The representation by computer of 3 dimensional forms is normally restricted to the projection onto a plane, namely the 2 dimensional computer screen or hardcopy device. The following is a procedure that transforms points in 3 dimensional space to screen coordinates given a particular coordinate system, camera and projection plane models. This discussion describes the mathematics required for a perspective projection including clipping to the projection pyramid with a front and back cutting plane. It assumes the projection plane to be perpendicular to the view direction vector and thus it does not allow for oblique projections.
Included in the appendices is source code (written in the C programming language) implementing all the processes described.
Coordinate system
In what follows a so called right handed coordinate system is used, it has the
positive x axis to the right, the positive z axis upward, and the positive y
axis forward (into the screen or page).
Conversion between this and other coordinate systems simply involves the swapping and/or negation of the appropriate coordinates..
Camera model
The camera is fundamentally defined by its position (from), a point along the
positive view direction vector (to), a vector defining "up" (up), and a
horizontal and vertical aperture (angleh, anglev).
These parameters are illustrated in the following figure.
One obvious restriction is that the view direction must not be collinear with the up vector. In practical implementations, including the one given in the appendices, the up vector need not be a unit vector.
Other somewhat artificial variables in the camera model used here are front and
back clipping planes, a perspective/oblique projection flag, and a
multiplicative zoom factor. The clipping planes are defined as positive
distances along the view direction vector, in other words they are
perpendicular to the view direction vector. As expected all geometry before the
front plane and beyond the back plane is not visible. All geometry which
crosses these planes is clipped to the appropriate plane. Thus geometry visible
to a camera as described here lies within a truncated pyramid.
Screen model
The projection plane (computer screen or hardcopy device) can be defined in
many ways. Here the central point, width and height are used. The following
will further assume the unfortunate convention, common in computer graphics
practice, that the positive vertical axis is downward. The coordinates of the
projection space will be referred to as (h,v).
Note that normally in computer windowing systems the window area is defined as
an rectangle between two points (left,top) and (right,bottom). Transforming
this description into the definition used here is trivial, namely
horizontal center = (left + right) / 2
vertical center = (top + bottom) / 2
width = right - left
height = bottom - top
The units need not be specified although they are generally pixel's, it is assumed that there are drawing routines in the same units. It is also assumed that the computer screen has a 1:1 aspect ratio, a least as far as the drawing routines are concerned
A relationship could be made between the ratio of the horizontal and vertical camera aperture and the horizontal and vertical ratio of the display area. Here it will be assumed that the display area (eg: window) has the same proportions as the ratio of the camera aperture. In practice this simply means that when the camera aperture is modified, the window size is also modified so as to retain the correct proportions.
Algorithm
The procedure for determining where a 3D point in world coordinates would
appear on the screen is as follows:
Transforming a line segment involves determining which piece, if any, of the
line segment intersects the view volume. The logic is shown below.
Clipping
Two separate clipping processes occur. The first is clipping to the front and
back clipping planes and is done after transforming to eye coordinates. The
second is clipping to the view pyramid and is performed after transforming to
normalised coordinates at which point it is necessary to clip 2D line segments
to a square centered at the origin of length and height of 2.
Source code
transform.c,
transform.h.