Taxicab Geometry Chapter 5

Distance On a number line On a plane with two dimensions Coordinate system skew () or rectangular

Axiom System for Metric Geometry Formula for measuring  metric Example seen on previous slide Results of Activity 5.4 Distance  0 PQ + QR  RP (triangle inequality)

Axiom System for Metric Geometry Axioms for metric space d(P, Q)  0 d(P, Q) = 0 iff P = Q d(P, Q) = d(Q, P) d(P, Q) + d(Q, R)  d(P, R)

Euclidian Distance Formula Theorem 5.1 Euclidian distance formula satisfies all three metric axioms Hence, the formula is a metric in Demonstrate satisfaction of all 3 axioms

Taxicab Distance Formula Thus, the taxicab distance formula is a metric in Consider this formula Does this distance formula satisfy all three axioms?

Application of Taxicab Geometry

Application of Taxicab Geometry A dispatcher for Ideal City Police Department receives a report of an accident at X = (-1,4). There are two police cars located in the area. Car C is at (2,1) and car D is at (-1,- 1). Which car should be sent? Taxicab Dispatch

Circles Recall circle definition: The set of all points equidistance from a given fixed center Or Note: this definition does not tell us what metric to use!

Taxi-Circles Recall Activity 5.5

Taxi-Circles Place center of taxi-circle at origin Determine equations of lines Note how any point on line has taxi-cab distance = r

Ellipse Defined as set off all points, P, sum of whose distances from F1 and F2 is a constant

Ellipse Activity 5.2 Note resulting locus of points Each point satisfies ellipse defn. What happened with foci closer together?

Ellipse Now use taxicab metric First with the two points on a diagonal

Ellipse End result is an octagon Corners are where both sides intersect

Ellipse Now when foci are vertical

Ellipse End result is a hexagon Again, four of the sides are where sides of both “circles” intersect

Distance – Point to Line In Chapter 4 we used a circle Tangent to the line Centered at the point Distance was radius of circle which intersected line in exactly one point

Distance – Point to Line Apply this to taxicab circle Activity 5.8, finding radius of smallest circle which intersects the line in exactly one point Note: slope of line - 1 < m < 1 Rule?

Distance – Point to Line When slope, m = 1 What is the rule for the distance?

Distance – Point to Line When |m| > 1 What is the rule?

Parabolas Quadratic equations Parabola All points equidistant from a fixed point and a fixed line Fixed line called directrix

Taxicab Parabolas From the definition Consider use of taxicab metric

Taxicab Parabolas Remember All distances are taxicab-metric

Taxicab Parabolas When directrix has slope < 1

Taxicab Parabolas When directrix has slope > 0

Taxicab Parabolas What does it take to have the “parabola” open downwards?

Locus of Points Equidistant from Two Points

Taxicab Hyperbola

Equilateral Triangle

Axiom Systems Definition of Axiom System: We have seen A formal statement Most basic expectations about a concept We have seen Euclid’s postulates Metric axioms (distance) Another axiom system to consider What does between mean?

Application of Taxicab Geometry

Application of Taxicab Geometry We want to draw school district boundaries such that every student is going to the closest school. There are three schools: Jefferson at (-6, -1), Franklin at (-3, -3), and Roosevelt at (2,1). Find “lines” equidistant from each set of schools

Application of Taxicab Geometry Solution to school district problem

Taxicab Geometry Chapter 5