19.1 Taxicab Geometry

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19.1 Taxicab GeometryThe student will learn about:circles and parabolas in taxicab geometry.

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Introduction

We are going to examine a variety of geometric figures that use distance in their definitions. But first let us revisit our ruler.

Remember from the last class.Ruler Postulate - Examples

A (x 1, y 1) ↔ (1 + |m| ) x 1 = a

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This multiple of (1 + |m|) is a little odd as well as only using the x coordinate in finding the coordinate a. Let’s examine it a bit.

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mDistance from the origin is (1 + |m| ) 1 or (1 + |m| )x if you are at the x coordinate of the x-axis.

Axiomatics – Ruler PostulateWhat does the ruler look like?

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Then A (x 1, y 1) ↔ (1 + |m| ) x 1 = a

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DefinitionsLet A (0, 0). Graph all the points P so that PA = 6.

We did this last class period.

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Nice circle!!! What is it’s equation?

x 2 + y 2 = 6 2 ?No

|x| + |y| = 6

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More PlayGraph all the points on the circle with center at (1, 3) and radius 4.

|x - 1| + |y - 3| = 4

What is it’s equation?

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DefinitionsGraph the circles with center (0, 0) radius 2 and center (3, 0) with radius 3.

Notice – two circles intersecting in two points!

What are the possibilities?

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DefinitionsGraph the circles with center (0, 0) radius 4 and center (4, 0) with radius 4.

Note that these two circles mark the points equidistant from the centers.

We are going to use circles to measure distances.

Definitions

Just as a circle is all the points equidistant from a fixed point the other conics may be defined with respect to distance.

A parabola is all the points equidistant from a fixed point (focus) and a fixed line (directrix).

Taxicab ParabolasConsider the line that is the x-axis and the point F(0, 2). Find the set of points P so that the taxicab distance from the line is equal to the distance PF.

Circle of radius 4.Line parallel to the directrix 4 units away.

Taxicab ParabolasFind all the points equidistant from the point and line given below.

Taxicab ParabolasFind all the points equidistant from the point and line given below.

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Summary.

• We learned about taxicab parabolas.

• We learned about taxicab circles.

Assignment: §19.1