perspective was not used until circa 15th century
technical explanation by Leon Battista Alberti
1436, De Pictura – Della Pittura
“Trovai adunque io questo modo ottimo cosi in tutte le cose seguendo quanto dissi, ponendo il punto centrico, traendo indi linee alle divisioni della giacente linea del quadrangolo.
”
perspective was not used until circa 15th century
technical explanation by Leon Battista Alberti
1436, De Pictura – Della Pittura
“Trovai adunque io questo modo ottimo cosi in tutte le cose seguendo quanto dissi, ponendo il punto centrico, traendo indi linee alle divisioni della giacente linea del quadrangolo.
Therefore, I found this so great as in all things according to what I said , putting the centric point , then drawing lines to the divisions of the line lying quadrangle.
”
ray tracing: image plane to object point
projection: object point to image plane
inverse process
maps 3D world points to 2D image plane positions
two stages:
any affine transform
useful to define one for our viewer model
computed by
in general, function that transforms points from \(m\)-space to \(n\)-space, where \(m > n\)
in graphics, maps 3D points to 2D image coordinates
the result of a projection
\((x,y)\) are image plane coordinates in \([-1,1] \times [-1,1]\)
keep around the \(z\) normalized in \([-1,1]\)
why introduce near/far clipping planes?
box view volume
viewing rays are parallel
center around the \(\z\) axis
\[\mat{x' \\ y' \\ z'} = \mat{x/r \\ y/t \\ (2z - n - f)/(n-f)}\]
in matrix form
\[O = \mat{1/r & 0 & 0 & 0 \\ 0 & 1/t & 0 & 0 \\ 0 & 0 & 2/(n-f) & -(n+f)/(n-f) \\ 0 & 0 & 0 & 1}\]
truncated pyramid view volume
viewing rays converge to a point
center around the \(\z\) axis
\[\mat{x' \\ y' \\ z'} = \mat{(nx)/(rz) \\ (ny)/(tz) \\ \cdots}\]
in matrix form
\[P = \mat{n/r & 0 & 0 & 0 \\ 0 & n/t & 0 & 0 \\ 0 & 0 & (f+n)/(n-f) & -2nf/(n-f) \\ 0 & 0 & 1 & 0}\]
\[O = \mat{ \frac{2}{r-l} & 0 & 0 & \frac{l+r}{l-r} \\ 0 & \frac{2}{t-b} & 0 & \frac{b+t}{b-t} \\ 0 & 0 & \frac{2}{n-f} & \frac{n+f}{n-f} \\ 0 & 0 & 0 & 1 }\]
\[P = \mat{ \frac{2n}{r-l} & 0 & \frac{l+r}{l-r} & 0 \\ 0 & \frac{2n}{t-b} & \frac{b+t}{b-t} & 0 \\ 0 & 0 & \frac{f+n}{n-f} & \frac{-2nf}{n-f} \\ 0 & 0 & 1 & 0 }\]