The Complex Helix

The Complex Helix

Grade 11 plotted a helix on a cylinder. Grade 12 adds helices on cones and helices of square or rectangular section, such as a square-thread screw.

A conical helix. The point still rotates uniformly and advances uniformly along the axis, but the radius shrinks as it rises. In plan the path is a spiral rather than a circle. Construction: divide the plan spiral into twelve, divide the pitch into twelve, and pair them exactly as before — the pairing rule does not change, only the plan shape.

A square-thread helix. Two helices of the same pitch, offset by the thread depth, with the flanks joining them. The crest helix and the root helix are parallel curves, not the same curve shifted vertically.

Multi-start threads. A two-start thread has two helices offset by half a pitch. The lead — the advance per revolution of the nut — is twice the pitch. This is why a two-start thread advances faster for the same turn.

Worked example. On a conical helix, equal angular steps cover shorter arc lengths as the radius shrinks, so the curve appears to tighten as it rises. That is correct, not an error.

Core checkpoint: The pairing rule is unchanged — equal divisions of rotation against equal divisions of pitch. Only the plan shape differs.