Taxicab Geometry II Math 653 Fall 2012 December 5, 2012

Schedule Next week: Do project presentations, turn in project write-up Next week: Hand out take-home final Take-home final is due to my email sspitzer@towson.edu or to admin assistants upstairs by 5pm on Wednesday, 12/19. (To drop off hard copy, walk in to the glass doors of math suite and hand to Scott or Diana or Charlotte) HARD DEADLINE! Open office hours December 17, 4-7pm. In Room 355. (After 5pm call me @301-275-7518 to get into math suite) Also available to answer questions by email. OR, schedule a phone or Skype meeting.

Taxicab Geometry – Recap Last week, we talked about the following ideas in taxicab geometry: New distance formula Idea that the length of a line is dependent on its position Applications to urban geography (apartment problems) Taxicab circles (value of π) Midset of two points (the points equidistant from each of those two points) Using midset ideas in urban applications (school district problem)

Taxicab Geometry – New Problem How can we find the distance from a point to a line in taxicab geometry? (Note: How do we find the distance from a point to a line in Euclidean geometry?) Definition: The distance from a point A to a line l is defined as the minimum distance between the A and a point on l. (i.e. we find the closest point on line l to point A and measure their distance)

Distance from a Point to a Line Use the next 10-15 minutes and determine a procedure for finding the distance between a point A and a line l in taxicab geometry. Remember, different positions for a point and a line might give you different results (just like we got different results for the midset depending on the relationship between the points) Try a variety of situations to test your conjectures.

Angles in Taxicab Geometry Angles in taxicab geometry are slightly complex. But, we can get a good sense of them by starting with the more basic angles, which will be enough to study triangles. Recall: We measure angles in Euclidean or Spherical geometry by specifying a unit of angle (which itself is an angle, measured in terms of how it intersects with a circle) and measure other angles in terms of that unit angle. In taxicab geometry, call the unit angle a Taxicab Radian, symbolized rT

Taxicab Angles A taxicab radian rT refers to the angle which intersects a unit taxicab “circle” to cut an “arc” length of 1 taxicab unit.

Taxicab Angles Given this definition, how could we label right angles and straight angles? How many taxicab radians are in a full circle? Be careful! Taxicab angles are dependent on their position and may change when rotated (just like other items). However, all right angles are congruent and all equal to 2 rT

A few further ideas Taxicab distance remains the same under the transformation of translation, as well as reflection over a horizontal or vertical line Other lines of reflection, and rotation, generally do not preserve distance. (So, the shape and size of items, including angles, will stay the same if we translate them, but not if we rotate them.)

A few further ideas So, if this is the case, what can we say about parallel lines and a transversal? Corresponding & alternate interior angles are congruent, vertical angles are also congruent.

Triangles What can we say about taxicab triangles? A triangle is the union of three non-collinear points and the segments which join them. What properties of triangles hold? What properties do not hold? Think about congruence conditions (SSS, SAS, etc) Similarity? Pythagorean theorem? Angle sums? Isosceles triangle theorem?

Triangles What can we say about taxicab triangles? A triangle is the union of three non-collinear points and the segments which join them. What properties of triangles hold? What properties do not hold?

What about the midset of two lines? Can we find the midset of two lines? That is, we are looking for the set of all points that are equidistant from two lines l and m. What is the midset of two lines in Euclidean geometry? How about in Taxicab geometry?