Free-form surface gear

A transmission gear having a tooth surface and a tooth bottom land, said transmission gear comprising:

• the tooth surface and the tooth bottom land having different free form surfaces;
• the free form surface of the tooth surface is formed of a first curved surface having a plurality of a radius of curvatures;
• the free form surface of the tooth bottom land is formed of a second curved surface having a radius of curvature;
• the first curved surface of the tooth surface has an area around a mating pitch point, wherein the smallest radius of curvature among a plurality of radii of curvatures in the area around the mating pitch point is maximized;
• the second curved surface of the tooth bottom land has a tooth root connecting to the tooth surface, wherein the radius of curvature around the tooth root is maximized;
• and
• wherein the first curved surface is formed of a compound surface of three different types of surfaces selected from one of the sets of:
• (a) a group of a cycloid helicoid surface, a trochoid helicoid surface, a circular helicoid surface, a limacon helicoid surface, a three-dimensional coordinate surface, an epitrochoid helicoid surface, a petritrochoid helicoid surface, and an involute helicoid surface;
• and (b) a group of a cycloid surface, a trochoid surface, an epitrochoid surface, a peritrochoid surface, a circular surface, a limacon surface, and an involute surface;
• and
• wherein the second curved surface is formed of a circular surface.