Parallel projections catagorize one of two major subclasses of planar geometric projections. Projections within this subclass have two characteristics in common. The first characteristic concerns the placement of the center of projection (PRP) which represents the camera or viewing position. In a parallel projection, the camera is located at an infinite distance from the viewplane (see Figure 3). By placing the camera at an infinite distance from the viewplane, projectors to the viewplane become parallel (the second characteristic of a parallel projection) in result forming a parallelepiped view volume. Only objects within the view volume are projected to the viewplane. Figure 3 shows the projection of line AB to the viewplane. In this case, the measurement of line AB is maintained in the projected line A'B'. While the measurements of an object are not preserved in all parallel projections, the parallel nature of projectors maintains the proportion of an object along a major axis. Therefore, parallel projections are useful in applications requiring the relative proportions of an object to be maintained.
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| Figure 3 Parallel projection defined by the Center of Projection (PRP) placed at an infinite distance from the viewplane. |
Orthographic projections are one of two projection types derived by subdivision of the parallel projection subclass. In addition to being parallel, projectors in an orthographic projection (shown in Figure 4) are also perpendicular to the viewplane (Hearn & Baker, 1996). Orthographic projections are further catorgorized as either multiview or axonometric projections, which are described below.
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| Figure 4 Direction of projectors for an orthographic projection. |
A multiview projection displays a single face of a three-dimensional object. Common choices for viewing a object in two dimensions include the front, side, and top or planar view. The viewplane normal differs in the world coordinate system axis it is placed along for each of the multiview projections. In the top view, the viewplane normal is parallel with the positive y-axis in a right-handed system. Figure 5a illustrates a top or planar view of the three-dimensional building shown in Figure 5b. To project the top view of the 3-D object, the y-coordinates are discarded and the x- and z-coordinates for each point are mapped to the viewplane. By repositioning the viewplane normal to the positive z-axis and selecting the x-, and y-coordinates for each point, a front view is projected to the viewplane (Figure 5c). Likewise, a side view (Figure 5d) results when the viewplane normal is directed along the positive x-axis and the y- and z-coordinates of a three-dimensional object are projected to the viewplane. These projections are often used in engineering and architectural drawings (Hearn & Baker, 1996). While they do not show the three-dimensional aspects of an object, multiview projections are useful because the angles and dimensions of the object are maintained.
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| Figure 5 Two and three dimensional projections; a) top or
planar view; b) three-dimensional view for reference; c) front view; and d) side view. |
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Unlike multiview projections, axonometric projections allow
the user to place the viewplane normal in any direction such that
three adjacent faces of a "cubelike" object are visible.
To avoid duplication of views displayed by multiview projections,
the viewplane normal for an axonometric view is usually not placed
parallel with a major axis (Hill, 1990). The increased versatility
in the direction of the viewplane normal positions the viewplane
such that it intersects at least two of the major axes. Lines
of a three-dimensional object that are parallel in the world coordinate
system are likewise projected to the viewplane as parallel lines.
In addition, the length of a line, or line preservation, is maintained
for lines parallel to the viewplane. Other receding lines maintain
only their proportion and are foreshortened equally with lines
along the same axes.
Axonometric projections are further divided into three classes
that depend upon the number of major axes which are foreshortened
equally (Hill, 1990). These axonometric views are defined as isometric,
dimetric, or trimetric projections.
An isometric projection is a commonly used axonometric projection (Foley et al., 1996; Hearn & Baker, 1996). In this view, all three of the major axes are foreshortened equally since the viewplane normal makes equal angles with the principal axes. To satisfy this condition, the viewplane normal n = (nx, ny, nz) has the requirement that |nx| = |ny| = |nz|. This limitation restricts n to only eight directions (Foley et al., 1996). Figure 6 shows an isometric projection of a cube. Isometric projections scale lines equally along each axis, which is often useful since lines along the principal axes can be measured and converted using the same scale.
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| Figure 6 Isometric Projection |
Dimetric projections differ from isometric projections in the direction of the viewplane normal. In this class of projections, n = (nx, ny, nz) is set so that it makes equal angles with two of the axes. Valid settings for a dimetric projection allow nx = |ny|, nx = |nz|, or ny = |nz| (Hill, 1990). In this class, only lines drawn along the two equally foreshortened axes are scaled by the same factor. Figure 7 shows a dimetric projection of a cube. When the viewplane normal is set so that the viewplane is parallel to a major axis, line measurements are maintained in the projection for lines which are parallel to this axis.
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| Figure 7 Dimetric Projection |
Trimetric projections, the third subclass of axonometric projections, allow the viewer the most freedom in selecting the components of n (Hill, 1990). In this class, the viewplane normal makes differing angles with each major axis since no two components of n have the same value. As with a dimetric view, a trimetric view displays different orientations by placing differing amounts of emphasis on the faces. Trimetric projections have a potential disadvantage in that measurement of lines along the axes is difficult because of a difference in scaling factors. Figure 8, a trimetric view of a cube, shows how this unequal-foreshortening characteristic affects line measurements along different axes. While disadvantageous in maintaining measurements, a trimetric projection, with the correct orientation, can offer a realistic and natural view of an object (Hill, 1990).
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| Figure 8 Trimetric Projection |
Oblique projections represent the second category of parallel projections. Oblique views are useful since they combine the advantageous qualities of both multiview and axonometric projections. Like an axonometric view, this class presents an object's 3D appearance. Similar to a multiview projection, oblique views display the exact shape of one face (Hill, 1990). As in an orthographic view, this class of projections uses parallel projectors but the angle between the projectors and the viewplane is no longer orthogonal. Figure 9 shows an example of the direction of the projectors in relation to the viewplane.
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| Figure 9 Direction of projectors for an oblique projection. |
Oblique projections are further defined as either cavalier or cabinet projections. Figure 10 shows the projection of a point (x, y, z) to the point (xp, yp) onto the viewplane. Cavalier and cabinet projections differ by the value used for the angle alpha. Angle alpha is defined as the angle between the oblique projection line from (x, y, z) to (xp, yp) and the line on the viewplane from (x, y) to (xp, yp) (see Figure 10). Two commonly used values for alpha = 45°, and alpha = 63.4°.
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| Figure 10 Conversion of a world coordinate point (x, y,z) to the position (xp,yp) on the viewplane for an oblique projection. |
When alpha = 45° as in Figure 11, the projection is a cavalier projection. In this projection, a cube will be displayed with all sides maintaining equal lengths (see Figure 11). This property is often advantageous since edges can be measured directly. However, the cavalier projection can make an object look too elongated.
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| Figure 11 Cavalier Projection |
In the second case, when alpha = 63.4°, the projection is labeled as a cabinet projection. For this angle, lines perpendicular to the viewplane are displayed one-half the actual length (see Figure 12). Because of the reduction of length in lines perpendicular to the viewplane, cabinet projections appear more realistic than cavalier projections (Hearn & Baker, 1996).
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| Figure 12 Cabinet Projection |
