Like parallel projections, perspective projections define a major subclass of planar geometric projections. Divisons within perspective projections are consistent in that the center of projection (PRP) is placed at a finite distance from the viewplane. Because of this finite distance between the camera and the viewplane, projectors are no longer parallel. By placing the camera near the viewplane, as shown for the perspective projection in Figure 13, projectors from the PRP to the edges of the projection window, located on the u-, v-plane, define a pyramidal view volume. As shown in Figure 13, the projectors from the center of projection to line AB form a much shorter line A'B' in the viewplane. The reduction in length of the projected line is attributed to the decreasing distance between the two projectors as the viewing surface becomes nearer to the center of projection.
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| Figure 13 Pyramidal view volume for a perspective projection. |
In comparison to parallel projections, perspective projections often provide a more natural and realistic view of a three-dimensional object. By comparing the viewplane of a perspective projection with the view seen from the lens of a camera, the underlying principals of a perspective projection can be easily understood. Like the view from a camera, lines in a perspective projection not parallel to the viewplane converge to a distant point (called a vanishing point) in the background. When the eye or camera position is close to the object, perspective foreshortening occurs with distant objects appearing smaller in the viewplane than closer objects of the same size.
Perspective projections are typically separated into three classes: one-point, two-point, and three-point projections. Each class differs in the orientation of the viewplane and the number of vanishing points the unit cube has.
In a one-point perspective, lines of a three-dimensional object along a major axis converge to a single vanishing point while lines parallel to the other axes remain horizontal or vertical in the viewplane (Hill, 1990). To create a one-point perspective view, the viewplane is set parallel to one of the principal planes in the world coordinate system. The viewplane normal is set parallel to a major axis and the viewplane noraml vector n is initialized such that two of its three components are zero (Hill, 1990). Figure 14 shows a one-point perspective view of a cube. In this projection, the viewplane is positioned in front of the cube and parallel to the x- and y-plane.
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| Figure 14 One-point perspective projection. |
A two-point perspective projects an object to the viewplane
such that lines parallel to two of the major axes converge into
two separate vanishing points. To create a two-point perspective,
the viewplane is set parallel to a principal axis rather than
a plane (Hill, 1990). In satisfying this condition, the viewplane
normal vector n should be set perpendicular to one of the
major world coordinate system axes. In this case, two of the components
of n = (nx, ny,
nz) are nonzero, while the third is zero
(Hill, 1990). Figure 15 shows a two-point perspective view of
a cube. In this figure, lines parallel to the x-axis converge
to vanishing point VP1 while lines parallel to the z-axis converge
to vanishing point VP2. Two-point perspective views often provide
additional realism in comparison to other projection types; therefore,
they are commonly used in architectural, engineering, industrial
design, and in advertising drawings (Foley et al., 1996).
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| Figure 15 Two-point perspective projection. |
A three-point perspective has three vanishing points (Hill,
1990). In this case, the viewplane is not parallel to any of the
major axes. To position the viewplane, each component of the viewplane
normal is set to a non-zero value so that the viewplane intersects
the three major axes (Hill, 1990). Vanishing points are often
used by artists for highlighting features or increasing dramatic
effects (Hill, 1990). However, many disagree as to the extent
of their utility. Artistic works such as the beguiling lithograph
Ascending and Descending by M. C. Escher (shown in Computer
Graphics by Hill, 1990) provides a creative example where
a three-point perspective increases the sense of scope. On the
contrary, Foley et al. (1996) note that three-point perspectives
are used infrequently since they offer little additional realism
to two-point perspective projections.