Affinity and Perspective Geometry

A mathematical theory for perspective (descriptive) and affine geometry has been developed. Theorems of affine geometry are identical to theorems of Euclidean geometry. And in both theories parallelism is important concept. In the perspective geometry initially parallel lines in general are not parallel.

An affine transformation is a combination of linear transformations, such as rotation and translation. For an affine transformation in the last column of the generalized 4×4 matrix is [0 0 0 1]. Otherwise, the transformed homogeneous coordinate h is not equal to 1 and is not correspondence between affine transformation and (4×4) matrix operator. Affine transformations form a useful subset of bilinear transformation, since the product of two affine transformations is also affine. This feature allows us to combine the total transformation of the set of points in an arbitrary coordinate system while keeping the value 1 for the homogeneous coordinates h.

Since Euclidean geometry has been taught in schools for many years, the methods of painting and drawing, based on Euclidean geometry, have become standard methods of graphic communication. When homogeneous coordinates used to specify the object, affine and perspective transformations are computed with equal ease.

Affine and perspective transformations are three-dimensional, i.e. are transformations from one three-dimensional space to another three-dimensional space. However, the observation results on two-dimensional surface projection of the required three-dimensional space. The result of this projection is called a flat geometric projection. Matrix projection of three-dimensional space always contains a column of zeros hence the determinant of this transformation is always zero.

Planar geometric projections of the objects form the intersection of straight lines, called projectors, with a plane called the plane of projection. Projectors are straight lines passing through an arbitrary point (called the center of projection) and each point of the object. If the projection center is located at the endpoint of a three-dimensional space we have perspective projection. If the center is located at infinity, then all the projectors are parallel and the result is a parallel projection. Planar geometric projections form the basis of descriptive geometry. Non-planar and non-geometrical projections are also useful and are widely used in cartography.

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