A point-conjugate worm drive is less constrained and has greater error tolerance compared to a line-conjugate worm drive. However, the lack of an effective solution method for multivariate nonlinear contact equation systems hinders the in-depth exploration of numerical results related to the contact characteristics of point-conjugate worm drives. In this paper, a seven-variable nonlinear underdetermined contact equation system is established at the contact points based on the tooth surface contact condition, and then an equivalent contact equation system containing only three unknowns is obtained using the elimination method. By the definition, an analytical expression for solving the parameters of the datum point is given. Considering the uncertainty of the endpoint position of the contact path, all possible scenarios for its intersection with the boundary of the worm pair are listed and analyzed. In addition, a numerical example is carried out and the contact path and conjugate area are investigated for the point-conjugate worm drive, and the result indicates that it has excellent contact characteristics.