PreCalc*B: More Polar Curves

Kieran Dias-Lalcaca, Jimmy Davey, Quintin Campbell

Table of Contents:

What are More Polar Curves?
Parameters
- General Equation
- a, b, n
General Rules
- The Sine and Cosine Functions
- The Effect of a and b
- The Effect of n
Family Tree
- Roses, Cardioids and Limacons, Circles, Nothing
Hypothetical Situations
Converting to Cartesian Form

What are "More Polar Curves?"

More Polar Curves is a Blanket Statement for a Group of Polar Curves

In this group we see a general heading for polar curves that don't quite fit in any of the other sections disscussed, yet follow a pattern similar to many that are seen throughout polar function groups. Specifically multi-pedaled curves, and complex shapes derving from them fall into this group.

Parameters

General Equation

General Equation is represented in two base forms:

r=a+bcos(nθ)

r=a+bsin(nθ)

Reguardless of whether it is sine or cosine, the varibles still work the same way.

a

a represents the stretching and shrinking (in terms of radius) of the "petals"

b

b represents the stretching and shrinking (in terms of radius) of the "petals"

n

n represents the total number of "petals" that the shape of the graph has, argubly the most intresting of the varibles

General Rules

The Base Sine and Cosine Functions

Similarly to sine and cosine functions on a rectangular plane, sine and cosine functions have different graphs on a polar plane.

Sine Function

Cosine Function

Notice: The cosine function has a rotation of -π/2 radians compared to the sine function

This is shown as the cosine function looks to be the sine function turned on its side, representing the rotation that it underwent

So: a + bsin(nθ+(π/2)) = a + bcos(nθ)

The Effects of a and b

The sum of abs(a+b) is equal to the radius of the circle which borders the resulting graph
When the absolute value of a and b are equal the petals meet at the origin
When the absolute value of b is greater than the absolute value of a more petals (which are smaller) appear
When the absolute value of b is less than the absolute value of a, there are no petals (more circular-like shape)
The radius of the smaller petals is the difference of abs(a-b)

The Effects of n

When n is even, the smaller petals are inside of the large petals
When n is odd, the smaller petals are outside of the large petals
n, represnts the number of petals at whole number intervals, a pattern that shows the importance of the smallest varation in these varibles

Family Tree

Roses

When a=0 and n≠0,1

Cardioids and Limacons

When a≠0 and b≠0 and n=1

Circles

When a=0 or b=0 and n=1,-1

Nothing

When n=0 and a=0 or a=0 and b=0

Hypothetical Situations

What happens as Theta, the angle of rotation, approaches infinity and negative infinity?

As θ approaches infinity, the curve repeats itself every period of 2π, as n=ℤ.
As θ approaches negative infinity, the curve also repeats itself every period of 2π as n=ℤ.

What happens as you change N to a multiple of 0.5(...1.5,2.5 etc)

A strange, yet intresting pattern can be found at the .5 interveles in the N value as a varible. Adding 1 from each iteration of this interval, the number of petals represnted increases by two, following a odd number patter from 3 at 1.5 to 5 at 2.5, 7 at 3.5, and so on and so forth. This is perticularly intresting as a pattern as we know that when n is equal to whole numbers that number is the number of petals that are shown on the graph, yet when we do the same intervel between two numbers yet start at a offset of .5 the number of petals completly shifts and becomes a pattern of odd numbers.

Converting to Cartesian Form

Due to the complex nature of the varible of N, and the idea of the drastic changes that occuer throughout the manipulation of this varible, solving for a cartesian equation is nearly impossible, and very impractical. However, specific equations are able to be converted into cartesian, the general case applies for most of these curves.

Changing just a

Changing just b

Changing just k

r = a + cos(3θ)

r = bcos(3θ)

r = 5cos(kθ)

k as a Fraction

While it is extremely difficult to predict the shape of the curve when k is a fraction, we can sum up what the general shape of the curves through qualitative observations. By our observations, when k is a fraction, the petals are much larger and overlap (as opposed to thinner petals that are symmetrically distributed about the pole).

r = 1 + cos(θ/3) on left; r = 1 + cos(3θ/4) in middle; r = 1 + cos(12θ/7) on right

k as an Irrational Number

Not many observations can be made about rose curves in which k is irrational, but we can say that there are probably infinite petals and there does seem to be some kind of symmetry at play.

r = 1 + cos((√2)θ) on left; r = 1 + cos(πθ) in middle; r = 1 + cos(eθ) on right

Page updated

Report abuse