Questions tagged [map-projections]

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How will the unit spheres volume change after mapping

Let $R^3$ be a vector space. Linear map $F$ will map base of $R^3$: $u_1=(1,2,-1)^T$, $u_2=(1,-3,3)^T$, $u_3=(-1,-2,2)^T$ on vectors $v_1=(-1,-3,5)^T$, $v_2=(2,5,-4)^T$, $v_3=(-2,-6,7)^T$. How will ...
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55 views

Prove that $\pi$ is a quotient map which is neither open nor closed

Exercise: Let $X:=\mathbb{R}^2\smallsetminus((-1,1)\times(-2,2)\cup[2,3]\times[-2,2])\subset\mathbb{R}^2$ be equipped with the subspace topology and consider the map $\pi:X\rightarrow\mathbb{R}$, ...
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2answers
40 views

Projection from n-fold cartesian product to coordinates indexed by a fiber.

Consider the example 1.3.2 (xi) of Emily Riehl (2016) Category theory in context: I am having trouble trying to understand the part where $M^f$ is described. As far as I have understood, $M^f:M\times\...
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80 views

How to create a map projection

Theoretically, there are infinitely many map projections. They are usually defined by precise mathematical formulae, although so-called comprise maps (most notably the Robinson projection) also exist, ...
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1answer
36 views

Norm convergence of a series of operators.

Let $\mathcal H$ be a infinite dimensional Hilbert space and $T \in \mathcal L (\mathcal H)$ be a compact normal operator. Let $\sigma (T) = \{\lambda_1,\lambda_2, \cdots\} \cup \{0\}$ be the spectrum ...
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157 views

What is the image of the circle $z=2n\pi e^{i\theta}$ under the map $w=1+e^z$

What is the image of the circle $z=2n\pi e^{i\theta}$ under the map $w=1+e^z$ Can somebody help me with this problem? In the book, it says that ${(i)}$ the region $0 \le x\le x_0$ and $z$ on the ...
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1answer
37 views

Pythagoras on Mercator's Map

Very stupid question, but I cannot help but wonder whether measuring (and finding true distance after correcting scale distortion) of the horizontal and vertical displacement, and using the Pythagoras ...
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1answer
54 views

Projecting Sphere to Rectangle/Square using Mercerator Projection

I am learning some projection technique where we can project a 3d object like globe to a 2d. I have the 3d coordinates of points on the surface of sphere same as globe. Here is a reference where a ...
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126 views

Does $y = f(x) = ax+b$ actually have two mappings inside it?

I’m just a high school student, so I may be somewhat logically flawed in understanding this. According to wikipedia, the definition of function requires an input $x$ with its domain $X$ and an output $...
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1answer
75 views

projection on pre-Hilbert space

Suppose $(X,*)$ is a pre-hilbert real space. Is it true that a linear projection $P:X\rightarrow X, P(X)=Y$, self-adjoint respect $*$, is the identity on $Y$? this means that $Px$ realize $\min_{y \...
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1answer
39 views

Mapping two superquadratics

Superquadrics are a family of geometric shapes defined by $$|\frac{x}{A}|^r + |\frac{y}{B}|^s + |\frac{z}{C}|^t =1$$ I have two superquadratics, namely SQ1 and SQ2, with parameters $\{A_1,B_1,C_1,r_1,...
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3answers
53 views

How does one calculate the projections onto sub-spaces of $C^3$?

Given the following two sub-spaces of $C^3$: $W = \operatorname{span}[(1,0,0)] \,, U = \operatorname{span}[(1,1,0),(0,1,1)].$ I want to find the linear operators $P_u , P_w$ which represent the ...
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1answer
64 views

Two-point equidistant projection of the sphere

According to this Wikipedia article, there is a projection from the $2$-sphere to a region in the plane that preserves distances to two given points. The article says that the projection was first ...
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12 views

How do you denote functions inherited by a product space?

Given a map $f:A\to B$, define the map $p_n(f):A^n\to B^n$ as $p_n(f)(a_1,...,a_i)=(f(a_1),...,f(a_i))$. Equivalently, you could say given $f$, $p_n(f)$ is such that $\mathrm{proj}_a\circ p_n(f)=f\...
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45 views

Can argmin act as a “projection”?

My professor is teaching us about a clustering algorithm called K-means clustering. He used the term projection in a way I'm unfamiliar with. Assume all vectors in this example are in $\mathbb{R}^d$....
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Which map projection should be used for preserving coordinates on a 2D image?

I have coordinates of a city on earth, say latitude: 57.7072326, longitude: 11.9670171. This is city A. I also have coordinates for another city, called city B. I want to place a marker on these 2 ...
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2answers
31 views

Mapping $2$ kinds of operators?

So $z$ from complex analysis can be mapped as: $$ z \to \hat z = \begin{pmatrix} \Re z \\ \Im z \end{pmatrix}$$ Now the $+$ in complex analysis seems to be the same operation as the $+$ in linear ...
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48 views

If $f:A\to B$ and $g :B\to C$ are 2 mappings then if $g\circ f:A\to C$ is injective then prove that $f$ is injective

My attempt : Let $f(x_1)=f(x_2)$ for some $x_1$ and $x_2$. $g\circ f(x_1)=g\circ f(x_2)$ implies that $x_1=x_2$ holds. Thus $f(x_1)=f(x_2)$ implies that $x_1=x_2$ and therefore $f$ is injective What ...
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54 views

Stereographic Projection - Example

to work with stereographic projections I want to know how to solve this example first: It is it about finding a map $f:\mathbb{R} \rightarrow S$ where $S=\{(x,y)\in \mathbb{R}^2|x^2+(y-1)^2=1\}$ (...
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43 views

Lat long projection to not distort distance

I have many circles which are lat long positions and a radius in meters. I need to check if two circles intersect and I can use this haversine distance function. However, to speed up performance I ...
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1answer
137 views

Latitude more than 90

I want to calculate mercator projection of points inside of sphere. First, the points are transformed to local system. Then the latitude and longitude are calculated based on following equations ...
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2answers
198 views

Conversion of coordinates (longitude ; latitude) to (X;Y)

We have an old mapping system we are needing to convert some data to and from. We need to convert from Lng/lat to XY and from XY to Lng/Lat. We can convert from Lng/Lat to XY Using the following: ...
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69 views

What is the formula for the Collignon projection (diamond form)?

So, I want to project a sphere onto a square with the poles at two opposite corners, preserving areas (long story, but I have a practical application in mind). Wikipedia and other online sources ...
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1answer
131 views

How can I project a curved surface onto another curved surface?

I don't really know where to look in the world wide web for this specific problem. So I hope you can point me into the right direction. Here is my specific problem: I have a deformed spherical ...
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2answers
350 views

Given a closed linear subspace, is there always a projection that maps onto it?

Given a closed linear subspace, is there always a projection that maps onto it? Here, a projection $P$ should be a linear and continuous mapping and satisfies $P^2 = P$.
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1answer
77 views

If projections $P$ and $Q$ are commutative, then $P+Q-PQ$ projects onto $\text{im}P+\text{im}Q$

Projection: Suppose $X$ is a normed vector space, we define a projection $P$ as a linear and continuous mapping from X to X, such that $P^2 = P$. im$P$: the image of $P$ commutative: $PQ=QP$ The ...
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15 views

Covering from Dense Projection

Let $V$ be a Banach space, $W$ a (strict) subspace of $V$, and $U$ a dense proper subset of $V$. When is does there exist a (linear) projection $P^V_W:V\twoheadrightarrow W$, such that $$ P_W^V(U)=W ....
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1answer
67 views

How to show that a map is linear in $C^n$?

Could someone tell me if I am on right way solving Problem b)? Problem: Let $U,V\subset\mathbb C^n$ be two subspaces, such that $\mathbb C^n = U+V$ and further assume $U\cap V = \{0\}$. a) Show that ...
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1answer
628 views

How to draw a globe in 2D? [duplicate]

I want to draw such globe: Let's say my input params are: center (x,y) radius number of meridians number of parallels. I know such concepts as: 2D ellipse equation 4x4 projection ...
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1answer
98 views

Howto calculate the latitude of a given y coordinate from a mercator projected map

Say I have a mercator projection map: I would like to calculation the latitude for different points with one formula. I have already resaerched several sites and wikipedia, where the hole math is ...
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1answer
206 views

Equal-area projection from sphere to tangent plane

I'm running into a problem trying to understand the work in the two following papers: Feasibility Study of a Quadrilateralized Spherical Cube Earth Data Base. and An Icosahedron-based Method for ...
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1answer
376 views

Non orthogonal projection of a point onto a plane

I have multiple points in a circle and I need to project them onto a finite plane that is angled in respect to the circle. This is actually a light cone, that originates some millimetres behind the ...
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2answers
53 views

Bounded linear operator

Linear bounded operator T on $l^2$ is given by : $T(x_1,x_2,x_3,..) := (x_1,x_1,x_1,x_2,x_2,x_2,x_3,x_3,x_3,..)$. Prove that $\frac{T^*T}{3}$ and $\frac{TT^*}{3}$ are orthogonal projections. Thanks ...
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55 views

Correcting for arbitrary distortion in experimental CNC Router

I have an issue which I would love any guidance at all with. The Background: I have been working on an experimental CNC router which looks like this: The X and Y position is controlled by ...
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2answers
71 views

Is it true that you need atleast 350 colors to color-in the boundries of a flat map where all the boundries are clearly defined?

Since a providence may build on a bridge, and since certain bridges may be placed as archways over land: it is an obvious fact that this Earth could be turned into a topological and territoried donut/...
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905 views

The inverse of projection function is a closed map?

I have this questione about the projection, be: $$\pi :X \times Y \to X$$ if we consider $ \pi^{-1}(x): x \to x \times Y $ I want to know if this is a closed map, is easy to see that $\pi$ is not, ...
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46 views

Simple and fast presentation of theory about projections and universality

I would to learn what happens in a projection about universality. I searched in google and i found this: https://en.wikipedia.org/wiki/Projection_(mathematics) Let's conside that we refer in $Set$ ...
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2answers
236 views

About the inverse function

Hello. Does it always an inverse map don't exist? I know that a mapping is considered as a function . And I know that there are some functions as sine that has an inverse that is arcsin . So why might ...
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42 views

What is the name of this map projection?

Is there a name for the map projection that makes a circle out of a sphere, where, in polar coordinates, the angle is the bearing angle of the point and the radius is the distance (on the surface of ...
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1answer
362 views

Positive operator on a Hilbert Space

Let $H$ be a Hilbert space. Let $L: H \rightarrow H$ be a linear bounded positive operator (i.e. $\langle L(u), u \rangle \geq 0$ for all $u \in H$). A) Prove that $I+aL$ is bijective for ...
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1answer
41 views

projection always yields to $0$? (what am I doing wrong)

Say we have vectors $x$ and $y$ in some inner product vectorspace. Then the projection of $x$ onto $y$ is given by: $\begin{aligned}P(x)=x-\frac{x\cdot y}{\Vert y\Vert ^2}y\end{aligned}$, but I would ...
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45 views

Points in Two surface Mapping Problem

First, this is 3d problem. I have two circle type line (closed), these line shape don't need to be Perfect Circle. They don't need to be in one plane. Based on the lines, we can get a surface and ...
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1answer
688 views

Projection in Banach Space

Let $X$ be a Banach Space and let $Y=\ker f \subset X$ be hyperplane in $X$. Prove that there exists a projection $P:X \to Y$ such that $||P||\leq 2$.
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1answer
1k views

Why restrict the domain of polar coordinates, cylindrical coordinates, spherical, etc?

For a change of variables one needs the mapping to be injective. In the book I'm reading, we restrict the mapping of polar coordinates $g(r,\theta)$ to the domain $r>0$ and $0<\theta<2\pi$. ...
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1answer
112 views

Distortions in a Subway Map

The NYC subway map - like most subway maps - isn't shown to scale. Is there a way to visually represent the distortion of the map in terms of area? More specifically, suppose you have the set of ...
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1answer
49 views

Approximation Theory, Projections and Hyperplanes [closed]

Let $X$ be a Banach space and let $Y = \ker f \subset X$ be a hyperplane in $X$ for some nonzero $f \in X'$ . Prove that if $P\colon X \longrightarrow Y$ is a continous projection, than there exists $...
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1answer
47 views

Working on a finite subspace

Let $X$ be a normed space and $L$ a finite dimensional linear subspace. I need to show that there exists finitely many $l_1,.....l_n \in L$ and $f_1,.....f_n \in X'$ such that $$ l = \sum_{i=1}^{n} ...
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1answer
155 views

I want to know the difference between metric projector and orthogonal projector?

Given a metric space $(X, \rho)$ and $A$ be its closed subset. Now for every $x \in X$ define $$P_A(x) = \{ y \in A : \rho(x, A) = \rho(y, x)\}$$ Now definition of metric projector is as follows: Let $...
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1answer
53 views

Surjectivity of a complex projection

Given is the projection $$u: \mathbb{C} \rightarrow \mathbb{R}: z \rightarrow Re(|iz|) - |iRe(z)|$$ which is $$u: \mathbb{C} \rightarrow \mathbb{R}: z \rightarrow |z| - |Re(z)|$$ when simplified....
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1answer
41 views

Help with surjectivity of a function

My function $f$ is as follows: $$f: \mathbb{R} \rightarrow \mathbb{R^2}: t \rightarrow (2cos(t), - sin(t))$$ Now, I'm fairly certain that the function isn't injective, as both sine and cosine are ...