## 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
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)=(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

This image shows all circles of the form

(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.

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