The line tfracx-12=tfracy-23=tfrac z1 is projected onto xOy…
The line $\tfrac{x-1}2=\tfrac{y-2}3=\tfrac z1$ is projected onto $xOy$ from $P(5,5,2)$. Find the projection.
Explanation of Method
The central projection of a line onto the xOy plane from the point P(5, 5, 2).
Step 1: Solution
All projecting rays lie in the plane $\sigma$ through $P$ and $\ell$: \[\vec n=\{2,3,1\}\times\overrightarrow{AP}=\{2,3,1\}\times\{4,3,2\}=\{3,0,-6\}\parallel\{1,0,-2\},\qquad \sigma:\ x-2z-1=0 .\]
\[\sigma\cap xOy:\qquad x=1,\quad z=0\]
Check: the points $(3,5,1)$ and $(-1,-1,-1)$ of $\ell$ project to $(1,5,0)$ and $(1,1,0)$.
Remark: the point of $\ell$ at height $z=2$ (the height of $P$) has no image, since its ray is parallel to $xOy$. Conversely, the point $(1,-1,0)$, where the ray from $P$ parallel to $\ell$ meets $xOy$, is the vanishing point and is not the image of any point of $\ell$. So the image is the line $x=1$ without the point $(1,-1,0)$.
Result
\[\boxed{\,\sigma\cap xOy:\qquad x=1,\quad z=0\,}\]