To spare you from finding vanishing points that occur waaay off the page, a diagram by Paul Rivoche:

When subdividing those vertical lines on each side (Step 3), I find it best to divide them into 10 equal parts, using centimeters. (Metric is just easier to use when plotting out page dimensions, since it makes the math easier.) So: a 6cm line on one side becomes ten 6mm segments, while a 16cm line on the other side becomes ten 16mm segments. Then you join the ten segments on one side to the ten on the other side (a la the broken lines in the diagram). This gives you a mathematically simple but thorough grid to work from. If you want more grid lines: mark twenty 3mm segments, and twenty 8mm segments.

To find lengths on each side that are easily divisible (such as my examples of 6 and 16), bridge your initial pair of perspective lines with your ruler and move it along them until you find two distances (one at the narrow end and one at the broad end) that are each made up of a divisible number of centimeters. If you can't find divisible distances where you need them, mark where the nearest divisible distance would be, and simply adjust one of the initial perspective lines to meet it. Your initial perspective line was just an estimate anyway, and keeping the numbers divisible will save you from tricky math.

Note that the subdivided vertical lines should always be perpendicular to the horizon line. So for "dutched" camera angles, when your horizon line is diagonal (instead of parallel to the top and bottom of your page), make sure the subdivided lines are perpendicular to that angled horizon line, instead of being parallel to the sides of the page as they usually are.

For more perspective tips, see this entry.

# Perspective.

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