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Image of region under a projective transformation

Problem goes like this Given the region ${F} = \{ (x, y) \in \mathbb{R}^2 \mid 0 \leq x \leq 1, y \geq 0 \}$ and the matrix $M$ of the projective transformation $f$ of the extended affine plane, $$ M =...
Milan's user avatar
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1 answer
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Orthogonal Projection in an Enlarged Hilbert Space

Let $(H, \langle \cdot, \cdot \rangle_H)$ and $(U,\langle \cdot, \cdot \rangle_U )$ be Hilbert Spaces such that $H$ embeds into $U$. Let $M$ be a closed subspace of $U$, and define $\mathcal{P}$ to be ...
RiaDoog's user avatar
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Can a Lambert conformal conic projection be constructed geometrically?

A gnomonic projection can be constructed geometrically by putting a light source in the middle of a semi-transparent globe to project an image of half of tyat globe onto a plane. A stereographic ...
HelloGoodbye's user avatar
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1 answer
54 views

The projection of an open ball is also an open ball

My problem is as follows: Let $\pi_i:\mathbb{R}^n\to\mathbb{R}$ be the projection onto the i-th coordinate. Prove that if $A \subset\mathbb{R}^n$ is open then its projection $\pi_i(A) \subset \mathbb{...
Kawanardo Queiroz's user avatar
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2 answers
51 views

Calculate "vanishing points" of line of latitude on orthographic map projection

For an orthographic map projection (of radius $r$) centered at latitude $\varphi_0$, I believe the ellipse defining the line of latitude $\varphi$ is centered at $y$ coordinate $r\cos(\varphi_0)\sin(\...
rvcx's user avatar
  • 23
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1 answer
44 views

Prove that if $p$ and $q$ are projections with the same kernel, then $p\circ q=p$ and $q\circ p=q$

Let $K$ be a field, and let $E$ be $K$-vector space. Let $p$ and $q$ be two endomorphisms of $E$. Prove the following proposition: ($p$ and $q$ are projections, and $\ker{p}$ = $\ker{q}$) $\...
virtualcode's user avatar
1 vote
0 answers
42 views

Help understand real and projected distance difference in UTM coordinates

I am learning about UTM coordinate system and reading in Wikipedia: In any zone a point that has an easting of 400000 meters is about 100 km west of the central meridian. For most such points, the ...
greatvovan's user avatar
2 votes
3 answers
184 views

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 $P$. ...
Ethan's user avatar
  • 376
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1 answer
35 views

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 ...
xyz's user avatar
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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,...
John's user avatar
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0 answers
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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'...
Gabrielek's user avatar
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1 answer
93 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 ...
Zeta.Investigator's user avatar
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1 answer
129 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 ...
Martin Clemens Bloch's user avatar
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54 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 $\Pi_k(x) = (x_1,\ldots,x_k)$...
Druizr's user avatar
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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 ...
nikibiki's user avatar
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1 answer
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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 +...
nikibiki's user avatar
1 vote
0 answers
35 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 ...
Narasimham's user avatar
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2 votes
2 answers
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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 ...
zabop's user avatar
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1 vote
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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 ...
Esteban's user avatar
  • 165
2 votes
1 answer
85 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 ...
SoraHeart's user avatar
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0 answers
151 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-...
SoySoy4444's user avatar
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33 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 ...
Rikib1999's user avatar
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2 votes
1 answer
855 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 $\pi:X\rightarrow\mathbb{R}$, ...
Laplace's Demon's user avatar
2 votes
2 answers
105 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\...
Arjonais's user avatar
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685 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, ...
SoySoy4444's user avatar
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1 answer
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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 ...
math maniac.'s user avatar
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3 votes
0 answers
224 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 ...
J.Dane's user avatar
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1 answer
238 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 ...
swang's user avatar
  • 65
1 vote
1 answer
1k 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 ...
Neil's user avatar
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7 votes
5 answers
188 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 $...
Timothy Chang's user avatar
3 votes
1 answer
197 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 \...
anto_zoolander's user avatar
0 votes
1 answer
53 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,...
Hossein's user avatar
  • 17
2 votes
3 answers
61 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 ...
Hartman's user avatar
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1 vote
1 answer
206 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 ...
rf1x's user avatar
  • 91
1 vote
0 answers
15 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\...
Cam White's user avatar
  • 490
1 vote
1 answer
279 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 $\mathbb{R}^d$....
Stan Shunpike's user avatar
1 vote
0 answers
48 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 ...
Sigfrid Stjärnholm's user avatar
0 votes
2 answers
34 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 ...
More Anonymous's user avatar
2 votes
4 answers
51 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 ...
Guria Sona's user avatar
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1 vote
0 answers
94 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 $S=\{(x,y)\in \mathbb{R}^2|x^2+(y-1)^2=1\}$ (...
emmotr's user avatar
  • 11
1 vote
0 answers
59 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 ...
David Callanan's user avatar
1 vote
1 answer
242 views

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 $\...
M Rezaei's user avatar
4 votes
2 answers
822 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: ...
LiamB's user avatar
  • 143
2 votes
1 answer
210 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 ...
Travis Reed's user avatar
2 votes
1 answer
266 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 ...
Mr Puh's user avatar
  • 101
4 votes
2 answers
1k 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 $P^2 = P$.
U2647's user avatar
  • 593
1 vote
1 answer
242 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 ...
U2647's user avatar
  • 593
1 vote
0 answers
18 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 ....
ABIM's user avatar
  • 6,759
1 vote
1 answer
73 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 ...
Kai's user avatar
  • 373
1 vote
1 answer
1k 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 ...
olha's user avatar
  • 121