Involute curves have always appeared so aesthetically pleasing to me. Turns out outside of aesthetics they also have practical applications especially in machine design. Involutes of a Circle are created by rolling a straight line on a circle. From the help of ChatGPT: "Imagine a taut string wrapped around a given circle. If you then unwind that string, keeping it taut, the path traced by the free end of the string is an involute of the original circle."
Figure 1. Circle and its involutes
What is unique about Involute Gears is that the profile of an involute gear depends only on the number of teeth on the gear (z), pressure angle (α), and pitch diameter (d0). And for a gear to properly mesh (engage) with another gear only the pressure angle of the gears and gear sizes (dependent on number of teeth) must match.
The following is grid of different gears with different pressure angle and number of gears(same pitch diameter). They all have been mathematically generated using parametric equations.
12 teeth, 10 degrees
24 teeth, 10 degrees
48 teeth, 10 degrees
12 teeth, 20 degrees
24 teeth, 20 degrees
48 teeth, 20 degrees
12 teeth, 30 degrees
24 teeth, 30 degrees
48 teeth, 30 degrees