# Questions tagged [map-projections]

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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 ...
1answer
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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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### 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 ...
1answer
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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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1answer
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### 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....
1answer
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### 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 ...