On perspective with 1, 2, and 3 vanishing points II, 11th of September 2013.

>> Last time I stopped at the perspective with two vanishing points. From >> the explanation of perspective with a single vanishing point, one can conclude that it is a somewhat "special" case, since the characteristic lines / directions present in the image are parallel and perpendicular to the line of sight. Therefore, the only noticeable vanishing point is directly "in front of" the viewer, and the other one is formally in infinity (the lines perpendicular to the line of sight). But, all other bundles of lines which make a certain angle to the line of sight have their vanishing points somewhere on the horizon if they are not inclining or declining with the depth, z-coordinate. The exact position of their vanishing point on the horizon depends on the angle they make with the line of sight.

Therefore, the perspective with two (characteristic) vanishing points is not really different from the perspective with a single vanishing point - it is simply the case when the noticeable, characteristic lines / directions are mutually perpendicular, but none of them is parallel to the line of sight. Therefore, each one of them has its vanishing point on the horizon. From the image, one can thus easily note the "presence" of two points of convergence of projected lines.

This type of projection is shown in the image above. The characteristic and mutually perpendicular lines discernible in the image are the lines along columns and rows. This time, both of these bundles of lines make a certain angle (which is not 0^o or 90 ^o) with the line of sight. One can easily see that the directions along columns (ping dashed lines) have their vanishing point which projects in the perspective projection window (denoted by F₁₂), for rows of characters (blue dashed lines) is situated far outside the projection window as shown in the lower part of the image above. This is often the case i illustration. In fact, most often, both vanishing points are situated outside the perspective (projection) window.

In perspective projection with a single vanishing point, the vanishing point is always in the perspective window, while in the case of the perspective projection with two vanishing points, neither point needs to be, and typically is not, in the perspective window.

This is a neat place to explain some effects of the perspective projection which need to be considered when illustrating. First, the height of the observer. From the image above one can conclude that the observers eyes are at the same height as are the waists of the characters - this is the height at which the horizon cuts the characters. We are therefore seeing a scene from a sitting position, or through the eyes of a child. What happens to the perspective projection when the height of the observer changes? This effect is shown in the image below.

The image (above) shows how the same scene looks like when viewed with the eyes of the observer whose height is approximately the same as the height of the characters in the scene. His view is straight i.e. the line of sight is at the certain height y, parallel to the ground. This time, the horizon cuts all of the characters approximately at their eye level which corresponds to the eye level of the observer.

The "law" which I describe states:

No matter where the characters of the same height are positioned, the horizon always cuts them at the same height. (e.g. all the characters at their waist level).

Someone may get the impression that this "law" somehow depends on the regularity of the character distribution so on the image below I show characters distributed in a completely random fashion (this time in the perspective with single vanishing point because the dominant and noticeable directions are this time defined by the square network that the characters stand on; the same, of course, holds regardless of the orientation of the network below the characters).

If the fact that all the characters in the scene are the same confuses you, in the image below I show the space populated with characters with two characteristic heights (children and women). Here we see that the horizon cuts the women at about the waist level, and the little girls at about the eye level. This is thus again the scene viewed by a child or a grown man sitting. This time, none of the characteristic directions is parallel to the line of sight thus we see the perspective projection with two vanishing points which can be read out only from the network that the characters stand on.

All that I explained thus far can be used to construct some simple scene in perspective projection, e.g. like the one shown in the image below. In that image, I kept the real lines of the scene, but also some of the construction lines. The main "catch" in this construction are the dashed construction lines which I used to construct the equal-width "modules" of the building. I will explain it in details in the next post. (Question: How tall is the observer of this scene?)

Yet, something in "not right" with this image. The building appears to be too "distorted" and it does not correspond to something we recognize in our visual memory. Why?

It is because both vanishing points are in the perspective window. But, why? And what does it really mean? Here we have to detour a little bit and return to mathematics of perspective projection, in particular to the meaning of parameter d, i.e. the distance of the observer from the perspective window.

Look in front of you. How big is the space that you can clearly and sharply see? It is within angle of view of about seventy degrees along the horizontal and approximately as much along the vertical - measure this angle by spreading your arms extended and seeking their position when they exit from the clear view. This is a "realistic" angle of view, which corresponds to the way we see nature (in the literature and on the internet one may find significantly larger angles, but they do not correspond to the angle of clear view but to the total angle of view which includes the field of peripheral vision). This angle is related to the distance of the observer from the perspective window, d, and to the size of the window, H (see below), as

tg (70^o / 2) ~ H / (2d),

i.e. the dimension of perspective projection is about 70 % of the distance of the observer from the perspective window,

H ~ 0.7 d

Let us now examine two perpendicular lines (1 and 2) which converge to two vanishing points. Let the lines be at the same height y₀, i.e.

x₁ = a₁z
x₂ = a₂z

As the lines are perpendicular, the following has to be fulfilled

a₁ a₂ = -1

The vanishing points of these two lines are at the projected coordinates (see >> the first part)

x'(F₁) = a₁ d
x'(F₂) = a₂ d

Let us now examine some typical case, i.e. when the perpendicular characteristic directions are both at an angle of 45^o with respect to the line of sight. Then

a₁ = tg (45^o) = 1
a₂ = tg (-45^o) = -1 = -1 / a₁,

and the (horizontal) distance between the two vanishing points of these directions is

x'(F₁) - x'(F₂) = (a₁ - a₂) d ~ 2 d

For approximately realistic angle of view, the width of the image is about 0.7 d, thus it is obvious that both vanishing points cannot in this case project in the perspective window.

Advice: In perspective constructions which approximately correspond to the angle of view typical for human eye, the separation of the two vanishing points should be about 2 to 4 times bigger than the width of the illustration i.e. perspective window.

The appearance of the same scene for different angles of view (i.e. distances d) is shown in the image below for angles of 45, 65, and 95 degrees. One can observe that both vanishing points project in the perspective window when the angle of view is 95^o (all the simulations with red-green square network were made in PovRay).

The image below is a "manual" construction of the perspective projection when the separation between the two vanishing points is about three times larger than the width of the perspective window. The image was constructed entirely manually, i.e. by positioning the horizon first, the two vanishing points, and by the construction of the rays from the vanishing points. In this case, the observer's eye level is at about two heights of the woman, thus the horizon does not cut the characters. In such a situation, when positioning characters one should consider the following advice:

When the horizon is above the characters, the distance from the top of the characters of the same height to the horizon is always an equal proportion of the height of the character (e.g. half of the character height, height of the head of the character and similar).

In the image below, the distance from the top of the woman's head to horizon is in all cases about the whole height of the character. For the child, the distance from the top of the child's head to the horizon is two heights of the child. The horizon is exactly at the bottom of the lowest balcony so you can measure it for yourself. The same mathematics holds also when the horizon is below the characters, but this is seldom the case.

I explained to you how to construct the two vanishing points and the rays emanating from them, and, in accordance with that, how to position the characters and objects in the perspective window. However, it is often necessary to determine distances in perspective projection. For example, one needs to position a character or object exactly half way between the two objects. Such a problem will typically appear e.g. when one positions windows on the building one illustrates (see above), or equally spaced cypresses. How to position a window exactly in the middle of the wall? Or, how to position the third window exactly half way between the two previously positioned windows, or, how to equally distribute the windows on the building in the perspective projection?

I will explain it in the next episode. Huh !

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Last updated on 11th of September, 2013.