Duality

Duality, Principle op, in Geometry, means that many theorems relating to points and lines in one plane must exist in pairs. The reason may be briefly stated thus: a line is completely determined if we know two points on it, and may therefore be regarded as the product of those two points. If the points are named A and B, the line is named AB. Conversely, a point is completely determined if we know two lines through it, and may, therefore, be regarded as the product of those two lines. If the lines are named a and b, the point is named ab. Hence, three points determine three lines, and three lines determine three points. Corresponding to a series of points in a line, we should speak of a series of lines through a point. The correlation of lines and points being thus suggested, we may cite Pascal's theorem and its correlative, Brianchon's theorem, in illustration of the principle of duality. The former states that if six points be taken on any conic section, and a six-sided figure be drawn by finding six consecutive lines that are determined by these points, then the intersections or products of opposite sides of the figure will be three points in a line. The second may be deduced from the first by interchanging the words "line" and "point" throughout. Thus, if six lines be taken on any conic section (i.e. six tangents), and a six-pointed figure be drawn by finding six consecutive points that are determined by these lines, then the products of opposite points of the figure will be three lines through a point. In solid geometry the principle of duality similarly applies to points and planes, a plane being determined by three points and a point by three planes.