Duality

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What is the duality?

A duality d in the plane assigns points to rects and vice versa, in a transformation that respects some interesting geometric property, such as the relative position above/below or inside/outside. It is called duality due to the fact that when applied twice it leaves the elements unchanged, whether these are points or rects. In other words, d2=Id. Although there are other dualities of the plane, the most used in computational geometry are defined by two conics: the parabola y = x2 / 2 and the circumference ax2 + by2 = 1.

Parabola duality

This duality takes as a reference the parabola y = x2 / 2 and the transformations follow the following rule:

p = (a,b) ↦ D(p) = p* : y = ax - b

l : y = ax + b ↦ D(l) = l* = (a,-b)

This duality has the following characteristics:

  1. d2 = Id.
  2. Every point p corresponds to a dual rect p*.
  3. Every non-vertical rect r has a dual point r*. On the other hand, vertical rects have no image for this duality.
  4. If p is a point above a rect r, then r* is a point above rect p*.
  5. Given a point p that belongs to the parabola, p* is the tangent rect to the parabola at the point p.
  6. Given a point q on the plane, let p be the vertical projection of q on the parabola. Then, q* is the rect parallel to p* whose vertical distance to p* is the opposite of the vertical distance between points q and p.
Circle duality

This duality takes as a reference the circle ax2 + by2 = 1 and the transformation are as follows:

p = (a,b) ↦ D(p) = p* : ax + by = 1

l : ax + by = 1 ↦ D(l) = l* = (a,b)

The characteristics of this duality are as follows:

  1. d2 = Id.
  2. Every point p different than the origin of coordinates (and center of the circumference) has a dual rect p*. In contrast, the point (0,0) cannot be dualized.
  3. Every rect r that does not pass through the origin corresponds a dual point r*. On the other hand, the rects by the origin do not have a dual.
  4. If p is a point of a dualizable rect r, then r* is a point of the rect p*.
  5. If p is a point different than the origin and r is a rect that does not go through the origin, and p belongs to the half plane determined by r that contains the origin, then r* is a point at the half plane determined by p* that does not contain the origin, and vice versa.
  6. Given a point p that belong to the circumference, p* is the tangent rect to the circumference at the point p.
  7. Given a point q on the plane, let p be the central projection of q to the circumference, according to a projection with center at the origin. Then, q* is the parallel rect to p* whose distance to the origin is the inverse to the distance to the origin between the points q and p.