Chapter 26 Duality

x I don't know why it should be, I am sure; but the sight of another man asleep in bed when I am up, maddens me. It seems to me so shocking to see the precious hours of a man's life-the priceless moments that will never come back to him again-being wasted in mere brutish sleep. – – Jerome K. Jerome, Three men in a boat Duality is a transformation that maps lines and points into points and lines, respectively, while preserving some properties in the process. Despite its relative simplicity, it is a powerful tool that can dualize what seems like " hard " problems into easy dual problems. 26.1 Duality of lines and points Consider a line ≡ y = ax + b in two dimensions. It is being parameterized by two constants a and b, which we can interpret, paired together, as a point in the parametric space of the lines. Naturally, this also gives us a way of interpreting a point as defining coefficients of a line. Thus, conceptually, points are lines and lines are points. Formally, the dual point to the line ≡ y = ax + b is the point = (a, −b). Similarly, for a point p = (c, d) its dual line is p = cx − d. Namely, p = (a, b) ⇒ p : y = ax − b : y = cx + d ⇒ = (c, −d). We will consider a line ≡ y = cx + d to be a linear function in one dimension, and let (x) = cx + d. A point p = (a, b) lies above a line ≡ y = cx + d if p lies vertically above. Formally, we have that b > (a) = ca + d. We will denote this fact by p. Similarly, the point p lies below if b < (a) = ca + d, denoted by p ≺. A line supports a convex set S ⊆ IR 2 if it intersects S but the interior of S lies completely on one side of. x This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit