Questions tagged [map-projections]
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How do I describe what I want to do with a planar graph?
The image shows a planar graph, the points of which are calculated by Python's NetworkX. (While, in this example, the graph also happens to be a triangulation, that is not always the case).
What I ...
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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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Defining an injective map from the algebraic numbers to the set of integer coefficient polynomials.
Let $P$ be the set of all polynomials with integer coefficients, one variable, and deg $n \ge 1$. A number is said to be algebraic, $\mathbb{A}$, if it is real and the solution to an element of $P$. ...
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Name of a map coupled with the identity
Given a map $\tau:\Omega\longrightarrow \mathbb{R}$, is there a standard name for the map $\tilde{\tau}:\Omega\longrightarrow\Omega\times\mathbb{R}$ that maps $\omega$ to $(\omega,\tau(\omega))$? It ...
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Closure of the inverse image under the projection map
Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
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Projection operator fixed up to a constant
Let $\Omega$ be a polygonal domain in $\mathbb{R}^2$, $V$ a closed and finite dimensional subspace of $H^1(\Omega)$ and $\mathbb{P}_k(\Omega)$ the usual space of polynomials of maximum degree $k$.
I'...
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In Spherical Mercator Tile coordinates, how to go one zoom-level deeper but span the same area?
For an arbitrary tile region defined by (z1, x1, y1) in Standard Web Mercator Tile format (Spherical Mercator) , how can we increase ...
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Is there an easy way to convert a partial arc length or movement along an elliptic curve into an X,Y position?
I had this idea to project the Earth onto a 2D map using elliptical cylinders that could be unrolled.
(The Earth can be well approximated by rotating an ellipse on its axis to create an oblate ...
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If $U \subset Y \subset \mathbb{R}^n$, and $U$ is open relative to $Y$, then is $\Pi_k(U)$ open relative to $\Pi_k(Y)$? ($\Pi_k$ is a projection map)
Here, $\Pi_k: \mathbb{R}^n \to \mathbb{R}^k$ denotes the projection map, mapping an element of $\mathbb{R}^n$ to its first $k$ coordinates. If $x = (x_1,\ldots,x_n)$, then $\Pi_k(x) = (x_1,\ldots,x_k)$...
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Proof of integral substitution rule
I have the proof of the integral substitution rule at the university. In order this rule to use, I must have some conditions. So $ f: I\to R$ and $g: I_0 \to I$ and $I,I_0$ are not trivial
intervals ...
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Basis of the polynomial with degree less or equal 2
Can u explain me one thing. We have
$P_2:= \{\text{all polynomial with degree}\leq 2\}$
and $U_0:=\{f \in P_2 \mid f(1)=0\}$
We have $f(1)=c_1+c_2+c_3$ (because every polymial has form of $ f(t)=c_1 +...
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Mapping of a unit circle to a lune of a unit sphere quarter
Uniform $\lambda $ spaced circles of unit radius through origin are given by
$$ (x-\lambda)^2 +(y-\lambda)^2 =1$$
include a net of constant curved differential length rhombuses.
What is the equal ...
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Is Mercator projection an affine trasformation?
A practical example lead me to believe that a geographical projection, such as the Mercator projection, is an affine transformation.
However, when I checked on Wikipedia:
More generally, an affine ...
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How can I extract the equations from this 3D projection graph?
I'm trying to transform these plots to functions, but I'm having a hard time figuring out a formulaic 2D->3D transformation on this type of mapping. They collapsed an axis flat on the abscissa, and ...
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Vector Projections
I am watching a youtube video on Principal Component Analysis. It is written that the projection of vector $x_i$ onto vector $u_1$ is
$proj_{u_{1}}(x_i) = {u_1}^T x_i u$
where $u$ is the unit vector ...
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Derivation of General form of Map Projections
In this paper, the following formulae are derived for the cylindrical and conic projections:
Cylindrical: $T(\phi, \lambda)= (\lambda, h(\phi))$ for some function $h$.
Examples include:
Equi-...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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To calculate Mercator projection of points inside of sphere. What if latitude is more than $90^\circ$?
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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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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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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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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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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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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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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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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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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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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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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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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