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Drawing Birds in Flight With Mathematics

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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

2016-01-06-1452077407-8282695-Parrot.jpg

(x-A(k))+(y-B(k))=(R(k)),

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

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

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

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

Stork

2015-12-25-1451066985-8579815-Stork.jpg

(x-A(k))+(y-B(k))=(R(k)),

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

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

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

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

Magpie

2016-01-04-1451909798-3756357-Magpie.jpg

(x-A(k))+(y-B(k))=(R(k)),

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

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

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

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

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