Drawing Birds in Flight With Mathematics

In this post you can see three images with their mathematical descriptions. I have defined them by trigonometric functions. In order to create such shapes, it is very useful to know the properties of the trigonometric functions. I believe these images show us an important fact:
We can draw with mathematical formulas.

In addition to the images in this post, you can see A Bird In Flight, Fish, Boat and three images in one of my previous posts: These Are Mathematical Sets.

Parrot

(x-A(k))²+(y-B(k))²=(R(k))²,

for k=-10000, -9999, ... , 9999, 10000, where

A(k)=(3k/20000)+(cos(37πk/10000))⁶sin((k/10000)⁷(3π/5))+(9/7)(cos(37πk/10000))¹⁶(cos(πk/20000))¹²sin(πk/10000),

B(k)=(-5/4)(cos(37πk/10000))⁶cos((k/10000)⁷(3π/5))(1+3(cos(πk/20000)cos(3πk/20000))⁸)+(2/3)(cos(3πk/200000)cos(9πk/200000)cos(9πk/100000))¹²,

R(k)=(1/32)+(1/15)(sin(37πk/10000))²((sin(πk/10000))²+(3/2)(cos(πk/20000))¹⁸).

Stork

(x-A(k))²+(y-B(k))²=(R(k))²,

for k=-4000, -3999, ... , 3999, 4000, where

A(k)=(3k/4000)+(cos(32πk/4000))⁶sin((k/4000)⁷(π/2)),

B(k)=-(cos(32πk/4000))⁶cos((k/4000)⁷(π/2))(1+(cos(πk/8000)cos(3πk/8000))⁴)+3(cos(πk/8000)cos(3πk/8000))¹⁶(cos(16πk/4000))⁹,

R(k)=(1/30)+(1/15)(cos(πk/8000)cos(5πk/8000))¹⁰(1-(1/2)(cos(32πk/4000))¹²)+(1/7)(sin(32πk/4000))⁴(sin(πk/4000))².

Magpie

This image shows all circles of the form

(x-A(k))²+(y-B(k))²=(R(k))²,

for k=-10000, -9999, ... , 9999, 10000, where

A(k)=(11k/100000)+(cos(41πk/10000))⁶sin((k/10000)⁷(π/2)),

B(k)=-(cos(41πk/10000))⁶cos((k/10000)⁷(π/2))(1+(5/2)(cos(3πk/100000)cos(9πk/100000))⁸)+(1/2)(cos(πk/40000)cos(3πk/40000)cos(3πk/20000))¹⁰,

R(k)=(1/50)+(1/20)(sin(41πk/10000)sin(πk/10000))².