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
Area of Triangle

• Area of the green trapezoid
• A12 (1/2)(y2 y1) (x2 x1)
• In the same way
• A23 (1/2)(y2 y3) (x3 x2)
• A13 (1/2)(y3 y1) (x3 x1)
• Area of the triangle
• T A12 A23 A13.

P2(x2,y2)
y2
y1
P1(x1,y1)
y3
P3(x3,y3)
O
x2
x1
x3
4
Area of Triangle
P2(x2,y2)
y2
y1
P1(x1,y1)
y3
P3(x3,y3)
O
x2
x1
x3
5
Triple of Collinear Points
P2(x2,y2)
y2
y1
P1(x1,y1)
y3
P3(x3,y3)
O
x2
x1
x3
The points P1(x1,y1), P2(x2,y2), P3(x3,y3), are
collinear if and only if T 0.
6
Point Configurations Line Arrangements

• Each point configuration S gives rise to a line
  arrangement A(S). The lines are determined by all
  pairs of points.
• Another line arrangement A3(S) is determined by
  triples of collinear points.

7
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
8
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 y(b-q) x(a-p) p(a-p) q(b-q) r2.
• Given
• p y kx n
• P(a,b)
• a -k/n
• b 1/n
• In general
• a p-kr2/(kp n q)
• b q r2/(kp n q)

9
Natural Parameters p,q,r

• For a given point configuration S the center of
  the circle(p,q) is determined as the barycenter
  of S while the radius is given as the average
  distance from the center.

10
Polarity in General

• In general polarity is given with respect to a
  conic section (ellipse, hyperbola, or parabola).

11
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.

12
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.

13
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.

14
Polar Duality of Points and Lines in the Affine
Space.

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

15
Homogeneous Coordinates

• Take the affine plane z 1. A point with
  Euclidean coordinates (x,y) can be extended to
  homogeneous coordinates (x,y,1). Ideal points get
  homogeneous coordinates (x,y,0).

16
Point on a Line

• Let (a,b,c) be homogeneous coordinates of a point
  P and let A,B,C be homogeneous coordinates of a
  line p.
• Then P lies on p if and only if aA bB cC 0.

• Let P(a,b,c) and P(a,b,c). The equation of a
  line through P Æ P. is defined by the cross
  product A,B,C (a,b,c) (a,b,c).
• Similarly we get the intersection of two lines.

17
Example

• Polarity of a point configuration consisting of
  10 10 grid points.
• Parameters of the circle are determined
  automatically.