Gear Tooth Geometry
I originally put this video together in order to help me understand the geometry of gears for the Kythera application on the Glenview Software web site. But I think these videos may be useful for others as well, so I’ve finished the video with some annotations and posted them here.
This video makes use of vector math. If you’re not familiar with the subject, you may wish to watch the excellent 3Blue1Brown video series on linear algebra. For our purposes we only need 2 dimensional vectors and rotational matrices.
The equations given in the video are:
(1) For a point passing along a straight line, given a angle in radians, and r the radius of the circle, the position of a point along the circle is:
(2) The equation giving the involute curve (that is, the equation above, rotated around our circle) is:
(3) The equation of an involute curve with an offset vector (x,y) is:
There are a number of techniques that can be used to calculate the depth map of ranges of segments as they cut into the gears. One technique borrows from polygon scan line rendering, building a horizontal bufffer (either using a z-buffer or by using span sorting in two dimensions) to calculate the maximum depth cut into a gear at a particular given angle a. This is the technique used by the graphics rendering engine used to draw the video above.
Typical gear dimensions
I’ve posted a spur gear reference. The circular pitch and pressure angle are the same as in the video (though it is represented as the pitch angle of the rack cutting two gears). The height of the rack is generally given as 2/DP, the diametrical pitch of the gear.