Questions tagged [map-projections]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
-1 votes
0 answers
16 views

Finding metric projection mapping [closed]

Let $X$ = $c_0$, with supremum norm and $W$ = $\{ \{x_n\}_{n \geq 1} \in X$ $|$ $x_1=0\}$. Then how to find $P_W(x)$. Note that, $P_W(x)$ = $\{y_0 \in W : ||x-y_0|| = \inf ||x-y||,$ $ y \in W \}$. In ...
user avatar
1 vote
0 answers
75 views

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'...
user avatar
  • 1,856
0 votes
0 answers
6 views

Transform distances in North and East to distances on Lambert projection

I think I read somewhere about a transformation between distances in North and East and distances on a Lambert conformal conic projection based on a series expansion. The goal was to get distances ...
user avatar
  • 135
0 votes
1 answer
15 views

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 ...
user avatar
0 votes
1 answer
112 views

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 ...
user avatar
0 votes
0 answers
22 views

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 $\Pi_k(x) = (x_1,\ldots,x_k)$...
user avatar
  • 51
0 votes
0 answers
18 views

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 ...
user avatar
0 votes
1 answer
38 views

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 +...
user avatar
1 vote
0 answers
27 views

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 ...
user avatar
  • 36.3k
2 votes
2 answers
94 views

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 ...
user avatar
  • 955
1 vote
0 answers
22 views

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 ...
user avatar
  • 155
2 votes
1 answer
66 views

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 ...
user avatar
  • 133
1 vote
0 answers
61 views

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-...
user avatar
0 votes
0 answers
13 views

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 ...
user avatar
2 votes
1 answer
220 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 $\pi:X\rightarrow\mathbb{R}$, ...
user avatar
2 votes
2 answers
53 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\...
user avatar
  • 123
0 votes
0 answers
301 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, ...
user avatar
0 votes
1 answer
51 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 ...
user avatar
  • 1,965
3 votes
0 answers
171 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 ...
user avatar
  • 1,027
0 votes
1 answer
77 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 ...
user avatar
  • 65
1 vote
1 answer
215 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 ...
user avatar
  • 11
7 votes
5 answers
134 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 $...
user avatar
3 votes
1 answer
100 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 \...
user avatar
0 votes
1 answer
48 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,...
user avatar
  • 17
2 votes
3 answers
58 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 ...
user avatar
  • 45
1 vote
1 answer
113 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 ...
user avatar
  • 91
1 vote
0 answers
14 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\...
user avatar
  • 450
1 vote
0 answers
81 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 $\mathbb{R}^d$....
user avatar
1 vote
0 answers
20 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 ...
user avatar
0 votes
2 answers
32 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 ...
user avatar
2 votes
4 answers
49 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 ...
user avatar
  • 1,495
1 vote
0 answers
68 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 $S=\{(x,y)\in \mathbb{R}^2|x^2+(y-1)^2=1\}$ (...
user avatar
  • 11
1 vote
0 answers
47 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 ...
user avatar
1 vote
1 answer
213 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 ...
user avatar
4 votes
2 answers
307 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: ...
user avatar
  • 143
2 votes
1 answer
117 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 ...
user avatar
2 votes
1 answer
154 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 ...
user avatar
  • 101
3 votes
2 answers
521 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 $P^2 = P$.
user avatar
  • 563
1 vote
1 answer
116 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 ...
user avatar
  • 563
1 vote
0 answers
17 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 ....
user avatar
1 vote
1 answer
68 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 ...
user avatar
  • 363
1 vote
1 answer
741 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 ...
user avatar
  • 121
0 votes
1 answer
132 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 ...
user avatar
4 votes
1 answer
255 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 ...
user avatar
0 votes
1 answer
411 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 ...
user avatar
  • 113
3 votes
2 answers
61 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 ...
user avatar
  • 149
1 vote
0 answers
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 ...
user avatar
  • 219
1 vote
2 answers
74 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/...
user avatar
0 votes
2 answers
1k 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, ...
user avatar
  • 959
0 votes
0 answers
54 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 $Set$ ...
user avatar
  • 21