Projection along an axis

Question

I have problem with understanding what projection along an axis means in practise. For example I have two paraboloids $H=\{(x,y,z)\in \Bbb{R}^3:z=2xy \} $ and $E=\{(x,y,z)\in \Bbb{R}^3:z=x^2 +y^2 \} $ and I would like to find a projection along $z$ axis: $g:H\to E$.

Answer 1

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Lines}{\mathcal{L}}$In the geometric sense of your question, projection (to an object $E$) refers to choosing a family $\Lines$ of lines (such as a family of parallels, or all lines through some point $p$) such that

• Every point of space (or, generally, the universe containing your geometric objects) lies on at least one line of the family $\Lines$;

• Aside from points forming a set $C$ of lower dimension (the center of projection), each point of space lies on at most one line of $\Lines$;

• Each point of $E$ lies on precisely one line of $\Lines$; particularly, $E \cap C = \varnothing$.

If $H$ is an object such that $H \cap C = \varnothing$, then projection along $\Lines$ is defined as follows: For each point $p$ of $H$, pick the (unique) line $\ell$ in $\Lines$ through $p$, and map $p$ to $\Pi(p) = \ell \cap E$, the (unique) point where $\ell$ hits $E$.

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"Projection along the $z$-axis" refers to taking $\Lines$ to be the set of lines in $\Reals^{3}$ parallel to the $z$-axis. For the surfaces you mention, projection $\Pi$ satisfies $$ \Pi(x, y, 2xy) = (x, y, x^{2} + y^{2}). $$