# WRITE-UP 11

Assignment 11 is to investigate some polar equations using theorist. I have chosen to investigate problem 1, r = a + b cos(kt) for a = b, a< b, and a> b

when k = integer values from 0 to 360 degrees inclusive

The first three rgaphs show changes when a = b = 3, and k = 3

k = 4

and k = 5

The next group of graphs show a = 2, b = 3 ( a < b) while k = 3

k = 4

and k = 5.

The next group show a = 3, and b = 2 ( a > b ) while k = 3,

k = 4

and k = 5.

Wh en a = b, the k leaf rose emerges. With a < b, a k leaf rose and a smaller k leaf rose appear. With a > b, a k 'finger' curve with a hole in the middle appears.
k = 1/2, 3/2, 5/2,....... for values from 0 to 720 degrees inclusive

The first group show the variance when a = b = 3, and k = 1/2,

k = 3/2

and k = 5/2

The next group shows a = 2, b = 3 ( a < b ) and k = 1/2,

k = 3/2

and k = 5/2

The next group show a = 3,b = 2, ( a > b ) and k = 1/2,

k = 3/2

and k = 5/2.

When a = b, the figure resembles 2k arcs with 2k petals in the arcs. When a < b, the figure is similar to when a = b with a smaller petal inside each of the first group of petals. When a > b, the arcs are enlarged so that the petals do not join as 2k > 1. As 2k gets very large, the arcs should form a nearly circular pattern near the outside of the grid.

k = 1/3,2/3,4/3,......for values from 0 to 1080 degrees inclusive.

The first group show a = b = 3, and k = 1/3,

k = 2/3

and k = 4/3

The next group show a = 2, b = 3 ( a < b ) and k = 1/3

k = 2/3

and k = 4/3

The next group show a = 3, b = 2 ( a > b ) and k = 1/3

k = 2/3

and k = 4/3.

When a = b, the graph shows intersecting spirals, with a < b, a second group of small spirals emerge outside the original set of arcs, and when a > b, the second set of small spirals appear inside the larger set of arcs.

The pattern shows a start of k leafs to variations where the leafs become intersecting arcs and many more points must be plotted to show a smooth curve.

RETURN