Projection

COS 350 - Computer Graphics

perspective projection in drawing

[ Marschner 2004 - original unknown ]

perspective projection in drawing

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 projection in drawing

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.

[ according to Google Translate ]

perspective projection in drawing

[ 1320–1325, Giotto [Web Gallery of Art, www.wgu.hu] ]

perspective projection in drawing

[ 1425–1428, Masaccio [Web Gallery of Art, www.wgu.hu] ]

perspective projection in photography

[ Marschner 2004 - original unknown ]

perspective projection in photography

Richard Zakia

raytracing vs. projection

ray tracing: image plane to object point


projection: object point to image plane


inverse process

viewing

maps 3D world points to 2D image plane positions


two stages:

viewing transform

any affine transform


useful to define one for our viewer model


computed by

projection

in general, function that transforms points from \(m\)-space to \(n\)-space, where \(m > n\)


in graphics, maps 3D points to 2D image coordinates

canonical view volume

the result of a projection


\((x,y)\) are image plane coordinates in \([-1,1] \times [-1,1]\)


keep around the \(z\) normalized in \([-1,1]\)

canonical view volume

why introduce near/far clipping planes?

taxonomy of projections

orthographic projection

box view volume

orthographic projection

viewing rays are parallel

orthographic projection

center around the \(\z\) axis

\[\mat{x' \\ y' \\ z'} = \mat{x/r \\ y/t \\ (2z - n - f)/(n-f)}\]

orthographic projection

in matrix form

\[O = \mat{1/r & 0 & 0 & 0 \\ 0 & 1/t & 0 & 0 \\ 0 & 0 & 2/(n-f) & -(n+f)/(n-f) \\ 0 & 0 & 0 & 1}\]

perspective projection

truncated pyramid view volume

perspective projection

viewing rays converge to a point

perspective projection

center around the \(\z\) axis

\[\mat{x' \\ y' \\ z'} = \mat{(nx)/(rz) \\ (ny)/(tz) \\ \cdots}\]

perspective projection

in matrix form

\[P = \mat{n/r & 0 & 0 & 0 \\ 0 & n/t & 0 & 0 \\ 0 & 0 & (f+n)/(n-f) & -2nf/(n-f) \\ 0 & 0 & 1 & 0}\]

projection matrices

projection matrices

general orthographic

\[O = \mat{ \frac{2}{r-l} & 0 & 0 & \frac{l+r}{l-r} \\ 0 & \frac{2}{t-b} & 0 & \frac{b+t}{b-t} \\ 0 & 0 & \frac{2}{n-f} & \frac{n+f}{n-f} \\ 0 & 0 & 0 & 1 }\]

general perspective

\[P = \mat{ \frac{2n}{r-l} & 0 & \frac{l+r}{l-r} & 0 \\ 0 & \frac{2n}{t-b} & \frac{b+t}{b-t} & 0 \\ 0 & 0 & \frac{f+n}{n-f} & \frac{-2nf}{n-f} \\ 0 & 0 & 1 & 0 }\]

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