Transcript Point Configuration

Point Configuration
• A point configuration
in R2 is a collection of
points afinely
spanning R2.
• In other words: not all
points are collinear.
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.
Polarity with Respect to a Circle
p
P
P
p
P
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.
Polarity with respect to the unit
circle
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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)
Polarity and Point Configurations
• Polarity maps a point configuration to a line
arrangement and vice versa.
• Exercise:Take 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.
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.
A Standard Affine Polar-Duality
• A standard affine polar
duality is a mapping
between non-vertical
lines and points of R2
associating the nonvertical line y = ax + b
with the point (a,-b)
and vice versa.
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 .