A single orthographic projection fails to illustrate the general three-dimensional shape of an object. Axonometric projections overcome this limitation. An axonometric projection is constructed by manipulating the object using rotations and translations, such that at least three adjacent faces are shown. The result is then projected from the center of projection at infinity on to one of the coordinate plane unless a face is parallel to the plane of projection, an axonometric projection does not show its true shape. However, the relative lengths of originally parallel lines remain constant, i.e., parallel lines are equally foreshortened.
Foreshortening factor-it is the ratio of the projected length of a line to its true length
Types of axonometric projections -
Trimetric projection is the least restrictive and isometric projection is the most restrictive
Trimetric Projection - A trimetric projection is formed by arbitrary rotations in arbitrary order, about any or all of the coordinate axes, followed by parallel projection on to the z=0 plane. The wide variety of trimetric projections precludes giving a general equation for these ratios
For any specific trimetric projection, the foreshortening ratios are obtained by applying transformation matrix to the unit vector along the principal axis specifically,
where [U] is the matrix of unit vectors along the untransformed x, y and z axes respectively, and [T] is the concatenated trimetric projection matrix. The foreshortening factors along the projected principal axes are then