It turns out that there are different types of geometries. Euclidean Geometry is
ID: 3109658 • Letter: I
Question
It turns out that there are different types of geometries. Euclidean Geometry is but one type of geometry based upon the assumption that given a line and and a point not on the line, there is one and only line that passes through that point and is parallel to the given line. When we compute distance in Euclidean Geometry, we use the Pythagorean Theorem. There is another type of geometry called Taxicab Geometry. The basic idea is that taxicabs cannot necessarily travel in a straight line path get from point A to point B but instead must use a triangular path (see below): To then calculate the distance between the points A and B, rather than use the Pythagorean Theorem, we use the following metric (for point A with coordinates (x_1, y_1) and B with coordinates (x_2, y_2)): d(A, B) = |x_1 - x_2| + |y_1 - y_2|. Give the definition of a circle (note that this definition is independent of the geometry being used and hence is NOT an equation necessarily!) Give the definition of an ellipse (note that this definition is independent of the geometry being used and hence is NOT an equation necessarily!) Find the equation of a circle in taxicab geometry that is centered at the point (h, k) and has a radius of r. Draw the graph of the circle. Discuss how you know the equation is what it is. Find the equation of an ellipse in taxicab geometry that is centered at (h, k) and has major and minor axes of length a and b respectively. Draw the graph of the ellipse. Then discuss how you the equation is what is.Explanation / Answer
The definition of circle is independent of any geometry being used and it is: the locus of the points whose distance from a fixed point is is constant. So according to taxicab geometry here it is D(A, B)=r. There fixed point is called center of the circle and the constant is called radius of circle.
B) the definition of ellipse is that it is locus of a point the sum of whose distances from two fixed points is constant. The two fixed points are called the foci of ellipse.
C) the equation of circle with center is point(h,k) and radius r is
|x-h| +|y-k|=r
D) the loci of ellipse are: (-a/2,0) and (a/2,0) so the equation of ellipse is:
(|x+a/2|+|x-a/2|) + (|y-0|+|y-0|) = 2a
Because 2a is the constant and can found out by summing the distances from the two foci from the intersection of ellipse with x axis.
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