Gear 1 and pinion 2 rotate about fixed axes A and B-B which cross each other at the angle δ. The pitch surfaces of the gears are circular hyperboloids of one sheet with radii of their narrowest cross sections equal to r₁ and r₂. The teeth of gear 1 and pinion 2 are located on certain selected conjugate portions of the pitch hyperboloids. Taking the sign of the angular velocities ω₁ and ω₂ of gears 1 and 2 into account, the transmission ratio of the mechanism is i₁₂=ω₁/ω₂=-(r₂/r₁)(cos(δ₂)/cos(δ₁)) where δ₁ and δ₂ are the angles between axes A and B-B and the straight line of contact of the pitch hyperboloids. The sum of radii r₁ and r₂ is r₁+r₂=a where a is the distance between axes A and B-B. If angle δ=90°, then the transmission ratio is i₁₂=-(r₂/r₁)tan(δ₁). If the portions of the pitch hyperboloids at the narrowest cross sections are approximated by circular cylinders, spiral (crossed helical) gearing is obtained; if the portions of the pitch hyperboloids near their end faces are approximated by cones, hypoid gearing is obtained.
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