Point Configuration

1
Point Configuration

• A point configuration in R2 is a collection of
  points afinely spanning R2.
• In other words not all points are collinear.

2
Line Arrangement

• A line arrangement is a partitioning of the plane
  R2 into connected regions (cells, edges, and
  vertices) induced by a finite set of lines.

3
Polarity with Respect to a Circle
p

• Let us consider the extended plane and a circle K
  in it. There is a mapping from points to lines
  (and vice versa). p p a P.
• p polar
• P pole
• Exercise Determine the polar of an ideal point
  and the pole of the ideal line.

P
p
P
p
P
4
Polarity with respect to the unit circle

• Given P(a,b) the equation of the polar is
• p y (-a/b)x (1/b)
• p by ax 1
• In general
• p b(y-q) a(x-p) r2.
• Given
• p y kx n
• P(a,b)
• a -k/n
• b 1/n
• In general
• a -kr2/(kp n q)
• b r2/(kp n q)

5
Polarity and Point Configurations

• Polarity maps a point configuration to a line
  arrangement and vice versa.
• ExerciseTake an equilateral triangle ABC with
  sides a,b,c. Find a polarity, such that a a A, b
  a B and c a C.
• Exercise Determine the polar figure of point
  configuration determined by the vertices of a
  regular n-gon with respect to its inscribed
  circle.

6
Polar Duality of Vectors and Central Planes in R3.

• Polar duality is a mapping associating a vector v
  2 R3 with an oriented central plane having v as
  its normal vector and vice versa.

7
A Standard Affine Polar-Duality

• A standard affine polar duality is a mapping
  between non-vertical lines and points of R2
  associating the non-vertical line y ax b with
  the point (a,-b) and vice versa.

8
Polar Duality of Points and Lines in the Affine
Space.

• General rule Take polar-duality of vectors and
  central planes and consicer the intersetion with
  some affine plane in R3 .