# Subdividing a line leading towards a vanishing point into equally long segments?

It is assumed that it is a one point perspective. I know that theoretically, it would be impossible, since the distance to the vanishing point is infinite. What I am interested in is a visual representation of an equally divided space. But I don't know how, e.g. given a distance of 10cm from the lower left corner of my picture to the vanishing point, I can calculate e.g. ten segments that represent the same distances towards the vanishing point in 3D space.

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I think I see the seeds of an interesting question somewhere here, but right now it's terribly vague. Can you provide motivation in a simple example? – rschwieb Oct 18 '12 at 13:57

How would you draw a picture of a road made of rectangular tiles (not necessarily yellow bricks) in a row leading to infinity? The left and right edge of the road become two lines meeting at the horizon. For simplicity, I assume the the vanishing point of this set of parallels is "straight ahead". This implies that all across edges of the tiles are simply parallel to the horizon. Now consider all diagonals through a lower right and upper left vertex of a tile. All these are parallel. Hence they all meet at a vanishing point on the horizon. By prescribing the first tile (i.e. drawing also its across edges) the rest is determined: draw the diagonal, extend it to the horizon, thus finding the vanishing point $P$ for all diagonals. Then draw the line from $P$ to the "upper right" vertex of the first tile. This produces the "upper left" vertex of the second tile as intersection point with the left road edge. Now draw the edge across to find the fourth vertex of the second tile and proceed by drawing a line from there to $P$ and so on.
If you want to use calculation instead of construction, the $n$th point of an equidistant sequence is at distance $\frac1{an+b}$ from the vanisching point. The values $a,b$ depend on the distance between 3D points, the relative position of the line to the eye and the "offset" where the points start.