Mathematical structures of line drawings of polyhedrons are studied and practical as well as theoretical solutions are obtained for several fundamental problems aroused in scene analysis and in man-machine communication. First, a necessary and sufficient condition for a line drawing to correctly represent a polyhedron is obtained in terms of linear algebra. Next, combinatorial structures are investigated and practical solutions are obtained to such problems as how to discriminate between correct and incorrect line drawings and how to correct vertex-position errors in incorrect line drawings. Lastly, distribution of the degree of freedom of a line drawing is elucidated and a method is proposed for interactive reconstruction of a polyhedron from a line drawing. The results obtained here enable us to make manmachine communication more ``flexible'' in the sense that a machine can reconstruct three-dimensional objects from hand-drawn pictures even if the pictures are not perfect.