How to Measure Distance
Above are the distance formulas for the different geometries. On the left you will find the usual formula, which is under Euclidean Geometry. On the right you will find the formula for the Taxicab distance. This difference here is that in Euclidean distance you are finding the difference between point 2 and point one. While in Taxicab you are adding the distance. The reason for this difference is that, in taxicab, you are finding the distance around the grid to get to the third point. In Euclidean you are finding the direct line from point a to point b. I guess you could say that Taxicab takes the "scenic route."
This example shows the difference in how the two distances are measured. The Taxicab distance is always going to be greater because it is the absolute value of both of the legs of the triangle, and then added together. The Euclidean distance is the difference in position from Point A to Point B (the hypotenuse of this right triangle).
CIRCLES!
To the left you will see an image of a Euclidean circle (green) and a Taxicab circle (red). Since distance is measured differently in Taxicab, many of the shapes and concepts while using Taxicab distance look and act differently. The radius for both of these circles is 5cm. You will see, in light purple, how the Taxicab circle still has a radius of 5cm. Since we measure along a grid in Taxicab, you can see that the distance from the center of the circle to the point on the perimeter of the circle is still 5cm when you find the distance from x1 to x2, and then the distance from y1 to y2.
Side Angle Side
In Euclidean Geometry, the Side Angle Side Axiom states that if two triangles have a set of corresponding side, angle, side measures that are the same, then the two triangles are congruent. Being that this is an axiom, accepted truth, many things in Euclidean Geometry are based off of this. Under Taxicab, however, this is proven false. Below you will find a counterexample to show that SAS does not prove congruency in Taxicab geometry.
If you take a look at the pictures on the left, this is a simple way to show congruency and the criteria for SAS. According to Euclidean Geometry, these two triangles are congruent. However, this does not necessarily work when measuring in Taxicab distance.
Let's look at the two triangles on the upper right-hand side. The two corresponding legs have the same measure, two units. Now, examine the hypotenuse of each of the two triangles. One measures two units, and one measures four units. When you measure along the grid in Taxicab geometry, these are the measures you get for all the sides of the triangles. This disproves SAS in Taxicab geometry because, if we are using the legs of the triangles and the right angle for the criteria, they are supposed to be congruent. Upon further examination, we found that they are not congruent figures. In Taxicab Geometry, you have to find every side and angle measure to prove congruency. |