slowly figuring - the gear speeds and sizes have the same ratio, because the angular tooth offsets must match for proper meshing
- this is why the pitch (contact) point distances must also have the same ratio, as the sizes and speeds - gears have a pressure angle, which is the angle of contact at the pitch point (as well as all other points), chosen by the designer - the surface normal equals the pressure angle in absolute coordinates. this draws the gear profile, but there are many approaches - involute gears have contact points following a straight line in absolute coordinates through the pitch point. involute gears retain their properties when their axes shift or they shrink or expand. - the involute curve is defined by the above properties; each point is defined by a surface normal; it is rooted on an imaginary involute base circle; each point as well as the pitch contact line is tangent to the involute base circle so: - the radius of the involute base circle is directly related to the relative speeds or sizes of the gears, defined by a tangent at the pressure angle going through the pitch point so if the pitch point is at r_p, we could draw a (90-20=70) deg angle from that, making a right triangle with the hypotenuse drawing p_r, and the involute base circle radius would be r_p cos(20) where 20 is the pressure angle i made https://www.desmos.com/calculator/ebvu4elizt where a_p is the pressure angle, r_p is the pitch radius, and r_b is the involute base radius. it draws two involute curves through the pitch circle, which is surprisingly close to their base at 20 degrees, i wonder if i did it right
