Here we analyze and illustrate the transformation effect of the maps.
Let us begin by analyzing how \(\PrjctnOnto{1}\) transforms \(\RNrSpc{2}\):
\[\PrjctnOntoOfCoord{1}{(x,y)} = x\]
So, the first coordinate of \((x,y)\) is retained, and the second coordinate is ‘forgotten’. Therefore all points of \(\RNrSpc{2}\) which share the same first coordinate get transformed to the same number in \(\RNrSpc{2}\):
All points with first coordinate \(a\) get transformed into the number \(a\).
For example, to determine how \(\PrjctnOnto{1}\) transforms an object \(T\) located above the \(x\)-axis, imagine parallel light rays shining vertically onto the \(x\)-axis. Then \(\PrjctnOnto{1}\) transforms the object into its shadow on the \(x\)-axis.
Now let us turn to analyzing how \(\PrjctnOnto{2}\) transforms \(\RNrSpc{2}\):
\[\PrjctnOntoOfCoord{2}{(x,y)} = y\]
So, the first coordinate of \((x,y)\) is ‘forgotten’, and the second coordinate is ‘retained’. Therefore all points of \(\RNrSpc{2}\) which share the same second coordinate get transformed into the same number in \(\RNrSpc{2}\):
All points with second coordinate \(b\) get transformed into the number \(b\).
For example, to determine how \(\PrjctnOnto{2}\) transforms an object \(T\) located to the right of the \(y\)-axis, imagine parallel light rays shining horizontally onto the \(y\)-axis. Then \(\PrjctnOnto{2}\) transforms the object into its shadow on the \(y\)-axis.
