Questions tagged [map-projections]
76 questions
7
votes
5 answers
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 $y$ with its domain $Y$, and the function $f$…
6
votes
1 answer
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 one interpret the difference in the two measure…
4
votes
1 answer
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 $\pi:X\rightarrow\mathbb{R}$, which is the restriction to $X\subset\mathbb{R}^2$ of…
Laplace's Demon
- 621
4
votes
2 answers
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:
MapWidth and MapHeight = 8192
x = (LngX + 180) *…
LiamB
- 143
4
votes
2 answers
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 $P^2 = P$.
U2647
- 593
4
votes
1 answer
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 Pixelizing the Celestial Sphere.
Both papers are about…
Samuel Powell
- 171
4
votes
1 answer
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 $||P||\leq 2$.
Melih Can
- 67
4
votes
1 answer
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 posite definite
$|\alpha V|^2 = |\alpha|^2 |V|$…
Carcassi
- 321
4
votes
1 answer
How do great circles project on the equirectangular projection?
Given a great circle connecting two points on a sphere, what is the function describing it's equirectangular projection? In other words, given two longitudes and latitudes $(\phi_1, \theta_1)$ and $(\phi_2, \theta_2)$, what is the function…
Nathaniel Bubis
- 33,425
3
votes
0 answers
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 circle $C_n$, the image points $w=1+e^z$ all lie to…
J.Dane
- 1,067
3
votes
1 answer
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 $\min_{y \in Y} \Vert x-y \Vert$
i know that…
anto_zoolander
- 129
3
votes
2 answers
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 in advance for the help
David
- 199
3
votes
1 answer
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 $0<\theta<2\pi$. However, why can't we also use $0\leq\theta<2\pi$ or…
An old man in the sea.
- 5,422
2
votes
3 answers
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 $P$. For a given $a \in \mathbb{A}$, let $f_a(x) = c_1…
Ethan
- 576
2
votes
2 answers
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 transformation is an automorphism of an
affine…
zabop
- 1,031