Perspective Affinity

Affine transformation or affinity is a special case of plane perspective collineation where the center is a point at infinity.

For affinity the following properties are valid, as well as the basic properties of the perspective collineation:

  • All affine rays are parallel.
    Line at infinity is also an affine ray.

  • Pairs of corresponding points lie on an affine ray.

  • Pairs of corresponding lines intersect at a point on the affine axis
    A line parallel to the affine axis is mapped onto a line also parallel to the axis.

  • Every affinity is uniquely determined by its axis o and a pair of corresponding points A1, A2.
    The affine ray is determined with the corresponding pair of points, hence the center of affinity is also determined.
    This way of setting an affinity will be denoted by (o, A1, A2).
    For the defining pair we can choose any pair of corresponding points with respect to the given affinity.

    Affine image of a point

    An affinity (o, A1, A2) and point B1 are given in Figure 25. Construct the point B2 i.e. the affine image of the point B1.
    (Although the ray zA determines the center of affinity in this figure the center is pointed out with an arrow. In the following this will not be specially pointed out.)

    Figure 25

    1st step: The ray zB of the point B1 is constructed so that it is parallel to the ray zA.

    2nd step: The affine image p2 of the line p1=A1B1 is constructed.

    3rd step: The point B2 is the intersection of the ray zB and line p2.
    - Move the point B1 and notice that with this construction, an affine image for any point of the plane can be found.
    - Change the chosen affinity by moving the point A1 or A2. Notice that the construction is valid for any affinity.

    - By moving the red point change the direction of the affine ray, i.e. the position of the center of affinity. Notice that the construction is valid for any direction of the affine ray.

    Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb, made with GeoGebra