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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{ }

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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!

P :d(P,C) = r,!where!r > 0!and !C !is! fixed{ }

= (xp , yp ) :! xp ! 0 + yp ! 0 = r{ }= (xp , yp ) :! xp + yp = r{ }

Taxicab Circles

Ellipses

• An ellipse is defined as the set of points P,the sum of whose distances from two fixedpoints, F1 & F2 is constant.

• Ellipse =

• The fixed points are called the foci(singular focus) of the ellipse.

P :d(P,F1) + d(P,F

2) = d,

where!d > 0!and !F1,!F

2!are! fixed ! po int s

!"#

$%&

Ellipses

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Taxi Ellipse

• Using the taxicab metric for distance,

instead of being rounded, taxi ellipses are

either octagonal or hexagonal. One or two

pairs of sides of the ellipse will be

horizontal and/or vertical and the

remaining four sides will follow the sides of

taxi-circles which are line segments with

slopes of ±1.

Taxi Ellipse

• If the foci lie on

diagonally

opposite corners

of a rectangle,

the taxicab

ellipse will be

octagonal.

Taxi Ellipse

• If the foci lie on the

same vertical or

horizontal segments,

one pair of vertical or

horizontal segments

disappears and the

ellipse is hexagonal.

Parabola

• Given a fixed line, k, and a fixed point, F, a

parabola is defined as the set of points P

that are equidistant from k and F. We write

this as:

• Parabola =

• Line k is called the directrix of the parabola

and point F is called the focus of the

parabola.

P :d(P,F) = d(P,k)!{ }

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Taxi Parabola

• Activity 9 has you investigate using the

taxicab metric to measure the distance.