Axonometry
Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coordinate axes play a crucial role. The result of an axonometric procedure is a uniformly-scaled parallel projection of the object. In general, the resulting parallel projection is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane), which in this context is called an orthogonal axonometry. In technical drawing and in architecture, axonometric perspective is a form of two-dimensional representation of three-dimensional objects whose goal is to preserve the impression of volume or relief. Sometimes also called rapid perspective or artificial perspective, it differs from conical perspective and does not represent what the eye actually sees: in particular parallel lines remain parallel and distant objects are not reduced in size. It can be considered a conical perspective conique whose center has been pushed out to infinity, i.e. very far from the object observed. The term axonometry is used both for the graphical procedure described below, as well as the image produced by this procedure. Axonometry should not be confused with axonometric projection, which in English literature usually refers to orthogonal axonometry. Principle of axonometry Pohlke's theorem is the basis for the following procedure to construct a scaled parallel projection of a three-dimensional object: Select projections of the coordinate axes, such that all three coordinate axes are not collapsed to a single point or line. Usually the z-axis is vertical. Select for these projections the foreshortenings, , and , where The projection of a point is determined in three sub-steps (the result is independent of the order of these sub-steps): starting at the point , move by the amount in the direction of , then move by the amount in the direction of , then move by the amount in the direction of and finally Mark the final position as point . In order to obtain undistorted results, select the projections of the axes and foreshortenings carefully (see below). In order to produce an orthographic projection, only the projections of the coordinate axes are freely selected; the foreshortenings are fixed (see :de:orthogonale Axonometrie). The choice of the images of the axes and the foreshortenings Notation: angle between -axis and -axis angle between -axis and -axis angle between -axis and -axis. The angles can be chosen so that The foreshortenings: Only for suitable choices of angles and foreshortenings does one get undistorted images. The next diagram shows the images of the unit cube for various angles and foreshortenings and gives some hints for how to make these personal choices. In order to keep the drawing simple, one should choose simple foreshortenings, for example or . If two foreshortenings are equal, the projection is called dimetric. If the three foreshortenings are equal, the projection is called isometric. If all foreshortenings are different,