Studies on the Axonometric axes in Parallel Projection

Parallel projection is classified into orthogonal projection and oblique projection. Therefore, the orthogonal projection and oblique projection must be considered as a whole. In this paper, the author shows a general formula in which the distortion factors belonging to axonometric axes and the angles between any two axonometric axes in an oblique projection are expressed by the direction cosines of the perpendicular of the image plane and the direction cosines of the projecting line. This formula is applicable to the well-known formula in the orthogonal projection as a particular case.Transformation of an image in parallel projections should be considered as a rotation of the image plane around one of the axes of axonometry. The author also shows a general formula of the axonometric axes which are projected onto any one of the planes holding an axis in common with the original plane. In order to clarify the distortion of an oblique projection from the orthogonal projection, the above-mentioned direction cosines, distortion factors and angles, expressed as the function of the rotation angle of the image plane, are described through the medium of the example on the isometric projection.