The following online calculator computes
the basic dimensions and tooth profiles
of a bevel gear pair (pinion and gear) based on their number of teeth
and angle between the shaft axes.

Conceptually, two meshing bevel gears can be represented by two touching cones, referred to as pitch cones or reference cones,
shown in thick lines on the left image below.

The diameter of a cone's base circle is simply the gear's number of teeth times module (*z*_{1}m
and *z*_{2}m). The angle between the shaft axes, *Σ*, is usually 90° but could be a different number.
Finally, the facewidth *b* is the length of the contact line of the two reference cones (see right image).
This value should not exceed ^{1}/_{3} of the Cone distance denoted *R* on the right image, which
is also the radius of the circle shown on the left image.

This calculator takes *m*, *z*_{1}, *z*_{2}, *Σ* and *b* as inputs,
and computes the parametric equations for the tooth profiles of both gears, the cone angles *δ1* and *δ2*,
cone distance *R*, a scaling factor *s* calculated as (R - b) / R, and distance from the cone base to apex.
The scaling factor makes it easy to model the top section of the gear by scaling down its bottom section, and the base-to-apex
height is used to correctly position the gear's origin, as shown in the tutorial.

Update: As of 2017-01-18, this calculator is equipped with two new buttons, *Show pinion script* and *Show gear script*.
These buttons generate Python scripts that instantly tranform flat gear outlines to saucer-like shapes that form the top and bottom of a bevel gear.