busman's holiday for perspectivists

A holiday for some. With no competition from the day job, a student perspectivist can buckle down, locate all the furniture in the interior perspective, figure out the height of the windows, indicate the ins-and-outs of door jambs, soffits and bay windows, bring it all to class tonight and -- heck, shed some light on it, too.

going deep

In the world of flat geometry, a right isosceles triangle has equal sized legs and a 45º angle. One can divide a vertically-striped plane into a grid of equal squares using the 45º diagonal.This same plane's set of parallel lines, when laid down into the geometry of linear perspective, appear as rays that converge as they recede. Their convergence is governed by the central vanishing point C in this diagram based on Kevin Forseth's useful book on perspective: Since we're no longer in flat space, a diagonal line is at a virtual 45º. A protractor laid on the drawing will not measure 45º.
The rate at which equal intervals diminish in size as they recede is a affected by how far the measured field/back wall/picture plane is from the observer. This process of representing space fixes the viewer's distance from the virtual scene. The distance between the viewer(SP) and the viewed object (in this case, the back wall) is related to the distance between C, the center of vision and the vpD, or diagonal vanishing point. A perspectivist can slide the back wall forward to make the viewer feel closer to it. 

Increasing the distance (on the HL) between the vpD and C, would flatten the virtual 45º diagonal, compress the receding intervals into a smaller space on the page, and give the impression that the scene is farther from the observer. Note that the size of the back wall is the same as in the drawing above. By moving the vpD away from the central VP, the "measured" distance, or depth, between that wall and the observer appears more compressed.

paraline drawings

Scaled orthographic drawings, such as plans, elevations and sections disclose the relative measure of only two dimensions at a time on a 2-d drawing surface. The viewer/user can put a ruler anywhere on the page to obtain information about the virtual object. Only 2 dimensions at a time can be related.
Machinists, cabinetmakers, architects and planners use paraline drawings to represent a third dimension on two-dimensional paper, while retaining the measured precision of 2-d orthographic plans, elevations and sections. Paraline drawings, while depicting objects "from above" or "from below," don't implicate any particular observer. Paraline drawings retain scalable lines useful to construction. Sets of lines that are parallel in the virtual object remain parallel in the 2-d paraline depiction.
Linear perspective drawings describe 3-d volumes and spatial relations on a 2-d surface as they appear to the fixed gaze of an observer --- the audience looking at the drawing. The observer is implicated in the virtual scenario.