PS2 Linux Programming
Viewing in Three Dimensions
Introduction
This tutorial will describe the background and main
techniques necessary to configure the matrices required for the viewing of
objects in three dimensions using the Perspective Transformation.
To compute the view of a 3D model, the 3D world
coordinates of the vertices in the model must be converted to 2D screen
coordinates, which determine which screen pixel represent each vertex. This
conversion depends on the position and orientation of the virtual camera in the
3D space and on the type of projection desired.
These notes will describe the steps required in
order to convert camera coordinates into corresponding screen coordinates.
During this conversion process, clipping of objects to the view frustum will
also be described.
Background
Figure 1 below shows the basic configuration for a
perspective viewing transformations.
Figure 1
First there is a viewing window or screen onto
which the 3D objects are projected. In order to limit the size of the
potentially infinite viewing volume two additional planes are introduced, the
near plane and the far plane. Only objects in the volume bounded by the near
and far planes and the other four sides of the view frustum will be projected
onto the screen. Objects outside the view frustum will be rejected.
The viewing volume must be defined relative to the
coordinate system (or axes) of the camera and this is illustrated in figure 2
below.
Figure 2
In Figure 2 the viewer or camera looks directly
along the negative z axis, the y axis points upwards and the x axis points to
the right. The z axis passes straight through the centre of the viewing volume
and the near and far planes.
A perspective projection viewing volume is defined
in Figure 3.
Figure 3
The near plane is situated n units from the camera
and the far plane f units from the camera. The viewing angle or field of view
(FOV) is defined as the angle alpha, which is an angle defined in the direction
of the y axis. The aspect ratio of the view frustum is defined as A = w/h.
Canonical Viewing Volume
In order to determine which objects are seen in the
viewing window, it is necessary to clip objects against the six planes that
form the view volume. Clipping against arbitrary 3D planes requires
considerable computation which in general can and should be avoided. For fast
clipping (which is supported by the hardware of the PS2), the viewing volume
can be transformed into a canonical viewing volume which is illustrated in
Figure 4.
Figure 4
This volume is a cube with side dimensions of 2
units. The cube extends between –1 and +1 in all of the three axes. Using the
canonical viewing volume it is a much simple matter to determine which objects
should be drawn or clipped out of the scene. Objects will only be drawn if
their vertices meet the following criteria: (–1 £
x £ 1); (–1 £ y £ 1); (–1 £ z £ 1). In general, the situation is more complex than
this and polygons or lines located partly inside and partly outside this volume
will need to be modified in a way such that only the part inside the canonical
view volume is kept. Such algorithms however are beyond the scope of this
present tutorial. Polygons that lie completely outside the canonical view
volume do not need further treatment and will be invisible to the viewer – they
will be rejected from the remainder of the drawing pipline. Polygons that lie
completely inside the canonical view volume can be drawn without requiring
further investigation.
Canonical Volume Perspective Projection
A 4x4 transformation matrix for homogeneous
coordinates is required which will map the truncated pyramid viewing frustum
described earlier into a standard cube in homogeneous coordinates such that:
(–1 £ x/w £ 1); (–1 £ y/w £ 1); (–1 £ z/w £ 1). First, the case
will be considered of a symmetrical perspective projection with a field of view
of 90 degrees and a square window with an aspect ration of unity. The centre of
projection is at the origin and the viewing window is located at z = -1 as illustrated
in figure 5.
Figure 5
Consider the following transformation:
Figure 6
The resulting points have affine coordinates (x/-z,
y/-z, -1) which is precisely the intersection of the line from the origin to
the point (x, y, z) with the plane z = -1. This transformation is sufficient if
there is no interest in the z coordinate since after division by w’ = -z the
result z = -1 is always obtained. Thus, due to the perspective division it is
no longer possible to determine the z value which is required for hidden
surface removal in the z-buffer.
However, it is still possible to construct a
projection matrix such that z is a monotonically increasing (non-linear)
function of the depth (-z) of the point in the range –1 to +1 which can be used
for visible surface determination and removal.
Z’ is determined by the coefficients in the third
row of the transformation matrix given in figure 6. A new transformation matrix
can be specified as shown in figure 7 which will have two additional parameters
(a and b) which will transform the z coordinate. Note that there is no need for
z’ to depend on x or y so the first two coefficients of the third row are zero.
Figure 7
Using the transformation in figure 7, the z
coordinate of an arbitrary point (x, y, z, 1) in eye coordinates is transformed
to the point:
It is required that the near clipping plane (n) is
mapped to 1 and the far clipping plane (f) is mapped to –1 which leads to the
two equations given below with the two unknowns a and b.
The solution is found to be:
Giving the following transformation matrix:
Figure 8
The z coordinate in the canonical volume which
results from the transformation given in figure 8 varies non-linearly with
increasing values of –z. This characteristic of the z buffer is illustrated in
figure 9 which shows the z buffer fill characteristics for near plane values of
0.1, 1 and 10 and a far plane value of 100.
Figure 9
It can be seen from figure 9 that changes in the z
position of a point close to the near plane result in large changes in the
transformed z coordinate, whilst changes in the z position of a point far from
the near plane result in only small changes in the transformed z coordinate.
Also, the closer the near plane is to the eye point the less uniformly the
z-buffer is filled. In practice, care must be taken in positioning the near
clipping plane: better z buffer performance can be obtained if the near plane
in placed as far as possible from the eye point.
A Generalised Canonical Volume Perspective
Projection
The vertical viewing angle alpha, as seen in figure
3 is also known as the Field of View (FOV). It is convenient to specify the
frustum by the vertical viewing angle (FOV) and the aspect ratio A = w/h. Both
of these parameters, FOV and A are independent of z. Now a symmetrical perspective
projection with a FOV not equal to 90 degrees and an non unity aspect ratio can
be reduced to the previous case by scaling x and y by the scaling factor:
The complete projection matrix is now:
Figure 10
Frustum Clipping
After points are transformed with the perspective
transformation matrix given in figure 10 it is then possible to perform
clipping to reject points from further processing. Any points which do not
satisfy all of the following conditions can be clipped and require no further
processing:
(–1 £ x’/w’ £ 1); (–1 £ y’/w’ £ 1); (–1 £ z’/w’ £ 1).
View Port Transformation
Figure 11
As discussed above, the projection transformation
matrix given in figure 10 transforms points onto a screen with dimensions of 2
units by 2 units as shown at the left hand side of figure 11. It is required
that the screen be mapped to a real screen with dimensions width and height and
offset (Xo, Yo) as illustrated. Also, it is necessary to expand the transformed
z coordinate to make use of the full range of the z buffer. In the PS2
configuration being used, there is a 24 bit z buffer which ranges from 0 to
16,777,215 (Zrange) with 0 being furthest away from the eye. Transforming from
the canonical view to the required view port can be accomplished with the
following transformation matrix.
This transformation will provide the correct screen
coordinates and z buffer value.
Example Program
The example program that accompanies this tutorial
is a simple illustration of setting up the view to screen matrix and the view
port transformation matrix. Following the view to screen transformation,
clipping against the canonical frustum is illustrated. A point in three
dimensional space is provided as input, and the program provided information on
whether this point will be drawn or clipped, the screen coordinate of the
transformed point and the corresponding z buffer value.
Conclusions
This tutorial has illustrated the use of the
perspective transformation to provide viewing of objects in three dimensions.
The z buffer value has been obtained to provide hidden surface removal and a
canonical transformation has been used to facilitate frustum clipping.
Dr Henry S Fortuna
University of Abertay Dundee