Homework SourseThe mathematical constant pi is defined as the ratio of the
The mathematical constant (pi) is defined as the ratio of the circumference of a circle to its diameter. What is the value of pi in the taxicab metric?
The mathematical constant (pi) is defined as the ratio of the circumference of a circle to its diameter. What is the value of pi in the taxicab metric?
Solution
pi=c/d
it would be because in taxicab geometry, the metric between vectors a and b with components a_i and b_i respectively is Sumision(i=1to n)|a_i -b_i|. in this geometry a circle is actually a square where the sides of square meet at 45deg angles with the coordinte axes.
Given that the sides of the square are lenghth s, using the Taxicab Metric, it is easy to verify that s=2r, r= radius.Since circumference is just perimeter then the circumference would be 4(2r). But 2r is the diameter, which means that the analogous value of pi is 4 in the Taxicab geometry.
