Lab 6: The Taxicab Metric

Write your name and your partner's name (if you have one) here.

The taxicab metric

The Euclidean metric (used in the Euclidean plane) is

. The distinct between two points is exactly the length of the line segment that connects them. The Taxicab metric on the Cartesian plane is

. This distance can be understood visually as the length of a shortest path taken between the two points that only moves horizontally or vertically.

The solid path from A to B is a taxicab path

Euclidean Distance

Find the Euclidean distance between and .

Taxicab distance

Find the distance between and in the taxicab metric.

Perimeter

In the graph below, move the points to form your own triangle and compute the perimeter of using the taxicab metric.

The set of points in our model of taxicab geometry is the usual Cartesian plane. The set of lines in this model is the usual set of lines in the plane. Let and be two points on the non-vertical line . Find . In particular, if you write , what is ?

Ruler Postulate

The ruler postulate says we must be able to assign real number coordinates to lines in a way that agrees with the given metric of the geometry. If

is the value you just found, then you can assign coordinates to the line

by giving a point

the coordinate

Protractor Postulate

Angle measure in taxicab geometry is the same as euclidean geometry. In particular, an angle in taxicab geometry is a right angle if and only if it is a right angle in Euclidean geometry.

Checking SAS

Triangle ABC

In the graph above, find the measure of ,, and angle

Triangle BCD

In the graph above, find the measure of ,, and angle

Your computations should show that under a certain correspondence of vertices, these two triangles satisfy the SAS criterion. Why are the two triangles not congruent in taxicab geometry?

Circles

The circle centered at with radius is the set . In other words, you want the set of all satisfying . Fix in the plot below. What does a circle look like in taxicab geometry?

Perpendicular Bisectors

Let and be distinct points. The perpendicular bisector of is the locus of points equidistant from and , that is, the set of points satisfying . Using the taxicab metric for , pick a pair of points in the graph below and plot the perpendicular bisector of . Is the perpendicular bisector a line?

Plot the perpendicular bisector of AB

Ellipses

An ellipse with foci and is the set of points such that the sum of distances from to and to is a fixed distance. This means you want the set of such that there is a fixed number such that . In the graph below, fix your points and choose a number . Plot the equation of this ellipse using the taxicab metric for . What does an ellipse look like in taxicab geometry?