Connecting Skew Axles with Hyperboloidal Gears
Gears shaped like hyperboloids can be used to build a transmission between skew axles. With a given "skew angle" (the angle between the axles), the eccentricity of two hyperboloids whose axes of symmetry coincide with these axles can be found such that there is a constant line of contact between them. These hyperboloids are created by rotating a straight line that is skew to the axle, at an angle that is half that of the skew angle. These lines then act as the gear teeth. If this is done correctly, the teeth on the gears will be in parallel orientations while in contact with one another, thus creating a smooth transmission.
Snapshot 1: in the simplest case, when the axles are parallel, the gears will be cylindrical (not hyperboloidal)
Snapshot 2: notice that the teeth of the gears are slanted in the opposite direction on the hyperboloids when the skew angle is negative
Snapshot 3: this setup can theoretically be used to connect any axles with a skew angle less than 90 degrees, but it becomes less practical for larger angles