To complete the transformation of a point to the view reference coordinate system, oblique and perspective projections require additional transformations to those previously described in the Converting to the View Reference Coordinate System section. Methods described in this section have been modified from the procedures described by (Hearn & Baker, 1996).
Oblique Projections
To finalize the transformation of a point to the view reference
coordinate system, oblique projections require the following additional
calculations
where (u, v, n) are the coordinates in the view reference coordinate system for a point determined by the methods described in Converting to the View Reference Coordinate System, and (u´, v´) are the coordinates which are mapped to the viewplane. Theta represents the angle between the horizontal direction in the viewplane and the line from (u, v) to (u´, v´) and alpha is defined as the angle between the oblique projection line from (u, v, n) to (u´, v´) and the line on the viewplane from (u, v) to (u´, v´) (see Figure 16). The length of the line between (u, v) to (u´, v´) can be calculated as (n / tan alpha). The transformation of a point from the world coordinate system to the view reference coordinate system is now complete for the oblique projection type.
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| Figure 16 Transformation of a point to the view reference coordinate system for an oblique projection. |
Perspective Projections
Using a procedure described by Hearn & Baker, (1996), the
projection of a point currently defined in the view reference
coordinate system requires two additional steps. The first step
includes calculations for shearing the pyramidal shaped view volume
(see Figure 17a for the initial location of the view volume).
When the eye or camera is not positioned on the n-axis, this transformation
shifts all positions that lie along the frustum centerline (defined
as a line from the eye position that intersects the view volume)
to a line perpendicular to the viewplane (see Figure 17b ).
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a.  |
b.  |
| Figure 17 The view volume for a perpsective projection with the eye positioned off the n-axis a) before the shearing transformation; and b) after the shearing transformation. |
Shearing parameters are defined as
where su represents shearing within the u-axis and sv represents the shearing effect within the v-axis. The location of the projection window shown in Figures ?? and ?? are defined by xmin, xmax, ymin, and ymax. The remaining values in this equation represent the location of the eye in the view reference coordinate system and the location of the VRP along the z-axis of the world coordinate system. To complete the shearing operation, the following equations are used
where (u´, v´, n´) represents a point in the sheared frustum and (u, v, n) represent a point in the original view volume.
The second step applies a scaling transformation so that the pyramidal view volume can be converted to a parallelepiped view volume. This formulas for completing the transformation of a point to the view reference coordinate system are
where (u´´, v´´) is the projected location of a point on the viewplane. By using this procedure, the eye can be positioned off of the n-axis in the view reference coordinate system. With the exception of selecting a position located on the viewplane or a position between the front and back faces of the object, the eye position can be located at any position in the viewing system.
References for Further Reading
Foley, J. D., van Dam, A., Feiner, S. K., and Hughes, J. F. (1996). Computer Graphics Principles and Practice (Second Edition in C). Addison-Wesley, Reading, Massachusetts.
Hearn, D. and Baker, M. P. (1996). Computer Graphics C Version (Second Edition). Prentice Hall, Upper Saddle River, New Jersey.
Hill Jr., F. S. (1990). Computer Graphics. Prentice Hall, Englewood Cliffs, New Jersey.
Watt, A. H. (1993). 3D Computer Graphics (Second Edition). Addison-Wesley, Reading, Massachusetts.