Questions tagged [map-projections]

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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
41 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
27 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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What is a good introduction to the math of map projections

I'm interested in learning the math behind map projections but often the explanations (wikipedia one for example) are way beyond my level, so I was wondering if someone know of a good book or internet ...
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17 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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12 views

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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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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4answers
43 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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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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40 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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41 views

How to map a Cartesian vector into equirectangular coordinates

I am working on a tectonic plate simulation for terrain generation. I have a plate in Cartesian coordinates mapped to an equirectangular and I need to map its drift vector onto the same ...
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40 views

Is the gnomonic projection the only one mapping great circles to straight lines?

The gnomonic projection maps from a hemisphere to the plane, with the equator at infinity. It is noted for having the property that great circles map to straight lines. As a result, any line segment ...
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1answer
80 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
83 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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42 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
106 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
165 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
53 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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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
64 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
507 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
78 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
163 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
320 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
46 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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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
68 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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651 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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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
128 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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37 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
265 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
39 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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38 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
527 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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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
102 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
45 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
117 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
40 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
38 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 ...
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1answer
1k views

What is the difference between moment projection and information projection?

Moment projection is defined as $$\text{arg min}_{q\in Q} D(p||q)$$ while information projection is defined as $$\text{arg min}_{q\in Q} D(q||p)$$. Aside from the difference in the formula, how should ...
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832 views

Difference between orthogonal projection and oblique projection physically.

Let $A$ and $B$ be $n \times n$ matrices in $\mathbb{R}^n$, $N(B)$ denotes nullity of $B$. Let $C $ and $N(B)$ are complementary subspaces of $\mathbb{R}^n$. Let $P_c$ denotes the oblique projector ...
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100 views

Are projection and norm enough to define an inner product?

Given an inner product, one can define a projection and a norm. Can we do the opposite? That is, suppose we have: a complex vector space V a norm $|V|^2 : V \rightarrow \mathbb{R}$ such that: is ...
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1answer
756 views

Converting from Mercator Projection to Latitude and Longitude

I have an image of (what I believe to be) a Mercator Projection map of Strangereal, from the Ace Combat game series. I have opened this map in GIMP and am reading pixel measurements for different ...
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3answers
3k views

Projection onto a plane

I was looking at this post ($3D$ projection onto a plane) in which the answer describes how to project a given set of points onto any arbitrary plane. However, this transformation is still of the ...
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58 views

Why is a projective module called “projective”? [duplicate]

Is it related to projections as in $P P = P$?
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1answer
742 views

How do great circles project on the mercator projection?

Given a great circle connecting two points on a sphere, what is the function describing it's Mercator projection? In other words, given two longitudes and latitudes $(\phi_1, \theta_1)$ and $(\phi_2, \...