Figure-construction problems on line drawings of polyhedra are studied, and practical as well as theoretical methods are proposed for deciding whether a figure-construction problem has a unique solution or not. A line drawing obtained as a projection of a three-dimensional polyhedron can represents an infinite number of different polyhedra, because it lacks information about distances from the projection plane to points on the objects. There exist, nevertheless, drawings on which some figure-construction operations have “invariant” meaning in the sense that the results of the operations do not depend on objects chosen from the family of all polyhedra the drawings can represent. Figure-construction problems with unique solutions in this sense are characterized algebraically, and a necessary and sufficient condition for a problem to possess a unique solution is given in terms of a degree of freedom in the choice of a polyhedron represented by the drawing. While the condition is not practical (because it requires evaluation of a rank of a certain large matrix), several practical methods for the discrimination of the uniquely solvable problems are also given with examples.