The "Paper Band" Method

Key-words: Central projection, the Diagonal Method, Extreme Oblique Line, Framing Shape, Halving Plane, Hexagon-Outline, the Object Picture Plane, Oblique Lines of In-Between, Orthogonal Picture, Preliminary Proportion of Lateral Planes, Principal Visual Ray, Proportion, Structural Web of Basic Lines, the Vanishing Point, View-Point, Visual Ray.

Summary

     The most sensitive part of assignments dealing with perspective reconstruction through orthogonal pictures is the sketching by freehand with a chosen view-point the starting cube. We need the first cube in order to create the structural grid of basic lines. Our first cube should be an extreme cube of which steeply converging edges do relate to remote Vanishing Points being frequently off paper size. Through the Diagonal Method the cubic grid might be afterward enlarged.
     Sketching our starting cube by heart can powerfully be supported by preliminary proportion of lateral planes taken from related orthogonal pictures. Distances between vertical edges might be metered from the top plan rotated into principal visual ray direction, reading data arising on the Object Picture Plane. We may take over the distortion of horizontal planes from the Halving Plane.
     While bringing into cover in our perspective sketch these two projective stripes belonging to the cube's edge in question, the extreme oblique line relating toward the Vanishing Point will soon become distinct.
     Through fixing our view-point we may preliminarily decide the visibility proportion between lateral planes of the cube. Choosing the picture distance the true value of "sight-from-below" will be determined. In order to evaluate the position of the extreme oblique edge we exploit these two previous decisions.

1. Terraced towers of cubes placed on a quadratic grid      Let's prepare a perspective view of the volume confirmed here by its orthogonal pictures!      Let us determine distortions belonging to the most remote square plane by using of proportion data taken from the plan (rotated into view-point direction) and from the halving plane which emanate the ratio of the "sight-from-below"!

3. Model-drawings help us to choose favorable view-points. During the preparatory stage let's create several volume sketches from every direction as well, as seen out of birth-eye level!

6. The "Halving Plane" derives from the plan turned into view-point-position.      We shall read extreme "sights-from-below" relations by examining the "Halving Plane". The Halving Plane is a special side view coordinated to our chosen view-point at the beginning. We imagine its position being perpendicular both to the ground plan and to the object's picture plane as well. Here in our example, within this special side view all vertical edges of the cube appear one by one. With the help of a paper band we may copy over their precise orthogonal position from the plan (turned into view-point position) to the Halving Plane.

7. Sections X

     The preliminary choice of lateral sides' sections follows also in the opposite direction. We revolve the square (inserted by its nearest corner to the object picture plane) on the plane as far as both visual rays launched from the view-point (NP) toward extreme corners (B, J) will dissect (within the object picture plane) the desired lateral sides' ratio.

8. Evaluating the stripe of the "big Y" (Y/S) We connect our view-point (NP) with the highest corner of the tower-edge (Smax) as well, as with the left corner of the top square (B). We understand the virtual height of this stripe within the line of the object picture plane which, in case of a remote view-point, doesn't differ much from the vertical. (The value of this section here is about 2/3 S.) Now the near vertical edge of our perspective cube under construction might be dissected into five parts. At the upper two-fifths we may mark the lower end of the stripe called "big Y".

9. Orthogonal stripes' overlapping zones will deliver the direction of the extreme oblique line. Next we may guess the position of the other (remote) vanishing point.

10. Position of corners - an "opposite ratio" If horizontal distances between vertical edges are here (as initially chosen): x / X = 1/4 then y / Y = 1/4, for this reason the virtual pending of the right corner is about a fourth of the left one. (small y = 1/4 big Y).

11. Extension of the structural grid Before extending the structural grid by the help of the "Diagonal Method", we ought to check from every direction our extreme cube's credibility. During this proceeding we should to a greater extent observe backside squares which reveal eventual errors sooner than frontal ones.

13. Enveloped into a big cube      The final figure shows the same task approached in a different way. Some times the volume waiting to be depicted needs a single enveloping cube of enormous size only.      Even in this case we accomplish our starting construction within the big cube's cover plane - that means at the largest distortion to be expected. Later on, by consequently applying of diagonals, going through mid-sections of border planes, we shall finally reach the grid's appropriate joints of in-between.