On perspective with 1, 2, and 3 vanishing points I, 6th of September, 2013.

I already wrote two posts on perspective and these are sufficient to learn from them all that an illustrator or a painter needs. The posts in question are >> Drawing with postscript III and >> Introduction to scientific visualization 2, where you can also hear my lecture explaining the basics of graphical perspective.

However, I am reading >> Andrew Loomis' books and I now more clearly comprehend the problems of perspective faced by the painter or illustrator working on large images / paintings, which often include large number of persons situated at different distances from the observer, in different positions, and of different heights (e.g. children and adults). People and objects, especially architecture should be positioned and scaled in accordance with perspective and Loomis gives some very useful advice on that. I have been thinking on his advice and I concluded that they are all in complete agreement with mathematical formulas which I exposed in the post >> Drawing with postscript III, and which I repeat below:

x' = x d / (z + d)	(1)
y' = y d / (z + d)	(2)

x' i y' are two-dimensional coordinates of the illustration (perspective window), and x, y, and z are Cartesian coordinates of a point in 3D space. d is the distance of the observer from the perspective window (see above).

Coordinate z represents the line of sight i.e. a line along which we view the space and which is perpendicular to the perspective (projection) window (spanned by x' and y' coordinates; see above). I will choose here the coordinate system (above) in such a way that x-coordinate coincides with left-right direction, and y-coordinate with up-down direction, while z-coordinate points into the space. I could have chosen the coordinate axes differently but nothing would change due to the symmetry of equations (1) and (2), as long as the z-coordinate represents the line of sight.

However, not all people are familiar with mathematical thinking so in the text which follows I will try to explain what the equations of perspective projection predict and how those predictions can be used by illustrators, even if they are not mathematically skilled.

In order to illustrate my subject, I have chosen female "characters" (pieces of plane, polygons) positioned on a square net (the image below). This is an idealized, bare bone representation of the problem faced by the illustrator representing scenes with many characters. In the image below, the characters were positioned in a "matrix", i.e. in columns (going towards the larger z-coordinate) and rows (left - right, parallel to the x-axis).

The image above represents what is known as a perspective with a single vanishing point. The vanishing point in question is denoted by F. This type of perspective is obtained when the scene contains two characteristic and mutually perpendicular lines, one of which is parallel to the line of sight, i.e. to the z-axis. One obtains such a scene e.g. when looking along a straight highway.

In our case, all the columns of characters appear as originating from point F, i.e. vanishing in it. One can see that by following the dashed pink lines drawn so to connect the highest points on characters in same columns. One can see that all the dashed lines cross (or originate) in the vanishing point F. The dotted yellow lines connect the characters in the same rows (i.e. at same z-coordinates, but different x-coordinates). We see that these lines are parallel to the frame of the perspective projection windows (to the x' axis). This is not entirely trivial finding, since it means that all the characters at the same z-coordinate will have the same projected height (y'), irrespectively of how distant from the observer or the perspective window they are, D = (x² + y² + z²)^1/2. One can see this from equation (2) which does not mix x- and y- coordinates, thus the projected height of a character (y') does not depend on its distance from the observer, only on its depth (z).

Here is a mathematical quickie which explains what we've found thus far. The lines parallel to the z-axis can be written as:

x = x0	(3)
y = y0,	(4)

where x₀ and y₀ are the parameter of the line - all x₀ and y₀ parameters form a bundle of lines parallel to the z-axis. We can also think of parameters (x₀, y₀) as of points in which the lines parallel to the z-axis cross the perspective window (i.e. when z=0).

Let us choose some line (x₀, y₀). It is transformed by the perspective projection to a line in (x', y') plane,

y' = (y0 / x0) x'.

(5)

The above equation can be obtained by eliminating the z-coordinate from the perspective transformation. We thus see that all the lines parallel to the z-axis cross in (or originate from) the origin of perspective coordinate system (i.e. a vanishing point), but their slopes (y₀ / x₀) are different.

But, what happens to the lines which are not parallel to the line of sight? The answer to this question can be also obtained from the image above, by examining e.g. the lines which do not connect the characters in the same columns, but in a "skewed" manner, e.g. one of the characters with the other in the row behind it, but it the columns just left from it (this is the observer's right as we are looking the characters in the faces). Two such lines are indicated in the image below by blue dashed lines. We may conclude that mutually parallel lines which are not parallel to the line of sight also cross in a single point in a perspective projection (denoted by F' in the image below), but this point does not coincide with the "main" vanishing point, F. But, both points are on the same height in the perspective projection, y'=0, and the line which connects them is called the horizon (see below).

It is important to remember that the horizon is at the same height as are the eyes of the observer (y=y'=0).

I will no longer derive mathematical relations which explain this finding, it's pretty easy so I leave it to you. The thing you should be thinking about when illustrating a scene is that all parallel lines have the same (their own) vanishing point - their vanishing point depends on the angle(s) they make with the line of sight.

When the lines in question are "flat", i.e. they do not ascend or descend with z-coordinate (i.e. they are given as x=az, y=y₀), their vanishing point is on the horizon (y'=0), at the perspective coordinates F' = (ad,0) (derive this, it's easy).

This is a good introduction to the perspective projection with two vanishing points shown in the image below. I already wrote a lot of text, so I leave this for the next post.

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On 1-, 2- and 3-point perspective II >>

Last updated on 6th of September, 2013.