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IMService makes available a Freeware Threading Program which calculates thread depth. Intrigued, I decided to figure out the math behind this.

For reference, here are same basic formulas for calculating The Area of Some Simple Polygons.

Basically, the area removed on each pass is equal. The area of a triangle is base times width divided by 2. So the total area = (0.875p x 0.875H) / 2 = (0.875p x 0.875 x 0.866p) / 2 = 0.3315 x p x p (or 0.3315 x p squared)

Using the formula from the previous step (which preserves the angle) we get total area = (0.875p x 0.875 x p/( 2 * tan( 30 ))) / 2 = 49 x p x p / ( 64 * 4 * tan( 30 )) = (49 x p x p) / (256 * tan(30)). I converted the 0.875 back into 7/8 to give integer numbers.

For a given depth d, d = base / (2 x tan(30)), so base = 2 x d x tan( 30 ), and the area = (base x depth) / 2 = (2 x d x tan(30)) x d / 2 = d x d x tan(30), so we can calculate d = sqrt( area / tan(30))

In this example, we're doing 5 passes, each with an equal area. So d1 = sqrt( 1/5 of total area / tan(30)). d2 = sqrt( 2/5 of total area / tan(30)), and so on.

The thread22 program allows a Ratio value to be specified to allow the last pass to be some percentage of the other passes. If we wanted 5 passes with Ratio set to 0.4, then the formula for the area of pass 1 would be 1/4.4, and the area for the 2nd pass would 2/4.4 an so on. Generalizing for pass i would give (i / ((num_passes - 1) + ratio) x total_area) except for the last pass which would use the total_area. Smaller...

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