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.

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Oblique axonometric projections such as cavalier projections or military projections are called frontal axonometric projections or frontal axonometries. The frontal axonometry is known to have been projected onto the image plane T parallel to one of the rectangular coordinate planes of an object. This paper is to show that such frontal axonometries are projected not only onto the above-mentioned image plane T, but also onto the image plane U crossing the three rectangular coordinate axes. In case of the frontal axonometry projected onto the image plane U, the origin of the axonometric axes is the point on which the origin of the rectangular coordinate axes of the object fails when rotated about one of the trace lines of the trace triangle of the plane U. The two image planes, that is the planes T and U, on which the same axonometric axes of frontal axonometry are produced when the axes are projected onto the two planes, are symmetric with respect to one of the straight lines of projection.

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Painting and perspective are sometimes considered as the same. The treatise points out that the two are different, because the former is an art subjective and the latter i s a science objective. The reason is that the painting i s made through human perceptual images constructed by many sight lines, and i t contains the personal visual constancies, on the contrary the perspective is made through a sight line which is assumed that it could see clearly the whole visual field. This conclusion is brought out of the psychological analysis of human depicting behavior.

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To draw a perspective, we have to draw many straight lines passing through vanishing point. If vanishing point exist outside of the drawing paper, it is difficult to draw them without instrument. We devised drafter useful in such a case. This drafter consists of two main parts. One of them is like a T-square, we slide it by our left hand on a straight guide perpendicular to HL. The another part is a ruler, it is fixed to the former at a point by a pin and turns led by a oblique guide. The oblique guide is fixed to the drawing board at a point and its inclination angler, which determines the position of vanishing point, is vanable. .We draw the straight line along the ruler, all of their elongation pass through a vanishing point.
The position of vanishing point is determined at all points on HL outside of drawing paper.

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