But there is also a more special principle of duality, according to which figures are reciprocal which lie both in a plane or both in a pencil. In the plane we take points and lines as reciprocal elements, for they have this fundamental property in common, that two elements of one kind determine one of the other. In the pencil, on the other hand, lines and planes have to be taken as reciprocal, and here it holds again that two lines or planes determine one plane or line.
Thus, to one plane figure we can construct one reciprocal figure in the plane, and to each one reciprocal figure in a pencil. We mention a few of these. At first we explain a few names:—
| A figure consisting of n points in a plane will be called an n-point. | A figure consisting of n lines in a plane will be called an n-side. |
| A figure consisting of n planes in a pencil will be called an n-flat. | A figure consisting of n lines in a pencil will be called an n-edge. |
It will be understood that an n-side is different from a polygon of n sides. The latter has sides of finite length and n vertices, the former has sides all of infinite extension, and every point where two of the sides meet will be a vertex. A similar difference exists between a solid angle and an n-edge or an n-flat. We notice particularly—
| A four-point has six sides, of which two and two are opposite, and three diagonal points, which are intersections of opposite sides. | A four-side has six vertices, of which two and two are opposite, and three diagonals, which join opposite vertices. |
| A four-flat has six edges, of which two and two are opposite, and three diagonal planes, which pass through opposite edges. | A four-edge has six faces, of which two and two are opposite, and three diagonal edges, which are intersections of opposite faces. |
A four-side is usually called a complete quadrilateral, and a four-point a complete quadrangle. The above notation, however, seems better adapted for the statement of reciprocal propositions.
§ 44.
| If a point moves in a plane it describes a plane curve. | If a line moves in a plane it envelopes a plane curve (fig. 15). |
| If a plane moves in a pencil it envelopes a cone. | If a line moves in a pencil it describes a cone. |
A curve thus appears as generated either by points, and then we call it a “locus,” or by lines, and then we call it an “envelope.” In the same manner a cone, which means here a surface, appears either as the locus of lines passing through a fixed point, the “vertex” of the cone, or as the envelope of planes passing through the same point.
| Fig. 15. |