Rayen Antillanca. Assignment 11

The polar coordinate system is a two dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.

The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle.

The equations which connect the Cartesian coordinates with the polar coordinates are the next:

Now, I am going to investigate the next equation

First Investigation

 Let a=1, b=1 and diferente values of k. Then When k varies from negative integer values to positives integer values this happens When k varies, the equation takes the form of a flower which has k leaves. If k>0 or k<0, it does not matter, the flower has the same number of leaf. This occurs when a=b and k is an integer number. Some textbooks called this n-leaf rose, where n=k. Actually the name is obvius, the form of the graph is like a rose where k is the number of leaf.

 This rose curve describe a family of curves where the length of the patals are the sum of a+b when a=b For example you can see the next image Where the purple rose has 5 petals and the length of its leaf is 2, because a=1 The length of the leaf of red one is 4, because a=2 The length of the leaf of blue one is 6, because a=6 The length of the leaf of green one is 8, because a=4

Second Investigation

 When a varies and b=1 and k=1. Then I am going to take small values for a When a takes values close to zero, the shape of the curve trends to a “smooth heart”. When the values begin to move away from zero, this curve trends to a take circular shape. When b=1 and k=1, these curves as known as limacon, which is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids.

Third Investigation

 Now, I am going to compare two different equations The blue curve has the form of circle, the red one also. Why? As b=1, and cos(0)=1 in the red circle the radius is 1, and in the red circle the radious is 2. The red curve is still a circle but the blue curve is a limacon.

 Another comparison As you can see, the both curves have a flower-like shape

Fourth Investigation

 If cos() is replaced by sin() If I replace cos() by sin() this two functions have the same flower-like shape. But, the blue flower is rotated around the origin with respect to the red flower.

 Note: All graphs of this webpage were made with Graphing Calculator 4.0