## 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))^{6}sin((k/10000)^{7}(3π/5))+(9/7)(cos(37πk/10000))^{16}(cos(πk/20000))^{12}sin(πk/10000),

B(k)=(-5/4)(cos(37πk/10000))^{6}cos((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))^{6}sin((k/4000)^{7}(π/2)),

B(k)=-(cos(32πk/4000))^{6}cos((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))^{6}sin((k/10000)^{7}(π/2)),

B(k)=-(cos(41πk/10000))^{6}cos((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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