2 Metric Geometry • Formula or rule for measuring distance is called a metric. Set of Axioms for a Metric Space Let P, Q and R be points, and let d(P,Q) denote the distance from P to Q. 1. d(P,Q) ! 0 and d(P,Q) = 0 iff P = Q 2. d(P,Q) = d(Q,P) 3. d(P,Q) + d(Q,R) ! d(P,R) Our ordinary distance formula satisfies the three axioms dP, Q ( ) = x p ! x Q ( ) 2 + y p ! y Q ( ) 2 Taxicab Distance? Let P, Q and R be points, and let d(P,Q) denote the distance from P to Q. 1. d(P,Q) ! 0 and d(P,Q) = 0 iff P = Q 2. d(P,Q) = d(Q,P) 3. d(P,Q) + d(Q,R) ! d(P,R) d T P, Q ( ) = x p ! x Q + y p ! y Q Circles • A circle is defined as the set of all points at a given distance, r, from a fixed center, C. • Circle = • The fixed point C is the center of the circle, and the length r is its radius. P : d (P, c) = r, where r > 0 and C is fixed { }
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Transcript
2
Metric Geometry
• Formula or rule for measuring distance is
called a metric.
Set of Axioms for a Metric Space
Let P, Q and R be points, and let d(P,Q) denote thedistance from P to Q.
1. d(P,Q) ! 0 and d(P,Q) = 0 iff P = Q
2. d(P,Q) = d(Q,P)
3. d(P,Q) + d(Q,R) ! d(P,R)
Our ordinary distance formula
satisfies the three axioms
d P,Q( ) = xp ! xQ( )2
+ yp ! yQ( )2
Taxicab Distance?
Let P, Q and R be points, and let d(P,Q) denote the
distance from P to Q.
1. d(P,Q) ! 0 and d(P,Q) = 0 iff P = Q
2. d(P,Q) = d(Q,P)
3. d(P,Q) + d(Q,R) ! d(P,R)
dT P,Q( ) = xp ! xQ + yp ! yQ
Circles
• A circle is defined as the set of all points
at a given distance, r, from a fixed center,
C.
• Circle =
• The fixed point C is the center of the circle,
and the length r is its radius.
P :d(P,c) = r,!where!r > 0!and !C !is! fixed{ }
3
Taxi Circles
The Taxi-circle centered at C = (0,0) with radius r > 0is the set:
Graph has flat sides! With line segments with slopesof ±1!