# perspective

Figure 2.

Two figures or solid bodies that arise from one another by a perspective transformation are said to be in perspective with respect to one another. The perspective image of a circle can, for example, be a parabola, and the parabola and the circle are then in perspective with respect to one another.

## Perspective transformation

### Plane perspective transformation

In a perspective transformation in a plane, the points of two different straight lines are associated with one another in the following way:

If *l* and *m* are given straight lines and *Z* is a point of the plane which lies neither on *l* or *m*, then a point *L* of *l* is associated with the point of intersection, *M*, of *ZL* with *M* (*M* is the image-point of *L*). *Z* is called the **perspective center** or the **center of perspectivity** (see Figure 1). Thus in a perspective, the image-points on *m* arise out of the points of *l* by central projection.

Figure 1. |

Such a perspective of the points of two straight lines possesses a fixed point (a point whose image coincides with the point itself), namely the point of intersection *P* of the two lines.

## Example of perspective transformation in space

If *E*_{1} and *E*_{2} are two different planes in space, and *Z* is a point outside both *E*_{1} and *E*_{2}, a perspective can be defined in the following way.

If *P*_{1} is a point of *E*_{1} the point of intersection *P*_{2} of the line *ZP*_{1} with the plane *E*_{2} is associated with *P*_{1} as its image-point. The points of the line of intersection *l* of *E*_{1} and *E*_{2} are fixed points of this transformation (see Figure 2).