# 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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### 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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### 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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### 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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### 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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### 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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