Perspective Transformations

Perspective transformation occurs when we have a non-zero value in any of the first three elements of the fourth column of the generalized (4×4) transformation matrix of homogeneous coordinates. As mentioned earlier, the perspective transformation is the transformation from one three-dimensional space to another one. In contrast with the parallel transformations discussed earlier, in this case parallel lines converge, the object size decreases with increasing distance from the center of projection, and there is non-uniform distortion of the object lines, depending on the orientation and distance from the object to the center of projection. All this helps our perception of depth, but does not keep the shape of the object.

One-point perspective transformation is given by:

(2.52)

Here h = rz + 1 <> 1. Normal coordinates are obtained by dividing by h:

(2.53)

Perspective projection onto a two dimensional plane can be obtained by combining the orthographic projection and perspective transformation. For example, the perspective projection on the plane z = 0 is produced by:

(2.54)

and

(2.55)

Ordinary coordinates are

(2.56)

To better understand the effect of perspective transformation, consider Fig. 2.5. It shows a perspective projection on the z = 0 plane of the segment AB, parallel to the axis z, to the segment A*B* at z = 0 plane with the center of projection located at the point -1/z on the axis z. The transformation can be divided into two stages. At the first stage the segment AB is transformed to the segment A'B'. Then we use the orthographic projection of the segment A'B' in three-dimensional space for transforming it to the segment A*B* on z=0 plane. Projection center is located at infinity.

Study of Fig. 2.5 shows that the lines A'B' and AB intersect the plane z=0 at the same point. Line A'B' also crosses the z axis at z = 1/r. Further, the perspective transformation maps transformed the infinity point of intersection of parallel lines AB and the axis z to the endpoint z = 1/r on the axis z. This point is called the vanishing point (see section 2.13). Note that the vanishing point lies at the same distance from the projection plane's center of projection only on the opposite side of the plane, for example, if z = 0 plane is a projection, and projection center is located at z = -1/r, then the vanishing point located at z = 1/r.

For completeness we mention the one-point perspective projection transformation with center and vanishing point is located on the axes x and y. One-point perspective transformation

(2.57)

with ordinary coordinates

(2.58)

has a center of projection [- 1/p 0 0 1 ] and the vanishing point located on the x axis [ 1/p 0 0 1 ].

Fig.2.5. Perspective projection of the segment AB on the plane z = 0

One-point perspective transformation

(2.59)

with ordinary coordinates

(2.60)

has a center of projection [ 0 - 1/q 0 1 ] and the vanishing point located on the x axis [ 01/q 0 1 ].

Fig.2.6. Example of the one-point perspective transformation of the unit cube

If in the fourth column (4×4) transformation matrix two elements of the first three are non-zero, then this transformation is called two-point perspective transformation. Two-point perspective transformation

(2.61)

with ordinary coordinates

(2.62)

has two centers of projection: the first on the x axis at the point [- 1/p 0 0 1 ] and the second on the y axis at [ 0 -1/q 0 1 ], and the two vanishing points: on the x axis at the point [ 1/p 0 0 1 ] and on the y axis at [ 0 1/q 0 1 ]. Note that the two-point conversion can be a union of two one-point one.

(2.63)

Three-point perspective is obtained if not equal to zero the first three elements of the fourth column (4×4) transformation matrix. This is a three-point perspective transformation:

(2.64)

with ordinary coordinates

(2.65)

There are three projection center: on the x axis at the point [ -1/p 0 0 1 ], on the y axis in the [ 0 -1/q 0 1 ] and the z axis [ 0 0 -1/r 1 ], as well as three vanishing points: on the x axis [ 1/p 0 0 1 ], on the y axis in [ 0 1/q 0 1 ] and the z axis [ 0 0 1/r 1 ].

Again, we note that the three-point perspective transformation can be obtained by the concatenation of three one-point perspective transformations, one for each coordinate axis.

Fig.2.7. Example of the two-point perspective transformation of the unit cube

Fig.2.8. Example of the tree-point perspective transformation of the unit cube

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