MichaelPrescott

I've turned what I learned about placing vanishing points into a little video lecture/tutorial.

www.youtube.com/watch?v=o6451u…

This is a question that's bugged me for a while.

Let's say you're doing a simple two-point perspective, with parallel vertical lines.  You draw your basic box, whose horizontal sides each line up with one of the two vanishing points.

Now you add a new box, stacked on top of it, but it's rotated with respect to the first one.  Where do the new vanishing points go?

Let's say you're looking north, and you have two vanishing points, northwest, and northeast.  They're equally spaced to either side of the center of the picture.  As we know, NW and NE are 90 degrees apart on the compass, so in a sense the two vanishing points are "90 degrees apart".

Let's say you want to add another vanishing point.  Maybe it's east (you've got a wide field of view, perhaps).  Maybe it's the 'up' or 'down' vanishing point.  Where should it go?

Naively, I thought I might just take the distance between the NW and NE vanishing points, which as I said earlier are 90 degrees apart, and use that as a ruler.  This turns out to be totally and utterly wrong - some of the time!

Classical perspective is based on the idea that you're tracing the three-dimensional scene before you onto a flat pane of glass.

Imagine yourself in a scene, facing north, with your pane of glass in front of you. With a moment's reflection (pun not intentional) you might realize a few things:

1. While a larger pane of glass allows you to draw a larger field of view, no matter how large your pane, you can never capture a field of view greater than 180 degrees.  It's just not possible.  In fact, just to draw the 'east' vanishing point on your pane of glass, you'd need an infinitely large pane, reaching all the way to the horizon!  (If you don't have one, try to look directly east through a north facing window. You can't.)

2. While 10 degrees east, 20 degrees east, and so on are relatively closely spaced, toward the edge of your picture the angles get further and further apart on the pane of glass.

But.. how much further?

It turns out the answer comes from geometry.  If north is 0 degrees (and northeast is 45 degrees), then the compass headings will be spaced on your canvas in proportion to tan(heading).

The distance, in pixels, to a compass heading on your canvas, is shown by this formula: (You can use inches, centimeters, it doesn't matter.)

canvasDistanceToBearing = canvasDistanceTo45Degrees * tan(bearing) / tan(45)

Let's north is the center of your image, and your north and northeast vanishing points are 500 pixels apart on your canvas.  This gives a formula of:

canvasDistanceToBearing = 500 * tan(bearing) / 1

or (conveniently)

canvasDistanceToBearing = 500 * tan(bearing)

So, where should 60 degrees go?  866 pixels to the right of the center of your image. (If you're working this out for yourself, just make sure you're using a calculator that lets you use the 'tan' function in degrees, not radians.)

This is time consuming, so you can just use the Perspective Crosshairs brush michaelprescott.deviantart.com… I've made to save you the time.

One thing to notice is this: in classical perspective, the bearings are not evenly spaced.  The distance between 0 (north) and 10 degrees east is 88 pixels, but the distance between a bearing of 70 and a bearing of 80 is 1462 pixels!  (The distance between 80 and 90 is infinite!)

Now, what I've said so far is only true if you're looking at the horizon.  If you're looking up or down, then things are slightly more complicated.  That's for the next journal entry!