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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.
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:
- d2 = Id.
- Every point p corresponds to a dual rect p*.
- Every non-vertical rect r has a dual point r*.
On the other hand, vertical rects have no image for this duality.
- If p is a point above a rect r, then r*
is a point above rect p*.
- Given a point p that belongs to the parabola, p*
is the tangent rect to the parabola at the point p.
- 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.
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:
- d2 = Id.
- 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.
- 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.
- If p is a point of a dualizable rect r, then r*
is a point of the rect p*.
- 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.
- Given a point p that belong to the circumference, p*
is the tangent rect to the circumference at the point p.
- 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.