathcal P_2(\Omega)$ (the Wasserstein-...
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Distance of a function (as a vector) from a certain kernel
Consider $C[-1,1]$, the normed linear space of real-valued continuous functions on $[-1,1]$ with the supremum norm. Let K be the kernel of the linear functional $I: f \to \int_{-1}^{1}f(x)dx $ on
$C[-...
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Clarification for a proof that a bounded projection $T$ with a norm of at most one is an orthogonal projection
I am trying to understand the accepted proof of this post: Orthogonal Projection, where $T$ is a bounded projection mapping with a norm of at most one. While the original question concerned finite ...
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If $T\in\mathcal{B}\left(\mathcal{H}\right):T^2=T\land ||T||\leq 1$ then why is the range of $T$ a closed subspace of $\mathcal{H}$?
I am trying to understand the accepted proof of this post: Orthogonal Projection, where $T$ is a bounded projection mapping with a norm of at most one. While the original question concerned finite ...
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How to Make the Minecraft World Look Spherical?
Note 1: I am aware of this video. It seems to me as though they used azimuthal equidistant projection to make the world look round. I was trying to use stereographic projection to make the world look ...
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Minimization of a quadratic function with both equality and inequality constraints using gradient projection method.
I should solve a quadratic programming problem, where I should minimize a function with respect to the given equality and inequality constraints using gradient projection algorithm. How can this be ...
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Showing that if $P\in\mathcal{B}(\mathcal{H}):||P||\leq 1$ and $P$ is a projection mapping, then $P$ is necessarily an orthogonal projection
Let $P \in \mathcal{B}(\mathcal{H})$ be a bounded projection mapping in an Hilbert space $H$. Suppose that $||P|| \leq 1$, i.e. $\forall x\in \mathcal{H}:||Px|| \leq ||x||$. I am trying to show that ...
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If $P$ is a projection mapping, $\mathcal{R}(P)$ is closed and $(v_k, P(v_k))\to(v, w)\in X^2$ then why $Pw = w$?
Let $X$ be a Banach space and $P$ a projection mapping. Suppose that $\mathcal{R}(P)$ (the range of $P$) is closed. Consider a sequence $(v_k, P(v_k))_{k=1}^\infty \subset X^2$ and suppose that $\lim_{...
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Why is $\mathrm{Ran}(P) = \mathrm{Ker}(I - P)$ for a projection mapping $P$?
Let $P$ be a projection mapping in a Banach space $X$. I am trying to understand why $\mathrm{Ran}(P) = \mathrm{Ker}(I - P)$: If $u \in \mathrm{Ker}(I - P)$ then $(I - P)u = 0 \Longleftrightarrow Iu = ...
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The range projection and monotone complete C*-algebra.
Let $A$ be a monotone complete C- algebra (i.e every increasing , norm bounded net in A has a supermum in A), this kind generlises many types of C-algebra (Von nuemann algebra...). Also, it generetes ...
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Prove that $P_{h}(f)$ satisfies $||f-P_{h}|| \leq Ch^{2}$
Consider a partition of the domain $\Omega=[a,b]$
$$ a=x_{1}<x_{2}<\cdots < x_{N}=b $$
with mesh size $h=\max\{x_{i+1}-x_{i}:i=1,...,N-1\}$.
Let V be an inner product space, with inner ...
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Projection of a 3D circle onto a 2D camera image
Assume that I have a 3D circle with a center at $(c_1, c_2, c_3)$ in the circle coordinate frame $C$. The radius of the circle is $r$, and there is a unit vector $(v_1, v_2, v_3)$ (also in coordinate ...
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Difficulties to understand projections (about PCA)
About PCA I read that we were looking for the direction $w_1 \in \mathbb{R^p}$ with $||w||=1$ such that the variance of our data projected onto this direction is maximal. The word "projection&...
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Theorem 11, Section 6.7 of Hoffman’s Linear Algebra
Let $T$ be a linear operator on a finite-dimensional space $V$.
If $T$ is diagonalizable and if $c_1,…, c_k$ are the distinct characteristic values of $T$, then there exist linear operators $E_1,…,E_k$...
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Exercise 1, Section 6.7 of Hoffman’s Linear Algebra
Let $E$ be a projection of $V$ and let $T$ be a linear operator on $V$. Prove that the range of $E$ is invariant under $T$ if and only if $ETE=TE$. Prove that both the range and null space of $E$ are ...
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A result about the separation of a point and a convex closed set
I would like to show the following result :
If $C\subset\mathbb{R}^n$ is a closed convex subspace that does not contains the null vector, then there exists a vector $Y\neq 0_{\mathbb{R}^n}$ such that
$...
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Exercise 10, Section 6.6 of Hoffman’s Linear Algebra
Let $F$ be a subfield of the complex numbers (or, a field of characteristic zero). Let $V$ be a finite-dimensional vector space over $F$. Suppose that $E_1,…,E_k$ are projections of $V$ and that $E_1+…...
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Do we have for any bounded projection $P$, $\left\Vert Px\right\Vert =\left\Vert P\right\Vert \left\Vert x\right\Vert $ for some non-zero $x\in H$?
Les $H$ be a Hilbert space and $P\in B\left( H\right) $ a bounded
projection. I made some drawing to try to find such an $x$, and I found that
$x~$is any non zero vector in the image of $P^{\ast }\...
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Exercise 9, Section 6.6 of Hoffman’s Linear Algebra
Let $V$ be a real vector space and $E$ an idempotent linear operator on $V$, i.e., a projection. Prove that $(I + E)$ is invertible. Find $(I + E)^{-1}$.
My attempt: Suppose $E:V\to V$ be a ...
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