The proposed abnormal hourglass worm gearing is a novel type of point-contact transmission that is insensitive to installation errors, has good meshing performance, favorable bearing contact and high manufacturing accuracy. Based on Differential Geometry and Gear Meshing Theory, the systematic meshing theoretical analysis and synthesis is carried out. By complementing concept of datum points and contact endpoints, the theory of tooth contact analysis (TCA) is developed, and the non-linear contact equation used to solve the momentary contact point is simplified into the equivalent form of one-variable by elimination method; A synthesis design is carried out from the desired meshing behavior, which allows more accurate acquirement of favorable parameter scheme; The influence of the contact trajectory direction on the eccentricity of the contact ellipse is further analyzed; The meshing performance with centering bearing contact and lower error sensitivity are demonstrated by tooth contact simulation. The described meshing theory and synthesis can also be applied effectively for other point-contact gearing.