tag:blogger.com,1999:blog-22619853616144993422024-01-22T19:28:12.568-08:00Fun math art (pictures) - benice equationMaking Art with Mathematical Equations, Inequalities and Functions (Mappings, Transformations)benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.comBlogger240125tag:blogger.com,1999:blog-2261985361614499342.post-14553566164273431292019-01-18T08:27:00.000-08:002019-01-26T16:49:42.604-08:00Nested Ellipses (Ellipse Whirl)<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-Twzs5ouzTNQ/XEHVskId7II/AAAAAAAAEPA/i_bFvFlrHv0KLFPxejJ3CIZzgKJ7SiWAQCLcBGAs/s1600/nested_ellipses_01.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" src="https://2.bp.blogspot.com/-Twzs5ouzTNQ/XEHVskId7II/AAAAAAAAEPA/i_bFvFlrHv0KLFPxejJ3CIZzgKJ7SiWAQCLcBGAs/s1600/nested_ellipses_01.gif" /></a></div>
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These animations are constructed by shrinking and rotating</div>
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a sequence of concentric and similar ellipses,</div>
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so that each ellipse lies inside the previous ellipse and is tangent to it.</div>
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這些動畫是藉由收縮和旋轉</div>
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一序列同心且相似的橢圓來構造的,</div>
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使得每個橢圓都位於前一個橢圓內並與之相切。</div>
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<a href="https://2.bp.blogspot.com/-dL3-gBMJ7ko/XEHWLGhIptI/AAAAAAAAEPI/hhIjGYsQXN0ut_bmeqYCSROgBWGnIm2OwCLcBGAs/s1600/nested_ellipses_02.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" src="https://2.bp.blogspot.com/-dL3-gBMJ7ko/XEHWLGhIptI/AAAAAAAAEPI/hhIjGYsQXN0ut_bmeqYCSROgBWGnIm2OwCLcBGAs/s1600/nested_ellipses_02.gif" /></a></div>
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All the angles of rotations are equal to 5°.</div>
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<a href="https://2.bp.blogspot.com/-7wxUkAZEzTk/XEHWhUtK2oI/AAAAAAAAEPQ/EHrG6ZwQTrAJG4T92PQW2qUwUd0bmaZEQCLcBGAs/s1600/nested_ellipses_03.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" src="https://2.bp.blogspot.com/-7wxUkAZEzTk/XEHWhUtK2oI/AAAAAAAAEPQ/EHrG6ZwQTrAJG4T92PQW2qUwUd0bmaZEQCLcBGAs/s1600/nested_ellipses_03.gif" /></a></div>
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All the angles of rotations are equal to 10°.</div>
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<a href="https://1.bp.blogspot.com/-UJ1bgdUo378/XEHW7LokjNI/AAAAAAAAEPY/Od6fqDKx8kYjcWv7b-xxc9kZXggl626RQCLcBGAs/s1600/nested_ellipses_04.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" src="https://1.bp.blogspot.com/-UJ1bgdUo378/XEHW7LokjNI/AAAAAAAAEPY/Od6fqDKx8kYjcWv7b-xxc9kZXggl626RQCLcBGAs/s1600/nested_ellipses_04.gif" /></a></div>
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All the angles of rotations are equal to 15°.</div>
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<a href="https://3.bp.blogspot.com/-J_h_g5JQLnk/XEHXK99mffI/AAAAAAAAEPc/aUoreZfJ1BQo1ggVGZFhubIr8YpZhNCMACLcBGAs/s1600/nested_ellipses_05.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" src="https://3.bp.blogspot.com/-J_h_g5JQLnk/XEHXK99mffI/AAAAAAAAEPc/aUoreZfJ1BQo1ggVGZFhubIr8YpZhNCMACLcBGAs/s1600/nested_ellipses_05.gif" /></a></div>
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The angles of rotations are 0.5°, 1°, 1.5°, 2°, .....</div>
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<a href="https://3.bp.blogspot.com/-USD8xrL-Fw4/XEz_hNqIsEI/AAAAAAAAEQA/SGYCNIdHyMANaBh81lTLZbVUzaKT1bapQCLcBGAs/s1600/nested_ellipses_08.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" src="https://3.bp.blogspot.com/-USD8xrL-Fw4/XEz_hNqIsEI/AAAAAAAAEQA/SGYCNIdHyMANaBh81lTLZbVUzaKT1bapQCLcBGAs/s1600/nested_ellipses_08.gif" /></a></div>
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All the angles of rotations are equal to 5°.</div>
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<a href="https://4.bp.blogspot.com/-IdP7_KosFIE/XEHXgv3eKgI/AAAAAAAAEPs/gXi8yu_O-38e7npAPpqmGgw9QwJQNiDVACLcBGAs/s1600/nested_ellipses_06.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" src="https://4.bp.blogspot.com/-IdP7_KosFIE/XEHXgv3eKgI/AAAAAAAAEPs/gXi8yu_O-38e7npAPpqmGgw9QwJQNiDVACLcBGAs/s1600/nested_ellipses_06.gif" /></a></div>
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All the angles of rotations are equal to 5°.</div>
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<a href="https://2.bp.blogspot.com/-1PIFRPw-YT0/XEHX8S0opWI/AAAAAAAAEP0/RDtKVloXIOow0puq9sH2ZYfuiJvTu727gCLcBGAs/s1600/nested_ellipses_07.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" src="https://2.bp.blogspot.com/-1PIFRPw-YT0/XEHX8S0opWI/AAAAAAAAEP0/RDtKVloXIOow0puq9sH2ZYfuiJvTu727gCLcBGAs/s1600/nested_ellipses_07.gif" /></a></div>
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All the angles of rotations are equal to the value of the animation parameter.</div>
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<b style="font-size: x-large;">Related posts:</b><br />
<ul>
<li><a href="http://benice-equation.blogspot.com/2013/06/nested-regular-polygons.html" target="_blank">Nested Regular Polygons</a></li>
<li><a href="http://benice-equation.blogspot.com/2013/06/nested-stars.html" target="_blank">Nested Stars</a></li>
</ul>
<br />
<br />
( Mathematical software used: GeoGebra )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com6tag:blogger.com,1999:blog-2261985361614499342.post-56158242164389735612017-12-12T08:18:00.000-08:002017-12-12T09:06:17.074-08:00Rotational Symmetry of Order 4Making Patterns with Rotational Symmetry of Order 4 Using a n by n Grid of Random Shaded Squares<br />
<br />
<span style="font-size: large;">(1) n is even</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-If09WcV98eE/Wi_0kdl35VI/AAAAAAAAEN0/aUgmJErIJVYlteYUJkEeMhl6O5t4e6gLQCLcBGAs/s1600/rotational_symmetry_of_order_4_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="600" data-original-width="600" height="400" src="https://2.bp.blogspot.com/-If09WcV98eE/Wi_0kdl35VI/AAAAAAAAEN0/aUgmJErIJVYlteYUJkEeMhl6O5t4e6gLQCLcBGAs/s400/rotational_symmetry_of_order_4_01.png" width="400" /></a></div>
<div style="text-align: center;">
(n = 12)</div>
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<div style="text-align: left;">
<u><b>gnuplot script:</b></u></div>
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<br /></div>
n = 12<br />
m = n/2<br />
max(x,y) = x>y?x:y<br />
a = 0.875<br />
f(t) = a*0.5*cos(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))<br />
g(t) = a*0.5*sin(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))<br />
funX(i,t) = f(t) + 0.5 + i<br />
funY(j,t) = g(t) + 0.5 + j<br />
set xrange [-m-1:m+1]<br />
set yrange [-m-1:m+1]<br />
unset key<br />
unset tics<br />
unset border<br />
set margins 0,0,0,0<br />
set size ratio -1<br />
set samples 1000<br />
set parametric<br />
set trange [0:1]<br />
Color = "#00cc00"<br />
set style fill solid noborder<br />
set terminal wxt 0 size 600,600 position 0,0 title "n = ".n<br />
seed = ceil(2147483646*rand(time(0)))<br />
random_test = rand((time(0),seed))<br />
set multiplot<br />
do for [i=0:m-1]{<br />
do for [j=0:m-1]{<br />
b = rand(0)>0.5?1:0<br />
if (!b) {continue}<br />
plot funX(i,t), funY(j,t) w filledc closed lw 1 lc rgb Color, \<br />
-funY(j,t), funX(i,t) w filledc closed lw 1 lc rgb Color, \<br />
-funX(i,t), -funY(j,t) w filledc closed lw 1 lc rgb Color, \<br />
funY(j,t), -funX(i,t) w filledc closed lw 1 lc rgb Color<br />
}<br />
}<br />
unset multiplot<br />
<div>
<br /></div>
<div style="text-align: left;">
<br /></div>
<span style="font-size: large;">(2) n is odd</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-TLVRVljCz_U/Wi_1qNzHaHI/AAAAAAAAEOA/3uwZKb-2rNwmCPWu7cA-od4pKGaRW0lsgCLcBGAs/s1600/rotational_symmetry_of_order_4_02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="600" data-original-width="600" height="400" src="https://3.bp.blogspot.com/-TLVRVljCz_U/Wi_1qNzHaHI/AAAAAAAAEOA/3uwZKb-2rNwmCPWu7cA-od4pKGaRW0lsgCLcBGAs/s400/rotational_symmetry_of_order_4_02.png" width="400" /></a></div>
<div style="text-align: center;">
(n = 11)</div>
<br />
<u><b>gnuplot script:</b></u><br />
<br />
n = 11<br />
m = (n+1)/2.<br />
max(x,y) = x>y?x:y<br />
a = 0.875<br />
f(t) = a*0.5*cos(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))<br />
g(t) = a*0.5*sin(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))<br />
funX(i,t) = f(t) + i<br />
funY(j,t) = g(t) + j<br />
set xrange [-n/2.-1:n/2.+1]<br />
set yrange [-n/2.-1:n/2.+1]<br />
unset key<br />
unset tics<br />
unset border<br />
set margins 0,0,0,0<br />
set size ratio -1<br />
set samples 1000<br />
set parametric<br />
set trange [0:1]<br />
Color = "#00cc00"<br />
set style fill solid noborder<br />
set terminal wxt 0 size 600,600 position 0,0 title "n = ".n<br />
seed = ceil(2147483646*rand(time(0)))<br />
random_test = rand((time(0),seed))<br />
set multiplot<br />
b = rand(0)>0.5?1:0<br />
if (b) {plot f(t), g(t) w filledc closed lw 1 lc rgb Color}<br />
do for [i=1:m-1]{<br />
do for [j=0:m-1]{<br />
b = rand(0)>0.5?1:0<br />
if (!b) {continue}<br />
plot funX(i,t), funY(j,t) w filledc closed lw 1 lc rgb Color, \<br />
-funY(j,t), funX(i,t) w filledc closed lw 1 lc rgb Color, \<br />
-funX(i,t), -funY(j,t) w filledc closed lw 1 lc rgb Color, \<br />
funY(j,t), -funX(i,t) w filledc closed lw 1 lc rgb Color<br />
}<br />
}<br />
unset multiplot<br />
<div>
<br /></div>
<br />
<span style="font-size: large;">(3) n is even (Non-Reflectional Symmetry)</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-zR4wZtVLLkg/Wi_2TOBglsI/AAAAAAAAEOI/J3Qag-ucgr8wfrnfbkjub_o_kw5577j-QCLcBGAs/s1600/rotational_symmetry_of_order_4_03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="600" data-original-width="600" height="400" src="https://3.bp.blogspot.com/-zR4wZtVLLkg/Wi_2TOBglsI/AAAAAAAAEOI/J3Qag-ucgr8wfrnfbkjub_o_kw5577j-QCLcBGAs/s400/rotational_symmetry_of_order_4_03.png" width="400" /></a></div>
<div style="text-align: center;">
(n = 12)</div>
<br />
<u><b>gnuplot script:</b></u><br />
<br />
n = 12<br />
m = n/2<br />
max(x,y) = x>y?x:y<br />
f(t) = 0.5*cos(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))<br />
g(t) = 0.5*sin(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))<br />
funX(i,t) = f(t) + 0.5 + i<br />
funY(j,t) = g(t) + 0.5 + j<br />
set xrange [-m-1:m+1]<br />
set yrange [-m-1:m+1]<br />
unset key<br />
unset tics<br />
unset border<br />
set margins 0,0,0,0<br />
set size ratio -1<br />
set samples 1000<br />
set parametric<br />
set trange [0:1]<br />
Color = "#00cc00"<br />
set terminal wxt 0 size 600,600 position 0,0 title "n = ".n<br />
seed = ceil(2147483646*rand(time(0)))<br />
random_test = rand((time(0),seed))<br />
set multiplot<br />
do for [i=0:m-1]{<br />
do for [j=0:i]{<br />
b = rand(0)>0.5?1:0<br />
if (b) {<br />
plot funX(i,t), funY(j,t) w filledc closed lw 1 lc rgb Color, \<br />
-funY(j,t), funX(i,t) w filledc closed lw 1 lc rgb Color, \<br />
-funX(i,t), -funY(j,t) w filledc closed lw 1 lc rgb Color, \<br />
funY(j,t), -funX(i,t) w filledc closed lw 1 lc rgb Color<br />
}<br />
if (j==i) {continue}<br />
if (!b) {<br />
plot funX(j,t), funY(i,t) w filledc closed lw 1 lc rgb Color, \<br />
-funY(i,t), funX(j,t) w filledc closed lw 1 lc rgb Color, \<br />
-funX(j,t), -funY(i,t) w filledc closed lw 1 lc rgb Color, \<br />
funY(i,t), -funX(j,t) w filledc closed lw 1 lc rgb Color<br />
}<br />
}<br />
}<br />
unset multiplot<br />
<div>
<br /></div>
<br />
<span style="font-size: large;">(4) n is odd (Non-Reflectional Symmetry)</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-IOfRIsC8x-Y/Wi_2n2MFzJI/AAAAAAAAEOQ/jl4ZvW9IkPkEhQtOtXu7lxt0vnKqkxA8wCLcBGAs/s1600/rotational_symmetry_of_order_4_04.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="600" data-original-width="600" height="400" src="https://3.bp.blogspot.com/-IOfRIsC8x-Y/Wi_2n2MFzJI/AAAAAAAAEOQ/jl4ZvW9IkPkEhQtOtXu7lxt0vnKqkxA8wCLcBGAs/s400/rotational_symmetry_of_order_4_04.png" width="400" /></a></div>
<div style="text-align: center;">
(n = 11)</div>
<br />
<u><b>gnuplot script:</b></u><br />
<br />
n = 11<br />
m = (n+1)/2.<br />
max(x,y) = x>y?x:y<br />
f(t) = 0.5*cos(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))<br />
g(t) = 0.5*sin(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))<br />
funX(i,t) = f(t) + i<br />
funY(j,t) = g(t) + j<br />
set xrange [-n/2.-1:n/2.+1]<br />
set yrange [-n/2.-1:n/2.+1]<br />
unset key<br />
unset tics<br />
unset border<br />
set margins 0,0,0,0<br />
set size ratio -1<br />
set samples 1000<br />
set parametric<br />
set trange [0:1]<br />
Color = "#00cc00"<br />
set terminal wxt 0 size 600,600 position 0,0 title "n = ".n<br />
seed = ceil(2147483646*rand(time(0)))<br />
random_test = rand((time(0),seed))<br />
set multiplot<br />
do for [i=0:m-1]{<br />
do for [j=0:i]{<br />
b = rand(0)>0.5?1:0<br />
if (b) {<br />
plot funX(i,t), funY(j,t) w filledc closed lw 1 lc rgb Color, \<br />
-funY(j,t), funX(i,t) w filledc closed lw 1 lc rgb Color, \<br />
-funX(i,t), -funY(j,t) w filledc closed lw 1 lc rgb Color, \<br />
funY(j,t), -funX(i,t) w filledc closed lw 1 lc rgb Color<br />
}<br />
if (j==0 || j==i) {continue}<br />
if (!b) {<br />
plot funX(j,t), funY(i,t) w filledc closed lw 1 lc rgb Color, \<br />
-funY(i,t), funX(j,t) w filledc closed lw 1 lc rgb Color, \<br />
-funX(j,t), -funY(i,t) w filledc closed lw 1 lc rgb Color, \<br />
funY(i,t), -funX(j,t) w filledc closed lw 1 lc rgb Color<br />
}<br />
}<br />
}<br />
unset multiplot<br />
<div>
<br /></div>
<div>
<br /></div>
<div>
<span style="font-size: large;">Graph files:</span></div>
<div>
<a href="http://forum.padowan.dk/viewtopic.php?id=1079" target="_blank"><span style="font-size: large;">Download page</span></a></div>
<div>
<br /></div>
<div>
<br /></div>
<div>
( Mathematical softwares used: <a href="http://www.padowan.dk/" target="_blank">Graph</a>, <a href="http://gnuplot.sourceforge.net/" target="_blank">gnuplot</a> )</div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-33460804784983096312017-11-12T21:53:00.000-08:002017-11-12T22:12:18.842-08:00Swastika Tiling<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-4bnwHayI738/WgktFyx0dEI/AAAAAAAAEMo/DKWkKpH0jKIoJnysAOITGYOIpBF4IolXgCLcBGAs/s1600/standard_swastika.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="400" src="https://1.bp.blogspot.com/-4bnwHayI738/WgktFyx0dEI/AAAAAAAAEMo/DKWkKpH0jKIoJnysAOITGYOIpBF4IolXgCLcBGAs/s400/standard_swastika.png" width="400" /></a></div>
<div style="text-align: center;">
An implicit curve: swastika(x,y) = 0</div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: center;">
The swastika here is an ancient Buddhist symbol, not the Nazi one.</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-qqAHSg8pT7E/Wgku_-E62oI/AAAAAAAAEM0/OBw_ZfvYtEImlnqAss-BLDJfvIP5ECaXgCLcBGAs/s1600/swastika_tiling_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="600" data-original-width="600" height="400" src="https://1.bp.blogspot.com/-qqAHSg8pT7E/Wgku_-E62oI/AAAAAAAAEM0/OBw_ZfvYtEImlnqAss-BLDJfvIP5ECaXgCLcBGAs/s400/swastika_tiling_01.png" width="400" /></a></div>
<div style="text-align: center;">
Composition of swastika(x,y), arctan(x) and tan(x)</div>
<div style="text-align: center;">
(k = 0.25)</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-rTsAKORYdZw/WgkvakdKXqI/AAAAAAAAEM4/iOtK5RKjBKcuRBL_qxec5u2GpoFbi_YwwCLcBGAs/s1600/swastika_tiling_02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="600" data-original-width="600" height="400" src="https://1.bp.blogspot.com/-rTsAKORYdZw/WgkvakdKXqI/AAAAAAAAEM4/iOtK5RKjBKcuRBL_qxec5u2GpoFbi_YwwCLcBGAs/s400/swastika_tiling_02.png" width="400" /></a></div>
<div style="text-align: center;">
Composition of swastika(x,y), arctan(x) and tan(x)</div>
<div style="text-align: center;">
(k = 0.125)</div>
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<a href="https://1.bp.blogspot.com/-Xena02hqHY8/WgkvuOo8wFI/AAAAAAAAENA/aQ4NXwQkb0kHovrGnZKQ_C6Z2r6CxPzBACLcBGAs/s1600/swastika_tiling_03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="600" data-original-width="600" height="400" src="https://1.bp.blogspot.com/-Xena02hqHY8/WgkvuOo8wFI/AAAAAAAAENA/aQ4NXwQkb0kHovrGnZKQ_C6Z2r6CxPzBACLcBGAs/s400/swastika_tiling_03.png" width="400" /></a></div>
<div style="text-align: center;">
Composition of swastika(x,y), arctan(x) and tan(x)</div>
<div style="text-align: center;">
(k = 0.4)</div>
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<a href="https://3.bp.blogspot.com/-A_Y0BzQbpy4/WgkwAzpk2dI/AAAAAAAAENE/4UGoV5Gcvj02VycanfpXgXbvQTTiFO49gCLcBGAs/s1600/swastika_tiling_04.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="600" data-original-width="600" height="400" src="https://3.bp.blogspot.com/-A_Y0BzQbpy4/WgkwAzpk2dI/AAAAAAAAENE/4UGoV5Gcvj02VycanfpXgXbvQTTiFO49gCLcBGAs/s400/swastika_tiling_04.png" width="400" /></a></div>
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Composition of swastika(x,y), arcsin(x) and sin(x)</div>
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(k = 0.25)</div>
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<div style="text-align: center;">
Composition of swastika(x,y), arcsin(x) and sin(x)</div>
<div style="text-align: center;">
(k = 0.4)</div>
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<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
Swastika of Swastikas</div>
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(k = 1/4, 5/24)</div>
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Swastika of Swastikas of Swastikas</div>
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(k = 1/4, 1/4, 5/24)</div>
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<a href="https://2.bp.blogspot.com/-k7g1AABEDcs/WgkxUzEFyII/AAAAAAAAENc/7Y4LdomcIYg3r-JXz7Tn3bqRXTis0QX4gCLcBGAs/s1600/swastika_tiling_08.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="565" data-original-width="527" height="400" src="https://2.bp.blogspot.com/-k7g1AABEDcs/WgkxUzEFyII/AAAAAAAAENc/7Y4LdomcIYg3r-JXz7Tn3bqRXTis0QX4gCLcBGAs/s400/swastika_tiling_08.png" width="372" /></a></div>
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Swastika of Swastikas of Swastikas</div>
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(Unicode of 卍: 0x534D)</div>
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<div style="text-align: center;">
A Maze of Swastikas</div>
<div style="text-align: center;">
(k = 0.1)</div>
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( Mathematical software used: <a href="http://www.padowan.dk/" target="_blank">Graph</a> )<br />
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<br />
<span style="font-size: large;">Related posts:</span><br />
<ul>
<li><a href="http://benice-equation.blogspot.com/2016/05/swastika.html" target="_blank">Swastika (卍)</a></li>
</ul>
<div>
<br /></div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com7tag:blogger.com,1999:blog-2261985361614499342.post-16188372063525043562017-09-08T17:32:00.000-07:002017-11-12T22:20:05.488-08:00Funny Scalar Fields (Heat Maps) on a Surface<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-cdMMQC_IGkU/WbM0PaStgZI/AAAAAAAAEME/RVNdIHTNicE0ZQqzQjWShEYJKQI10zW2QCLcBGAs/s1600/funny_scalar_field_on_a_surface_01_1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="400" src="https://1.bp.blogspot.com/-cdMMQC_IGkU/WbM0PaStgZI/AAAAAAAAEME/RVNdIHTNicE0ZQqzQjWShEYJKQI10zW2QCLcBGAs/s400/funny_scalar_field_on_a_surface_01_1.gif" width="400" /></a></div>
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Heat Map on a Torus</div>
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Heat Map on a Torus</div>
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<a href="https://3.bp.blogspot.com/-dB0YyKzoyck/WbM0jjIDNAI/AAAAAAAAEMM/bvXpYAUmfRYrmjZ5W-CkS7H1tVoNd0vnwCLcBGAs/s1600/funny_scalar_field_on_a_surface_02_1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="400" src="https://3.bp.blogspot.com/-dB0YyKzoyck/WbM0jjIDNAI/AAAAAAAAEMM/bvXpYAUmfRYrmjZ5W-CkS7H1tVoNd0vnwCLcBGAs/s400/funny_scalar_field_on_a_surface_02_1.gif" width="400" /></a></div>
<div style="text-align: center;">
Scalar Field on a Torus</div>
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Scalar Field on a Möbius Strip</div>
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<a href="https://2.bp.blogspot.com/-3yjG7t1EiUo/WbM1EwbXoBI/AAAAAAAAEMU/t9k7HCcX0xUKITluP7Ad8Y5Fqie7g7k6ACLcBGAs/s1600/funny_scalar_field_on_a_surface_02_3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="400" src="https://2.bp.blogspot.com/-3yjG7t1EiUo/WbM1EwbXoBI/AAAAAAAAEMU/t9k7HCcX0xUKITluP7Ad8Y5Fqie7g7k6ACLcBGAs/s400/funny_scalar_field_on_a_surface_02_3.gif" width="400" /></a></div>
<div style="text-align: center;">
Scalar Field on a Möbius Strip</div>
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( Mathematical software used: <a href="http://gnuplot.sourceforge.net/" target="_blank">gnuplot</a> )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com1tag:blogger.com,1999:blog-2261985361614499342.post-64772885221885266172017-09-08T17:10:00.000-07:002017-09-08T17:15:01.454-07:00Funny Inequalities on a Torus<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
(1)</div>
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<div style="text-align: center;">
(2-1)</div>
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<a href="https://1.bp.blogspot.com/-c_F2eg-H2SY/WbMtlDwdJ3I/AAAAAAAAELM/HaHasnppYJE4mKe3u3GKF5O1ms8-AA5cwCLcBGAs/s1600/funny_inequality_on_a_torus_02_2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="400" src="https://1.bp.blogspot.com/-c_F2eg-H2SY/WbMtlDwdJ3I/AAAAAAAAELM/HaHasnppYJE4mKe3u3GKF5O1ms8-AA5cwCLcBGAs/s400/funny_inequality_on_a_torus_02_2.gif" width="400" /></a></div>
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(2-2)</div>
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<a href="https://3.bp.blogspot.com/-IcdkREuSWEU/WbMtunNdhNI/AAAAAAAAELQ/a36P79-SnYsfzZAGZgkENdispGnWgcF_wCLcBGAs/s1600/funny_inequality_on_a_torus_03.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="400" src="https://3.bp.blogspot.com/-IcdkREuSWEU/WbMtunNdhNI/AAAAAAAAELQ/a36P79-SnYsfzZAGZgkENdispGnWgcF_wCLcBGAs/s400/funny_inequality_on_a_torus_03.gif" width="400" /></a></div>
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(3)</div>
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(4)</div>
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(5)</div>
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(6)</div>
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(7)</div>
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(8)</div>
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(9)</div>
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(10)</div>
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(11)</div>
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(12)</div>
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( Mathematical software used: gnuplot )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com1tag:blogger.com,1999:blog-2261985361614499342.post-90494325388759803942017-01-27T06:04:00.000-08:002017-01-27T06:47:56.738-08:00Plotting Slope Fields using Graph<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
y' = x² + y² - r² (r = 0, 1, 2, ..., 7)</div>
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<div style="text-align: center;">
y' = (x + y)/(x - y) (method 1)</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-DfybxVfRzZg/WItN9Zw94gI/AAAAAAAAEKI/JEx8CZd4dHgnYZDLoftpy1szpb3wpyjhgCLcB/s1600/slope_field_03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-DfybxVfRzZg/WItN9Zw94gI/AAAAAAAAEKI/JEx8CZd4dHgnYZDLoftpy1szpb3wpyjhgCLcB/s400/slope_field_03.png" width="400" /></a></div>
<div style="text-align: center;">
y' = (x + y)/(x - y) (method 2)</div>
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<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-45DWbKiKu1U/WItOEP4hdUI/AAAAAAAAEKM/a8XbONNLeIEY-6KtMMfe0pb4ErcVAa2OwCLcB/s1600/slope_field_04.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://2.bp.blogspot.com/-45DWbKiKu1U/WItOEP4hdUI/AAAAAAAAEKM/a8XbONNLeIEY-6KtMMfe0pb4ErcVAa2OwCLcB/s400/slope_field_04.png" width="400" /></a></div>
<div style="text-align: center;">
y' = sin(x)/y + sin(y)/x</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-uT6huuhLoPk/WItONlfkmSI/AAAAAAAAEKQ/0EiclIe-qtYYoPVKeieDdFYxLo5JYYxlgCLcB/s1600/slope_field_05.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-uT6huuhLoPk/WItONlfkmSI/AAAAAAAAEKQ/0EiclIe-qtYYoPVKeieDdFYxLo5JYYxlgCLcB/s400/slope_field_05.png" width="400" /></a></div>
<div style="text-align: center;">
y' = cos(x) + cos(y)</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-7cC5P2EVYAk/WItOYictXEI/AAAAAAAAEKY/-QW0nuPgecUv6Cc944PsKAARg__1ypRKwCLcB/s1600/slope_field_06.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-7cC5P2EVYAk/WItOYictXEI/AAAAAAAAEKY/-QW0nuPgecUv6Cc944PsKAARg__1ypRKwCLcB/s400/slope_field_06.png" width="400" /></a></div>
<div style="text-align: center;">
y' = cos(x) sqrt(1+x²) + cos(y) sqrt(1+y²)</div>
<br />
<br />
<b>Download Examples:</b><br />
<ul>
<li><a href="http://depositfiles.com/files/d4my1g3ic" target="_blank">slope_field.zip</a></li>
</ul>
<br />
<br />
( Mathematical software used: <a href="http://www.padowan.dk/" target="_blank">Graph</a> )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com2tag:blogger.com,1999:blog-2261985361614499342.post-84918046944481202042017-01-07T00:10:00.000-08:002017-01-07T08:44:35.352-08:00A Monster Equation<span style="font-size: large;">x² + y² - a[exp(sin³(x-y)) + exp(sin³(-x-y))]² = c</span><br />
<ul>
<li>x and y are variables.</li>
<li>a and c are parameters.</li>
</ul>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-giNLkinz888/WHCaiNfPCkI/AAAAAAAAEJk/abzCgaBBm_wxiorVX8EELjWiC3y7fm_IwCLcB/s1600/image1_monster.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://2.bp.blogspot.com/-giNLkinz888/WHCaiNfPCkI/AAAAAAAAEJk/abzCgaBBm_wxiorVX8EELjWiC3y7fm_IwCLcB/s400/image1_monster.png" width="400" /></a></div>
<div style="text-align: center;">
(a = 3, c = 0.5)</div>
<br />
<span style="font-size: large;"><b>Scilab script:</b></span><br />
<br />
clear;<br />
clf();<br />
xset("wdim",400,400);<br />
xset("fpf"," ");<br />
xset("thickness", 2);<br />
square(-8.2,-8.2,8.2,8.2);<br />
<br />
a = 3;<br />
c = 0.5;<br />
function z = fun(x,y)<br />
z = x^2 + y^2 - a*(exp(sin(x-y)^3) + exp(sin(-x-y)^3))^2 - c;<br />
endfunction<br />
data = linspace(-8.2, 8.2, 2000);<br />
<br />
contour2d(data, data, fun, [0,0], style=color("red"), axesflag=0);<br />
<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-eKm-gtq23fQ/WHCatTlssII/AAAAAAAAEJo/MsqPXjGf2wE4GZwsBMB4amMFEmjoRFI6ACLcB/s1600/image2_dog.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://2.bp.blogspot.com/-eKm-gtq23fQ/WHCatTlssII/AAAAAAAAEJo/MsqPXjGf2wE4GZwsBMB4amMFEmjoRFI6ACLcB/s400/image2_dog.png" width="400" /></a></div>
<div style="text-align: center;">
(a = -1, c = 24)</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-vb8wnl5msUA/WHCa1SFWz7I/AAAAAAAAEJs/6RReJ7b53fgXGiiEYOETH5wYz1CMFExGQCLcB/s1600/image3_face.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-vb8wnl5msUA/WHCa1SFWz7I/AAAAAAAAEJs/6RReJ7b53fgXGiiEYOETH5wYz1CMFExGQCLcB/s400/image3_face.png" width="400" /></a></div>
<div style="text-align: center;">
(a = -2, c = 50)</div>
<div style="text-align: left;">
<br /></div>
<br />
( Mathematical software used: Scilab )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com5tag:blogger.com,1999:blog-2261985361614499342.post-91987620238982108112016-12-14T05:02:00.000-08:002016-12-14T11:53:35.126-08:00Five Interlaced Moons<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-GgJnKLziLDk/WFExpFTdGDI/AAAAAAAAEIk/yyHvAxkMYzc5WNJjTLa-gfl-xwaiwrR9ACLcB/s1600/five_interlaced_moons_01.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-GgJnKLziLDk/WFExpFTdGDI/AAAAAAAAEIk/yyHvAxkMYzc5WNJjTLa-gfl-xwaiwrR9ACLcB/s1600/five_interlaced_moons_01.gif" /></a></div>
<div style="text-align: center;">
Five Interlaced Crescents</div>
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<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-Pd3-iwTD1Iw/WFEx3nLVS0I/AAAAAAAAEIo/2QEliyFCa1YaW0gwuc01UgmRupYSYow9wCLcB/s1600/set_difference_of_two_disks.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-Pd3-iwTD1Iw/WFEx3nLVS0I/AAAAAAAAEIo/2QEliyFCa1YaW0gwuc01UgmRupYSYow9wCLcB/s1600/set_difference_of_two_disks.gif" /></a></div>
<h3 style="text-align: center;">
Generalization:</h3>
<div style="text-align: center;">
Let A and B be two disks with radii a and b.</div>
<div style="text-align: center;">
The set difference A\B is used to plot the moons </div>
<div style="text-align: center;">
in the following animations.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-5BzGtLMieYg/WFEyFnMEqzI/AAAAAAAAEIs/VTXFpl9_Ba8sNU4JLxVUvmORhm8vyB5NgCLcB/s1600/five_interlaced_moons_02.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-5BzGtLMieYg/WFEyFnMEqzI/AAAAAAAAEIs/VTXFpl9_Ba8sNU4JLxVUvmORhm8vyB5NgCLcB/s1600/five_interlaced_moons_02.gif" /></a></div>
<div style="text-align: center;">
Animation parameter: the position of B</div>
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<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-7OvGbQPqvUA/WFEyXbReyAI/AAAAAAAAEIw/MG1vfzwN5sk_qSFIUvxJ_n8WWKToewtqACLcB/s1600/five_interlaced_moons_03.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-7OvGbQPqvUA/WFEyXbReyAI/AAAAAAAAEIw/MG1vfzwN5sk_qSFIUvxJ_n8WWKToewtqACLcB/s1600/five_interlaced_moons_03.gif" /></a></div>
<div style="text-align: center;">
Animation parameter: b (the radius of B)</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-E2-6vZzNIT8/WFEylJnjR0I/AAAAAAAAEI0/c9f03-X-cM0imYyxXEBvc2hmM8SKLM1HQCLcB/s1600/five_interlaced_moons_04.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-E2-6vZzNIT8/WFEylJnjR0I/AAAAAAAAEI0/c9f03-X-cM0imYyxXEBvc2hmM8SKLM1HQCLcB/s1600/five_interlaced_moons_04.gif" /></a></div>
<div style="text-align: center;">
Animation parameter: the position of A</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-8IcTXvdtfSQ/WFEyvBerUqI/AAAAAAAAEI4/Zaj9EGCREJQB-I2GT4v7McfK6UaxTUOIwCLcB/s1600/five_interlaced_moons_05.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-8IcTXvdtfSQ/WFEyvBerUqI/AAAAAAAAEI4/Zaj9EGCREJQB-I2GT4v7McfK6UaxTUOIwCLcB/s1600/five_interlaced_moons_05.gif" /></a></div>
<div style="text-align: center;">
Animation parameter: a (the radius of A)</div>
<br />
<br />
( Mathematical softwares used: gnuplot, GeoGebra )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com1tag:blogger.com,1999:blog-2261985361614499342.post-62630741332078216392016-12-14T03:37:00.000-08:002016-12-21T16:18:45.500-08:00Four Interlaced Crescents<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-5Th9v3lalGI/WFEtLSvcE2I/AAAAAAAAEIQ/IK7SSkQptdwNhrl8l6SNidkFmYU2vGMYgCLcB/s1600/four_interlaced_crescents_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-5Th9v3lalGI/WFEtLSvcE2I/AAAAAAAAEIQ/IK7SSkQptdwNhrl8l6SNidkFmYU2vGMYgCLcB/s400/four_interlaced_crescents_01.png" width="400" /></a></div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-phzYz3z3wWQ/WFEtTs6dSiI/AAAAAAAAEIU/wUhlnD0ZhWYG_mRNY2oKCSjQoynmIVDwwCLcB/s1600/four_interlaced_crescents_02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-phzYz3z3wWQ/WFEtTs6dSiI/AAAAAAAAEIU/wUhlnD0ZhWYG_mRNY2oKCSjQoynmIVDwwCLcB/s400/four_interlaced_crescents_02.png" width="400" /></a></div>
<div style="text-align: center;">
A cross pattée!</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-Vdjj-Ck3mSM/WFEthUKsv4I/AAAAAAAAEIY/hI4e-Iqzrs8jPQWMBDbzHR2wAIuamEVigCLcB/s1600/four_interlaced_crescents_03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-Vdjj-Ck3mSM/WFEthUKsv4I/AAAAAAAAEIY/hI4e-Iqzrs8jPQWMBDbzHR2wAIuamEVigCLcB/s1600/four_interlaced_crescents_03.png" /></a></div>
<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-_1kDD0szcjw/WFV3D0-2PkI/AAAAAAAAEJM/EzoGvFQi61o1aER334hZV4XH__j9JA1MACLcB/s1600/four_interlaced_crescents_04.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-_1kDD0szcjw/WFV3D0-2PkI/AAAAAAAAEJM/EzoGvFQi61o1aER334hZV4XH__j9JA1MACLcB/s1600/four_interlaced_crescents_04.gif" /></a></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-AjdYw7Rbbxc/WFV3L4r8wwI/AAAAAAAAEJQ/raNGmmH2L7k83S7NVJSHD8sqnnqfUp_ugCLcB/s1600/four_interlaced_crescents_05.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-AjdYw7Rbbxc/WFV3L4r8wwI/AAAAAAAAEJQ/raNGmmH2L7k83S7NVJSHD8sqnnqfUp_ugCLcB/s1600/four_interlaced_crescents_05.gif" /></a></div>
<br />
<br />
( Mathematical softwares used: Graph, gnuplot )<br />
<br />
<br />
<span style="font-size: large;">Related posts:</span><br />
<ul>
<li><a href="http://benice-equation.blogspot.com/2016/11/cross-patty.html">Cross Patty (cross pattée)</a></li>
</ul>
<div>
<br /></div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com1tag:blogger.com,1999:blog-2261985361614499342.post-55064560436894871272016-12-14T02:30:00.000-08:002016-12-14T02:30:59.634-08:00Triple Crescent Moon<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-ZS9FIyfkdr0/WFEPzVjkm2I/AAAAAAAAEHk/4CJOHIrNPxY3C5OO1oUiquPXQDG1QwfmACLcB/s1600/triple_crescent_moon.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-ZS9FIyfkdr0/WFEPzVjkm2I/AAAAAAAAEHk/4CJOHIrNPxY3C5OO1oUiquPXQDG1QwfmACLcB/s400/triple_crescent_moon.png" width="400" /></a></div>
<div style="text-align: center;">
Triple Crescent Moon Symbol</div>
<div style="text-align: center;">
(Boundary curves: circles)</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-hjXyrDMWVc8/WFEP-GtaV2I/AAAAAAAAEHo/RI8tf8-ybuUWLEN0pRPlfNr1WVvue8lCwCLcB/s1600/triple_crescent_super_moon_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-hjXyrDMWVc8/WFEP-GtaV2I/AAAAAAAAEHo/RI8tf8-ybuUWLEN0pRPlfNr1WVvue8lCwCLcB/s400/triple_crescent_super_moon_01.png" width="400" /></a></div>
<div style="text-align: center;">
Boundary curves: superellipses with exponent 2.5</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-EM2OWnNus5w/WFEQGuigkeI/AAAAAAAAEHs/6f4USdoL_5gZ1g0WP2ZucrvguzACeKpTQCLcB/s1600/triple_crescent_super_moon_02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-EM2OWnNus5w/WFEQGuigkeI/AAAAAAAAEHs/6f4USdoL_5gZ1g0WP2ZucrvguzACeKpTQCLcB/s400/triple_crescent_super_moon_02.png" width="400" /></a></div>
<div style="text-align: center;">
Boundary curves: superellipses with exponent 4</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-EBcuP93w-Dg/WFEQOtzovFI/AAAAAAAAEHw/R-Df1DJMmVkvOXz6qqQ7Jp-VWBNwT1U_gCLcB/s1600/monster_face_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://2.bp.blogspot.com/-EBcuP93w-Dg/WFEQOtzovFI/AAAAAAAAEHw/R-Df1DJMmVkvOXz6qqQ7Jp-VWBNwT1U_gCLcB/s400/monster_face_01.png" width="400" /></a></div>
<div style="text-align: center;">
A monster face!</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-L9WKeCsjvEc/WFEQW3Bjn6I/AAAAAAAAEH4/QIke2N6nsmkrmC-iPvulLhP9p5iK0Sk-gCLcB/s1600/monster_face_02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-L9WKeCsjvEc/WFEQW3Bjn6I/AAAAAAAAEH4/QIke2N6nsmkrmC-iPvulLhP9p5iK0Sk-gCLcB/s400/monster_face_02.png" width="400" /></a></div>
<div style="text-align: center;">
Hi!</div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-YUsgRwC4E0w/WFEQdxuKpGI/AAAAAAAAEH8/78rQKziSIa492XuRAJ1lpOrloO25EIuOQCLcB/s1600/monster_face_03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-YUsgRwC4E0w/WFEQdxuKpGI/AAAAAAAAEH8/78rQKziSIa492XuRAJ1lpOrloO25EIuOQCLcB/s400/monster_face_03.png" width="400" /></a></div>
<div style="text-align: center;">
Exponents: 1.5, 2, 2.5, 3, 3.5, 4</div>
<br />
<br />
( Mathematical software used: gnuplot )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-55554477659224025092016-11-28T02:31:00.000-08:002016-11-28T02:51:42.260-08:00Cross Patty (cross pattée)<div style="text-align: left;">
<br /></div>
<h3 style="text-align: center;">
max( f(abs(x), abs(y)), f(abs(y), abs(x)) ) > 0</h3>
<br />
<br />
<h3>
(1) f(x,y) = ax^2 - y + b (parabola)</h3>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-gF7faqzU358/WDv9UxlTj-I/AAAAAAAAEGk/E2i8GM0fP5UHz90RsWpC9lIleN7brjTaACLcB/s1600/cross_patty_1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-gF7faqzU358/WDv9UxlTj-I/AAAAAAAAEGk/E2i8GM0fP5UHz90RsWpC9lIleN7brjTaACLcB/s1600/cross_patty_1.png" /></a></div>
<div style="text-align: center;">
a = 0.035, b = 0.85</div>
<br />
<br />
<h3>
(2) f(x,y) = x^2 + (y - a)^2 - b^2 (circle)</h3>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-jRGvsL_gfhs/WDv9jo1zekI/AAAAAAAAEGo/xRT0b-TalDgPSH4RZEE4yLYRJ6LVziNSACLcB/s1600/cross_patty_2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-jRGvsL_gfhs/WDv9jo1zekI/AAAAAAAAEGo/xRT0b-TalDgPSH4RZEE4yLYRJ6LVziNSACLcB/s1600/cross_patty_2.png" /></a></div>
<div style="text-align: center;">
a = 15.6, b = 14.75</div>
<br />
<br />
<h3>
(3) f(x,y) = abs(x)^p + abs(y - a)^p - b^p (superellipse)</h3>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-Wdm61ScX6Ig/WDv9ueaE2ZI/AAAAAAAAEGs/gqoK4sKbXCkKEnhduORNky9depNuWQqzACLcB/s1600/cross_patty_3_1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-Wdm61ScX6Ig/WDv9ueaE2ZI/AAAAAAAAEGs/gqoK4sKbXCkKEnhduORNky9depNuWQqzACLcB/s1600/cross_patty_3_1.gif" /></a></div>
<div style="text-align: center;">
a = 15.6, b = 14.75</div>
<div style="text-align: center;">
p = 2.5, 2.4, 2.3, ...,1.5</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
p = 4, a = 10, b = 8</div>
<br />
<br />
<h3>
(4) f(x,y) = a[exp(-abs(x+b)^p) + exp(-abs(x-b)^p)] - y + c</h3>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
p = 1.5, a = 1, b = 7, c = 2</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-nnFrR9TIXRY/WDv-YBgUZ0I/AAAAAAAAEG8/qVKxrIDuGec9hcX3R6A42ea0UqwBKQkRgCLcB/s1600/cross_patty_4_2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-nnFrR9TIXRY/WDv-YBgUZ0I/AAAAAAAAEG8/qVKxrIDuGec9hcX3R6A42ea0UqwBKQkRgCLcB/s1600/cross_patty_4_2.png" /></a></div>
<div style="text-align: center;">
p = 3, a = 1, b = 7, c = 2</div>
<br />
<br />
<h3>
(5) f(x,y) = a cos(bx)^p - y + c</h3>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
p = 1, a = -1, b = π/7, c = 2</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
p = 2, a = 1, b = π/7, c = 2</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
p = 3, a = -1, b = π/7, c = 2</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-3kQX5_e4uss/WDv-1AODX7I/AAAAAAAAEHM/EFXDC3ypEksP1PviC4S49RSdD1-5H_R6QCLcB/s1600/cross_patty_5_4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-3kQX5_e4uss/WDv-1AODX7I/AAAAAAAAEHM/EFXDC3ypEksP1PviC4S49RSdD1-5H_R6QCLcB/s1600/cross_patty_5_4.png" /></a></div>
<div style="text-align: center;">
p = 4, a = 1, b = π/7, c = 2</div>
<br />
<br />
( Mathematical software used: Graph )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-63516848957779727842016-10-21T02:58:00.000-07:002016-10-21T03:39:50.181-07:00Rounded Cuboid with Superquadric Corners<br />
The corners of the rounded cuboids on my <a href="http://benice-equation.blogspot.com/2016/10/equation-of-rounded-rectangle.html" target="_blank">previous post</a> are spherical corners.<br />
<br />
We can generalize the spherical corners (power = 2) to superquadric corners (power > 0).<br />
<br />
<br />
<h3>
Equation 1.</h3>
<br />
The equation for a rounded cuboid with superquadric corners can be given by:<br />
<br />
max(|x| - a, 0)<sup>p</sup> + max(|y| - b, 0)<sup>p</sup> + max(|z| - c, 0)<sup>p</sup> = d<sup>p</sup>,<br />
<br />
where a, b, c, d > 0, and p ≧ 1.<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-CIfwDg73VlY/WAnhAGdT8rI/AAAAAAAAEGA/TRicannIC5oczH9rwCIJRlFRMzr_uzOagCLcB/s1600/rounded_cuboid_with_superquadric_corners_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="340" src="https://1.bp.blogspot.com/-CIfwDg73VlY/WAnhAGdT8rI/AAAAAAAAEGA/TRicannIC5oczH9rwCIJRlFRMzr_uzOagCLcB/s400/rounded_cuboid_with_superquadric_corners_01.png" width="400" /></a></div>
<div style="text-align: center;">
p = 1</div>
<br />
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<div style="text-align: center;">
p = 1.5</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
p = 2</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
p = 4</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
p = 10</div>
<br />
<br />
<h3>
Equation 2.</h3>
<br />
Furthermore, the above equation can be generalized as follows:<br />
<br />
max(|x| - a, 0)<sup>p</sup> + max(|y| - b, 0)<sup>q</sup> + max(|z| - c, 0)<sup>r</sup> = d,<br />
<br />
where a, b, c ≧ 0, d > 0, and p, q, r > 0.<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
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<br />
<br />
<h3>
Related posts:</h3>
<ul>
<li><a href="http://benice-equation.blogspot.com/2016/10/equation-of-rounded-rectangle.html">Equation of a Rounded Rectangle</a></li>
</ul>
<div>
<br /></div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-57936804044466924812016-10-17T11:57:00.000-07:002016-10-21T03:47:10.496-07:00Equation of a Rounded Rectangle<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-uKgXIDdCEkY/WAUM2KeLMBI/AAAAAAAAEFU/f9N9uIewXCcG20c4scrs_WCqT3dFxbHOACLcB/s1600/rounded_rectangle_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-uKgXIDdCEkY/WAUM2KeLMBI/AAAAAAAAEFU/f9N9uIewXCcG20c4scrs_WCqT3dFxbHOACLcB/s400/rounded_rectangle_01.png" width="400" /></a></div>
<br />
<br />
<h3>
Equations:</h3>
<br />
The equation of a rounded rectangle centered at (0,0) with width 2(a+c), height 2(b+c)<br />
and corner radius c can be expressed as<br />
<br />
max(|x| - a, 0)² + max(|y| - b, 0)² = c² ...... (I)<br />
<br />
or<br />
<br />
(|x| - a + │|x| - a│)² + (|y| - b + │|y| - b│)² = (2c)² ...... (II)<br />
<br />
, where a, b, and c are positive numbers.<br />
<br />
<br />
<br />
<br />
<h3>
Proof:</h3>
<br />
The rounded rectangle can be broken down into four parts (see the image below):<br />
╭<br />
│ (|x| - a)² + (|y| - b)² = c², if |x|≧a and |y|≧b ...... (1)<br />
│ |x| = a + c , if |x|≧a and |y|<b ...... (2)<br />
│ |y| = b + c , if |x|<a and |y|≧b ...... (3)<br />
│ no graph , if |x|<a and |y|<b ...... (4)<br />
╰<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-R1cRzYW5En8/WAUPNPhI5aI/AAAAAAAAEFc/ZrseaJEO2IIKmZ0Zo0j1y8ThpaT7rVw0QCLcB/s1600/rounded_rectangle_02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-R1cRzYW5En8/WAUPNPhI5aI/AAAAAAAAEFc/ZrseaJEO2IIKmZ0Zo0j1y8ThpaT7rVw0QCLcB/s400/rounded_rectangle_02.png" width="400" /></a></div>
<br />
Since |x|≧a in part (2) and |y|≧b in part (3),<br />
the above expression is equivalent to<br />
╭<br />
│ (|x| - a)² + (|y| - b)² = c², if |x|≧a and |y|≧b;<br />
│ (|x| - a)² = c² , if |x|≧a and |y|<b;<br />
│ (|y| - b)² = c² , if |x|<a and |y|≧b;<br />
│ 0 = c² , if |x|<a and |y|<b.<br />
╰<br />
<br />
Now make the equations of the four parts look similar:<br />
╭<br />
│ (|x| - a)² + (|y| - b)² = c², if |x|≧a and |y|≧b;<br />
│ (|x| - a)² + (b - b)² = c², if |x|≧a and |y|<b;<br />
│ (a - a)² + (|y| - b)² = c², if |x|<a and |y|≧b;<br />
│ (a - a)² + (b - b)² = c², if |x|<a and |y|<b.<br />
╰<br />
<br />
Since max(|x|, a) =<br />
╭<br />
│ |x|, if |x|≧a<br />
│ a, if |x|<a<br />
╰<br />
and max(|y|, b) =<br />
╭<br />
│ |y|, if |y|≧b<br />
│ b, if |y|<b,<br />
╰<br />
the four equations can be combined into a single one as follows:<br />
<br />
(max(|x|, a) - a)² + (max(|y|, b) - b)² = c²,<br />
<br />
or equivalently,<br />
<br />
max(|x| - a, 0)² + max(|y| - b, 0)² = c² ...... (I)<br />
<br />
<br />
Using the formula max(z, 0) = (z + |z|) / 2,<br />
the above equation can be expanded to<br />
<br />
(|x| - a + │|x| - a│)² + (|y| - b + │|y| - b│)² = (2c)² ...... (II)<br />
<br />
<br />
<br />
<br />
<h3>
Note:</h3>
<br />
<ul>
<li>When a or b is zero (but not both), the equation (I) describes a stadium (curve).</li>
<li>When c is zero (a and b are positive), the equation (I) describes a rectangular region rather than a rectangle.</li>
<li>The equation max(|x| - a, |y| - b) = 0 describes a rectangle.</li>
<li>The equation max(|x| - a, 0)² + max(|y| - b, 0)² + max(|z| - c, 0)² = d² describes a rounded cuboid.</li>
</ul>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-3JdCc5sztEc/WAUQr5bRQ4I/AAAAAAAAEFk/HbaGS3DnhHIyeSLRNr4KyAtlJp5Kib4eQCLcB/s1600/stadium_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://2.bp.blogspot.com/-3JdCc5sztEc/WAUQr5bRQ4I/AAAAAAAAEFk/HbaGS3DnhHIyeSLRNr4KyAtlJp5Kib4eQCLcB/s400/stadium_01.png" width="400" /></a></div>
<div style="text-align: center;">
max(|x|, 0)² + max(|y| - 1, 0)² = 2²</div>
<div style="text-align: center;">
(a = 0)</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: center;">
max(|x| - 3, 0)² + max(|y|, 0)² = (1.5)²</div>
<div style="text-align: center;">
(b = 0)</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-wedvYy12cZ8/WAURDbBaNlI/AAAAAAAAEFs/DpOm66EWpRMNAhU9qTSkMYim0N4QudYswCLcB/s1600/rectangular_region.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-wedvYy12cZ8/WAURDbBaNlI/AAAAAAAAEFs/DpOm66EWpRMNAhU9qTSkMYim0N4QudYswCLcB/s400/rectangular_region.png" width="400" /></a></div>
<div style="text-align: center;">
max(|x| - 3, 0)² + max(|y| - 2, 0)² = 0</div>
<div style="text-align: center;">
(c = 0)</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-td4LwXGZ7Tw/WAUROwpuCZI/AAAAAAAAEFw/Ci-178L_aqUmLXm8V77V1ly9i8XetCOBwCLcB/s1600/rounded_cuboid.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="340" src="https://4.bp.blogspot.com/-td4LwXGZ7Tw/WAUROwpuCZI/AAAAAAAAEFw/Ci-178L_aqUmLXm8V77V1ly9i8XetCOBwCLcB/s400/rounded_cuboid.png" width="400" /></a></div>
<div style="text-align: center;">
max(|x| - 4, 0)² + max(|y| - 3, 0)² + max(|z| - 1, 0)² = 1</div>
<div>
<br /></div>
<div>
<br /></div>
<h3>
Related posts:</h3>
<div>
<ul>
<li><a href="http://benice-equation.blogspot.com/2016/08/rounded-cross.html">Rounded Cross</a></li>
<li><a href="http://benice-equation.blogspot.com/2016/10/rounded-cuboid-with-superquadric-corners.html">Rounded Cuboid with Superquadric Corners</a></li>
</ul>
<div>
<br /></div>
</div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-878263841338886192016-10-11T19:01:00.001-07:002016-10-11T19:01:52.763-07:00Funny Boxing Champion (2)<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-6gwbY5GT9vc/V_2ZPd5a06I/AAAAAAAAEFA/gkwdc1ZlyMozo_djx1PaHANnA7YoiGj9wCLcB/s1600/funny_boxing_champion_02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-6gwbY5GT9vc/V_2ZPd5a06I/AAAAAAAAEFA/gkwdc1ZlyMozo_djx1PaHANnA7YoiGj9wCLcB/s400/funny_boxing_champion_02.png" width="400" /></a></div>
<br />
<br />
( Mathematical software used: gnuplot )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-16381271085772586672016-10-05T06:16:00.000-07:002016-10-11T18:59:25.005-07:00Funny Boxing Champion (1)<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-XZI88FfxraM/V_T3mGO7xLI/AAAAAAAAEEs/Bq5c_-9hgvAtqh88j9ng1ZZIJ49w7GAwACLcB/s1600/funny_boxing_champion.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-XZI88FfxraM/V_T3mGO7xLI/AAAAAAAAEEs/Bq5c_-9hgvAtqh88j9ng1ZZIJ49w7GAwACLcB/s400/funny_boxing_champion.png" width="400" /></a></div>
<br />
<br />
<br />
<b><span style="font-size: large;">Equations (方程式):</span></b><br />
<br />
<br />
(1) body (身体):<br />
<div style="text-align: center;">
x² + y² - 72[exp(1/(sin(y+x) - 1)) ± exp(1/(sin(y-x) - 1))]² = 5/2</div>
<br />
<br />
(2) eye sockets (眼眶):<br />
<div style="text-align: center;">
(abs(x) - π/4)² + (y - 3π/2)² = (π/8)²</div>
<br />
<br />
(3) eyes (眼睛):<br />
<div style="text-align: center;">
arcsin²(k(abs(x) - π/4)) = arcsin²(k(y - 3π/2))</div>
<br />
<br />
(4) mouth (嘴巴):<br />
<div style="text-align: center;">
arcsin²(kx) = arcsin²(k(y - 19π/16))</div>
<br />
<br />
(5) navel (肚臍):<br />
<div style="text-align: center;">
arcsin²(kx) = arcsin²(k(y + π/3))</div>
<br />
<br />
<br />
<div style="text-align: center;">
(8 ≦ k ≦ 10)</div>
<br />
<br />
<br />
( Mathematical software used: gnuplot )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-25654813106895691232016-08-21T21:41:00.001-07:002016-08-21T21:55:55.371-07:00Tiling using Rounded Crosses<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-yPmYnDo9n54/V7qAoemdB8I/AAAAAAAAED8/_dZUvnEebpEjBRk1P5eYaG5zlxjZE1j3ACLcB/s1600/tiling_using_rounded_crosses_01.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-yPmYnDo9n54/V7qAoemdB8I/AAAAAAAAED8/_dZUvnEebpEjBRk1P5eYaG5zlxjZE1j3ACLcB/s400/tiling_using_rounded_crosses_01.png" width="400" /></a></div>
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<a href="https://2.bp.blogspot.com/-rZvO9UGfZXs/V7qA51jvAEI/AAAAAAAAEEE/yfWU--m-YO8HDL6f9rcN2oqvRCBig4IHQCLcB/s1600/tiling_using_rounded_crosses_03.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://2.bp.blogspot.com/-rZvO9UGfZXs/V7qA51jvAEI/AAAAAAAAEEE/yfWU--m-YO8HDL6f9rcN2oqvRCBig4IHQCLcB/s400/tiling_using_rounded_crosses_03.png" width="400" /></a></div>
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( Mathematical software used: Graph )<br />
<br />
<br />
<span style="font-size: large;"><b>Related posts:</b></span><br />
<ul>
<li><a href="http://benice-equation.blogspot.com/2016/08/rounded-cross.html" target="_blank">Rounded Cross</a></li>
<li><a href="http://benice-equation.blogspot.com/2013/09/tiling-using-nested-crosses.html" target="_blank">Tiling using Nested Crosses</a></li>
</ul>
<div>
<br /></div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com1tag:blogger.com,1999:blog-2261985361614499342.post-24405120489209190762016-08-17T22:53:00.000-07:002016-10-21T03:45:02.331-07:00Rounded Cross - cross with rounded corners (圓角十字形)<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-SJMaKiViZcs/V7VJQkZR_RI/AAAAAAAAECw/jrPKijeyjSAUoFrP0VsTVgKmq6aDusoOQCLcB/s1600/rounded_cross.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-SJMaKiViZcs/V7VJQkZR_RI/AAAAAAAAECw/jrPKijeyjSAUoFrP0VsTVgKmq6aDusoOQCLcB/s1600/rounded_cross.png" /></a></div>
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<a href="https://1.bp.blogspot.com/-F-TWucOxmak/V7VJkpzmOsI/AAAAAAAAEC0/w2w4vYMwEMkNstc2VzOF8ShUnu5M2IrdQCLcB/s1600/rounded_cross_tw.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-F-TWucOxmak/V7VJkpzmOsI/AAAAAAAAEC0/w2w4vYMwEMkNstc2VzOF8ShUnu5M2IrdQCLcB/s1600/rounded_cross_tw.png" /></a></div>
<br />
<span style="font-size: large;"><b>gnuplot script</b>:</span><br />
<br />
a = 2<br />
b = 1<br />
c = 1<br />
w = 8<br />
max(x,y) = x < y ? y : x<br />
min(x,y) = x < y ? x : y<br />
fun(x,y) = \<br />
min(min(max(abs(x)-b,0)**2 + max(abs(y)-(a+b+2*c),0)**2 - c**2, \<br />
max(abs(x)-(a+b+2*c),0)**2 + max(abs(y)-b,0)**2 - c**2), \<br />
max((abs(x)-(b+c))**2 + (abs(y)-(b+c))**2 - c**2, \<br />
c**2 - (abs(x)-(b+2*c))**2 - (abs(y)-(b+2*c))**2))<br />
set xrange [-w:w]<br />
set yrange [-w:w]<br />
set size ratio -1<br />
set samples 512<br />
set isosamples 512<br />
set contour base<br />
set cntrparam levels discrete 0.0<br />
unset key<br />
unset surface<br />
set table 'fun.dat'<br />
splot fun(x,y)<br />
unset table<br />
plot 'fun.dat' w l lw 1.5 lc rgb "red"<br />
<div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-pPZ7BnG6i_4/V7VKcRI5LlI/AAAAAAAAEDA/ouC6EnoUw0YULm_icNFHxs-gYd-VO3IwgCLcB/s1600/rounded_cross_a2_b1_c1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-pPZ7BnG6i_4/V7VKcRI5LlI/AAAAAAAAEDA/ouC6EnoUw0YULm_icNFHxs-gYd-VO3IwgCLcB/s1600/rounded_cross_a2_b1_c1.png" /></a></div>
<div style="text-align: center;">
(a = 2, b = 1, c = 1)</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-ihsrkBJa374/V7VKxFZhmKI/AAAAAAAAEDE/ilrBA4hv4A400ytMONfCXDYFI7AVLer6QCLcB/s1600/rounded_cross_a4_b05_c05.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-ihsrkBJa374/V7VKxFZhmKI/AAAAAAAAEDE/ilrBA4hv4A400ytMONfCXDYFI7AVLer6QCLcB/s1600/rounded_cross_a4_b05_c05.png" /></a></div>
<div style="text-align: center;">
(a = 4, b = 0.5, c = 0.5)</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-X1R603IRC-Y/V7VLAGSHCdI/AAAAAAAAEDI/9H_2_QRQYjsoHy1XsYMFWfnVcX_FDRjtACLcB/s1600/rounded_cross_a0_b0_c2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-X1R603IRC-Y/V7VLAGSHCdI/AAAAAAAAEDI/9H_2_QRQYjsoHy1XsYMFWfnVcX_FDRjtACLcB/s1600/rounded_cross_a0_b0_c2.png" /></a></div>
<div style="text-align: center;">
(a = 0, b = 0, c = 2)</div>
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<a href="https://4.bp.blogspot.com/-CySDWHZOwLo/V7p8f1AEubI/AAAAAAAAEDo/VqrRKkWqPfQ6aa1sKzbazJNy5xsMctl_gCLcB/s1600/rounded_cross_a0_b2_c4_3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-CySDWHZOwLo/V7p8f1AEubI/AAAAAAAAEDo/VqrRKkWqPfQ6aa1sKzbazJNy5xsMctl_gCLcB/s1600/rounded_cross_a0_b2_c4_3.png" /></a></div>
<div style="text-align: center;">
(a = 0, b = 2, c = 4/3)</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-WB98TaWF1Ho/V7p86UofhEI/AAAAAAAAEDs/UzVhkn0ClS8maatpVk3Gv3ejnlSAznanQCLcB/s1600/rounded_cross_a2_b0_c4_3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-WB98TaWF1Ho/V7p86UofhEI/AAAAAAAAEDs/UzVhkn0ClS8maatpVk3Gv3ejnlSAznanQCLcB/s1600/rounded_cross_a2_b0_c4_3.png" /></a></div>
<div style="text-align: center;">
(a = 2, b = 0, c = 4/3)</div>
</div>
<div>
<br />
<br /></div>
<div>
( Mathematical softwares used: Graph, gnuplot )<br />
<br />
<br />
<span style="font-size: large;"><b>Related posts:</b></span><br />
<ul>
<li><a href="http://benice-equation.blogspot.com/2016/08/tiling-using-rounded-crosses.html" target="_blank">Tiling using Rounded Crosses</a></li>
<li><a href="http://benice-equation.blogspot.com/2016/10/equation-of-rounded-rectangle.html">Equation of a Rounded Rectangle</a></li>
</ul>
<div>
<br /></div>
</div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com1tag:blogger.com,1999:blog-2261985361614499342.post-17777855889007663362016-08-07T00:42:00.000-07:002016-08-28T21:35:37.965-07:00How to Draw the Olympic Rings Mathematically?<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-bxUAoJW9gdo/V6bexC_jR8I/AAAAAAAAECg/eupCntv3-b4QX5-MZST0S_v0OG0PNMjSwCLcB/s1600/olympic_rings.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="222" src="https://3.bp.blogspot.com/-bxUAoJW9gdo/V6bexC_jR8I/AAAAAAAAECg/eupCntv3-b4QX5-MZST0S_v0OG0PNMjSwCLcB/s400/olympic_rings.png" width="400" /></a></div>
<br />
<h3>
Inequalities (Relations) for the Olympic Rings:</h3>
<ol>
<li>(Blue Ring) min(max(R<sub>1</sub>(x,y), -y), max(R<sub>1</sub>(x,y), y, -R<sub>4</sub>(x,y))) < 0</li>
<li>(Black Ring) min(max(R<sub>2</sub>(x,y), -y, -R<sub>4</sub>(x,y)), max(R<sub>2</sub>(x,y), y, -R<sub>5</sub>(x,y))) < 0</li>
<li>(Red Ring) min(max(R<sub>3</sub>(x,y), -y, -R<sub>5</sub>(x,y)), max(R<sub>3</sub>(x,y), y)) < 0</li>
<li>(Yellow Ring) min(max(R<sub>4</sub>(x,y), -y, -R<sub>1</sub>(x,y)), max(R<sub>4</sub>(x,y), y, -R<sub>2</sub>(x,y))) < 0</li>
<li>(Green Ring) min(max(R<sub>5</sub>(x,y), -y, -R<sub>2</sub>(x,y)), max(R<sub>5</sub>(x,y), y, -R<sub>3</sub>(x,y))) < 0</li>
</ol>
<div>
<br />
<b>Functions:</b></div>
<div>
<ul>
<li>R<sub>1</sub>(x,y) = (sqrt((x+2d)² + (y-h)²) - a)² - b²</li>
<li>R<sub>2</sub>(x,y) = (sqrt(x² + (y-h)²) - a)² - b²</li>
<li>R<sub>3</sub>(x,y) = (sqrt((x-2d)² + (y-h)²) - a)² - b²</li>
<li>R<sub>4</sub>(x,y) = (sqrt((x+d)² + (y+h)²) - a)² - b²</li>
<li>R<sub>5</sub>(x,y) = (sqrt((x-d)² + (y+h)²) - a)² - b²</li>
</ul>
</div>
<br />
<b>Constants:</b><br />
<ul>
<li>a = 49.25</li>
<li>b = 4.75</li>
<li>d = 58.5</li>
<li>h = 25</li>
</ul>
<div>
<br /></div>
<div>
<b>Download Examples:</b></div>
<div>
<ul>
<li><a href="http://depositfiles.com/files/7cufxlnfp" target="_blank">olympic_rings.zip</a></li>
</ul>
</div>
<div>
<br /></div>
<div>
<br /></div>
<div>
( Mathematical softwares used: Graph, gnuplot )</div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-35065873692268799132016-07-05T23:09:00.000-07:002016-07-11T02:16:38.473-07:00Using Trigonometry to Express the Equations of Piecewise Spirals<span style="font-size: large;">
Circle Spirals</span> (spirals made of semicircles):<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-o98JCiX_1xA/V3yZpc7TIGI/AAAAAAAAEB8/rh6NQ-4zvn0bn-qakY0bvacz2B3Lha4UwCKgB/s1600/circle_spirals_sin.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-o98JCiX_1xA/V3yZpc7TIGI/AAAAAAAAEB8/rh6NQ-4zvn0bn-qakY0bvacz2B3Lha4UwCKgB/s1600/circle_spirals_sin.gif" /></a></div>
<div style="text-align: center;">
Equations: sin(π sqrt((x + k sgn(y)/2)² + y²)) = 0</div>
<div style="text-align: center;">
k = 1, 2, 3, ..., 8</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-FMBuurF7yxI/V3yZt3vaRgI/AAAAAAAAEB8/T2KKrh_b7fM3bNAY2dnSNLLnOHBTT1f7ACKgB/s1600/circle_spirals_cos.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-FMBuurF7yxI/V3yZt3vaRgI/AAAAAAAAEB8/T2KKrh_b7fM3bNAY2dnSNLLnOHBTT1f7ACKgB/s1600/circle_spirals_cos.gif" /></a></div>
<div style="text-align: center;">
Equations: cos(π sqrt((x + k sgn(y)/2)² + y²)) = 0</div>
<div style="text-align: center;">
k = 1, 2, 3, ..., 8</div>
<br />
<br />
<span style="font-size: large;">
Square Spirals</span> (spirals made of half-squares):<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-Rw1yHq8IigA/V3yZxuQSXNI/AAAAAAAAEB8/22xuFl2Rcxg10gqrWSto0SRWcGqrZtH1gCKgB/s1600/square_spirals_sin.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-Rw1yHq8IigA/V3yZxuQSXNI/AAAAAAAAEB8/22xuFl2Rcxg10gqrWSto0SRWcGqrZtH1gCKgB/s1600/square_spirals_sin.gif" /></a></div>
<div style="text-align: center;">
Equations: sin(π(abs(x + k sgn(y)/2) + abs(y))) = 0</div>
<div style="text-align: center;">
k = 1, 2, 3, 4</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-LVHLJIhqQyE/V3yZ0_9Y31I/AAAAAAAAECQ/IedM70k5gQkzMINVwF1V8XPEfsEVBgBzQCKgB/s1600/square_spirals_cos.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-LVHLJIhqQyE/V3yZ0_9Y31I/AAAAAAAAECQ/IedM70k5gQkzMINVwF1V8XPEfsEVBgBzQCKgB/s1600/square_spirals_cos.gif" /></a></div>
<div style="text-align: center;">
Equations: cos(π(abs(x + k sgn(y)/2) + abs(y))) = 0</div>
<div style="text-align: center;">
k = 1, 2, 3, 4</div>
<br />
<br />
<span style="font-size: large;">
Regular Polygon Spirals</span> (spirals made of regular semi-polygons):<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-G5wCxJp161E/V3yaCrhchAI/AAAAAAAAECQ/sw_ztyNC4EQEA_FtRodJMHqpAxBmKAYfwCKgB/s1600/hexagon_spirals_sin.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-G5wCxJp161E/V3yaCrhchAI/AAAAAAAAECQ/sw_ztyNC4EQEA_FtRodJMHqpAxBmKAYfwCKgB/s1600/hexagon_spirals_sin.gif" /></a></div>
<div style="text-align: center;">
Equations: sin(π p(x + k sgn(y)/2, y)) = 0</div>
<div style="text-align: center;">
k = 1, 2, 3, ..., 8</div>
<div style="text-align: center;">
p(x,y) = sqrt(x² + y²) sec(π/n) cos(mod(arctan(y/x),2π/n) - π/n)</div>
<div style="text-align: center;">
n = 6</div>
<div>
<br /></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-ZeJo16gmKuE/V3ya7RdxiUI/AAAAAAAAECQ/Snicpch7vFQbFepgw29oNOdO0mBQldRegCKgB/s1600/hexagon_spirals_cos.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-ZeJo16gmKuE/V3ya7RdxiUI/AAAAAAAAECQ/Snicpch7vFQbFepgw29oNOdO0mBQldRegCKgB/s1600/hexagon_spirals_cos.gif" /></a></div>
<div style="text-align: center;">
Equations:cos(π p(x + k sgn(y)/2, y)) = 0</div>
<div style="text-align: center;">
k = 1, 2, 3, ..., 8</div>
<div style="text-align: center;">
p(x,y) = sqrt(x² + y²) sec(π/n) cos(mod(arctan(y/x),2π/n) - π/n)</div>
<div style="text-align: center;">
n = 6</div>
<br />
<br />
( Mathematical softwares used: Graph, gnuplot )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com2tag:blogger.com,1999:blog-2261985361614499342.post-22496724829620597912016-06-22T21:42:00.000-07:002016-06-22T22:10:24.904-07:00Folding Star Polygons using Equations (用方程式摺星形多邊形)<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-HoS_TY5GRI0/V2tkvtbK4vI/AAAAAAAAEA8/VxaZSvTD7a87k_TdI1CuHaqLBKWyjUKogCKgB/s1600/folding_star_polygons_n8_k3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-HoS_TY5GRI0/V2tkvtbK4vI/AAAAAAAAEA8/VxaZSvTD7a87k_TdI1CuHaqLBKWyjUKogCKgB/s1600/folding_star_polygons_n8_k3.gif" /></a></div>
<div style="text-align: center;">
Regular star polygon {8/3}</div>
<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-yQOmxM5ZjKg/V2tkyzmf5PI/AAAAAAAAEA8/5WZAc0PaXwoj6R7tNL76Wi74oMQiFS_PACKgB/s1600/folding_star_polygons_n12_k4.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-yQOmxM5ZjKg/V2tkyzmf5PI/AAAAAAAAEA8/5WZAc0PaXwoj6R7tNL76Wi74oMQiFS_PACKgB/s1600/folding_star_polygons_n12_k4.gif" /></a></div>
<div style="text-align: center;">
Regular star polygon {12/4}</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-_QYWCw_FM54/V2tk3OxbD1I/AAAAAAAAEA8/pzgR5iIMOCktD-kg7_PMBaJ8rYL081Y2ACKgB/s1600/folding_star_polygons_n12_k5.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-_QYWCw_FM54/V2tk3OxbD1I/AAAAAAAAEA8/pzgR5iIMOCktD-kg7_PMBaJ8rYL081Y2ACKgB/s1600/folding_star_polygons_n12_k5.gif" /></a></div>
<div style="text-align: center;">
Regular star polygon {12/5}</div>
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<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-moIMZBb4VBw/V2tmK7yRRQI/AAAAAAAAEBA/v8eBR13ImyEj1TiVIvHP9SqMBx5xwOi7ACLcB/s1600/folding_star_polygons_n10_k4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-moIMZBb4VBw/V2tmK7yRRQI/AAAAAAAAEBA/v8eBR13ImyEj1TiVIvHP9SqMBx5xwOi7ACLcB/s400/folding_star_polygons_n10_k4.png" width="400" /></a></div>
<div style="text-align: center;">
Regular star polygon {10/4}</div>
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<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-cGScCe6G6mo/V2tmV7XRW6I/AAAAAAAAEBI/XiL-lmIDutg44FbBAY-iAulqiJgVHNNQwCLcB/s1600/folding_star_polygons_n15_k6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://2.bp.blogspot.com/-cGScCe6G6mo/V2tmV7XRW6I/AAAAAAAAEBI/XiL-lmIDutg44FbBAY-iAulqiJgVHNNQwCLcB/s400/folding_star_polygons_n15_k6.png" width="400" /></a></div>
<div style="text-align: center;">
Regular star polygon {15/6}</div>
<br />
<br />
( Mathematical softwares used: Graph, gnuplot )<br />
<br />
<br />
<span style="font-size: large;">
Related posts:</span><br />
<ul>
<li><a href="http://benice-equation.blogspot.com/2016/06/folding-stars-using-equations.html">Folding Stars using Equations</a></li>
<li><a href="http://benice-equation.blogspot.com/2011/10/regular-star-polygons.html">Regular Polygons and Regular Star Polygons</a></li>
</ul>
<div>
<br /></div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-74674754901931864492016-06-17T14:23:00.000-07:002016-08-28T21:40:30.508-07:00Folding Stars using Equations (用方程式摺星星)<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-EKYX3-cH6sE/V2RkvJ9SuOI/AAAAAAAAD-0/uXhsanPP7CkAJjZKOvXhDnVKUh8oVE7pACKgB/s1600/folding_stars_n5_01.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-EKYX3-cH6sE/V2RkvJ9SuOI/AAAAAAAAD-0/uXhsanPP7CkAJjZKOvXhDnVKUh8oVE7pACKgB/s1600/folding_stars_n5_01.gif" /></a></div>
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<a href="https://2.bp.blogspot.com/-KOErtHV1omk/V2Rk1huawGI/AAAAAAAAD_M/0_7Jvddm3Fg0uZ3ypHKsb6g4kevIK7jXgCKgB/s1600/folding_stars_n5_04.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-KOErtHV1omk/V2Rk1huawGI/AAAAAAAAD_M/0_7Jvddm3Fg0uZ3ypHKsb6g4kevIK7jXgCKgB/s1600/folding_stars_n5_04.gif" /></a></div>
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<a href="https://3.bp.blogspot.com/-S8Wa56ZnPvQ/V2Rk6QBTWKI/AAAAAAAAD_o/qNcIUdKfO00ymUdGlDqBqPW9D7zbyyv3ACKgB/s1600/folding_stars_n6_02.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-S8Wa56ZnPvQ/V2Rk6QBTWKI/AAAAAAAAD_o/qNcIUdKfO00ymUdGlDqBqPW9D7zbyyv3ACKgB/s1600/folding_stars_n6_02.gif" /></a></div>
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<h2>
Example 1.</h2>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-ozA1w23f004/V2RnY4vFV7I/AAAAAAAAD_s/RekEI5uND1QL-njlBJFq5tdNGWQ2BsZlACLcB/s1600/folding_stars_n5_example.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-ozA1w23f004/V2RnY4vFV7I/AAAAAAAAD_s/RekEI5uND1QL-njlBJFq5tdNGWQ2BsZlACLcB/s1600/folding_stars_n5_example.png" /></a></div>
<div style="text-align: center;">
(<span style="text-align: start;">n = 5, </span>ratio = 1)<br />
<div style="text-align: left;">
<br /></div>
</div>
<h3>
gnuplot script 1:</h3>
<br />
n = 5<br />
ratio = 1.0<br />
A = 2*pi/n<br />
c = 2*cos(A)<br />
d = ratio*c*(1 - cos(A))/(1 + cos(A))<br />
max(x,y) = x < y ? y : x<br />
f1(x,y) = max(max(abs(y - c) - d, y*cos(A) - x*sin(A) - c/cos(A)), \<br />
max(d - abs(y*cos(A) - x*sin(A) - c), y*cos(3*A) - x*sin(3*A) - c + d))<br />
f2(x,y) = f1(y*sin(A) + x*cos(A), y*cos(A) - x*sin(A))<br />
f3(x,y) = f1(y*sin(2*A) + x*cos(2*A), y*cos(2*A) - x*sin(2*A))<br />
f4(x,y) = f1(y*sin(3*A) + x*cos(3*A), y*cos(3*A) - x*sin(3*A))<br />
f5(x,y) = f1(y*sin(4*A) + x*cos(4*A), y*cos(4*A) - x*sin(4*A))<br />
set xrange [-2.5:2.5]<br />
set yrange [-2.5:2.5]<br />
set size ratio -1<br />
set samples 256<br />
set isosamples 256<br />
set contour base<br />
set cntrparam levels discrete 0.0<br />
unset key<br />
unset surface<br />
set table 'f1.dat'<br />
splot f1(x,y)<br />
unset table<br />
set table 'f2.dat'<br />
splot f2(x,y)<br />
unset table<br />
set table 'f3.dat'<br />
splot f3(x,y)<br />
unset table<br />
set table 'f4.dat'<br />
splot f4(x,y)<br />
unset table<br />
set table 'f5.dat'<br />
splot f5(x,y)<br />
unset table<br />
plot 'f1.dat' w l lw 2 lc rgb "#0099ff", \<br />
'f2.dat' w l lw 2 lc rgb "#0099ff", \<br />
'f3.dat' w l lw 2 lc rgb "#0099ff", \<br />
'f4.dat' w l lw 2 lc rgb "#0099ff", \<br />
'f5.dat' w l lw 2 lc rgb "#0099ff"<br />
<div>
<br /></div>
<div>
<br /></div>
<h2>
Example 2.</h2>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-1luQeCcxF8w/V2Rn-PmL4JI/AAAAAAAAD_0/OHwpnMYgmnwIjRAVOiHtYJx9LzPieWLnQCLcB/s1600/folding_stars_n6_example.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-1luQeCcxF8w/V2Rn-PmL4JI/AAAAAAAAD_0/OHwpnMYgmnwIjRAVOiHtYJx9LzPieWLnQCLcB/s1600/folding_stars_n6_example.png" /></a></div>
<div>
<div style="text-align: center;">
(<span style="text-align: start;">n = 6, </span>ratio = 1)<br />
<div style="text-align: left;">
<br /></div>
</div>
</div>
<h3>
gnuplot script 2:</h3>
<div>
<div>
<br />
n = 6<br />
ratio = 1.0</div>
<div>
A = 2*pi/n</div>
<div>
c = 1</div>
<div>
d = ratio*c/3</div>
<div>
max(x,y) = x < y ? y : x</div>
<div>
f1(x,y) = max(max(abs(y - c) - d, y - sqrt(3)*x - 4*c), \</div>
<div>
max(d - abs(y*cos(A) - x*sin(A) - c), y*cos(4*A) - x*sin(4*A) - c + d))</div>
<div>
f2(x,y) = f1(y*sin(A) + x*cos(A), y*cos(A) - x*sin(A))</div>
<div>
f3(x,y) = f1(y*sin(2*A) + x*cos(2*A), y*cos(2*A) - x*sin(2*A))</div>
<div>
f4(x,y) = f1(y*sin(3*A) + x*cos(3*A), y*cos(3*A) - x*sin(3*A))</div>
<div>
f5(x,y) = f1(y*sin(4*A) + x*cos(4*A), y*cos(4*A) - x*sin(4*A))</div>
<div>
f6(x,y) = f1(y*sin(5*A) + x*cos(5*A), y*cos(5*A) - x*sin(5*A))</div>
<div>
set xrange [-2.5:2.5]</div>
<div>
set yrange [-2.5:2.5]</div>
<div>
set size ratio -1</div>
<div>
set samples 256</div>
<div>
set isosamples 256</div>
<div>
set contour base</div>
<div>
set cntrparam levels discrete 0.0</div>
<div>
unset key</div>
<div>
unset surface</div>
<div>
set table 'f1.dat'</div>
<div>
splot f1(x,y)</div>
<div>
unset table</div>
<div>
set table 'f2.dat'</div>
<div>
splot f2(x,y)</div>
<div>
unset table</div>
<div>
set table 'f3.dat'</div>
<div>
splot f3(x,y)</div>
<div>
unset table</div>
<div>
set table 'f4.dat'</div>
<div>
splot f4(x,y)</div>
<div>
unset table</div>
<div>
set table 'f5.dat'</div>
<div>
splot f5(x,y)</div>
<div>
unset table</div>
<div>
set table 'f6.dat'</div>
<div>
splot f6(x,y)</div>
<div>
unset table</div>
<div>
plot 'f1.dat' w l lw 2 lc rgb "#0099ff", \</div>
<div>
'f2.dat' w l lw 2 lc rgb "#0099ff", \</div>
<div>
'f3.dat' w l lw 2 lc rgb "#0099ff", \</div>
<div>
'f4.dat' w l lw 2 lc rgb "#0099ff", \</div>
<div>
'f5.dat' w l lw 2 lc rgb "#0099ff", \</div>
<div>
'f6.dat' w l lw 2 lc rgb "#0099ff"</div>
</div>
<div>
<br /></div>
<div>
<br /></div>
<div>
<h2>
Example 3.</h2>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-WXMqCk-53cA/V2fxRwP1UUI/AAAAAAAAEAM/5kBwSGHP26wbJY1QhfS97q1qum-QzL9NACLcB/s1600/folding_stars_n8_example.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-WXMqCk-53cA/V2fxRwP1UUI/AAAAAAAAEAM/5kBwSGHP26wbJY1QhfS97q1qum-QzL9NACLcB/s1600/folding_stars_n8_example.png" /></a></div>
<div style="text-align: center;">
(<span style="text-align: start;">n = 8, </span>ratio = 1)<br />
<div style="text-align: left;">
<br /></div>
</div>
<h3>
Download Graph files:</h3>
<h3>
<ul>
<li><a href="http://depositfiles.com/files/bkst5xq0f" target="_blank">folding_stars.zip</a></li>
</ul>
</h3>
<br />
<br /></div>
<div>
( Mathematical softwares used: Graph, gnuplot )<br />
<br />
<br />
<span style="font-size: large;">Related posts:</span><br />
<ul>
<li><a href="http://benice-equation.blogspot.com/2016/06/folding-star-polygons.html">Folding Star Polygons using Equations</a></li>
</ul>
<div>
<br /></div>
</div>
benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-44300419509886413312016-06-12T07:20:00.000-07:002016-06-12T07:26:21.755-07:00Momotaro Peach Boy (桃太郎)<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-pKft7FbEJv4/V11tCoZ7CNI/AAAAAAAAD-Y/_7IiLN4IZDMMZ-FpTJaSHo6p2LdvezpqwCLcB/s1600/momotaro_peach_boy.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-pKft7FbEJv4/V11tCoZ7CNI/AAAAAAAAD-Y/_7IiLN4IZDMMZ-FpTJaSHo6p2LdvezpqwCLcB/s1600/momotaro_peach_boy.png" /></a></div>
<div style="text-align: center;">
<b>Inequalities:</b></div>
<div style="text-align: center;">
(x² + 4(y+1)²) [(x² + y² - 3)sqrt(x² + y²) - a(1 - by/sqrt(x² + y²))^ln(c + abs(x)/sqrt(x² + y²))] < d</div>
<div style="text-align: center;">
a = 5, -1.25, -2.1, -2.2, -2.36, -2.6</div>
<div style="text-align: center;">
b = 0.84, c = 0.5, d = 0.1</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-fmjx98lvUhI/V11sobB9YFI/AAAAAAAAD-g/WxHaGE-8Fk870-Mzb6eI_OmZTckLeSfNwCKgB/s1600/momotaro_peach_boy.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-fmjx98lvUhI/V11sobB9YFI/AAAAAAAAD-g/WxHaGE-8Fk870-Mzb6eI_OmZTckLeSfNwCKgB/s1600/momotaro_peach_boy.gif" /></a></div>
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( Softwares used: Graph, PhotoScape )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-16445075499477265492016-06-10T04:30:00.000-07:002016-06-10T06:55:51.397-07:00Lion Man<div class="separator" style="clear: both; text-align: center;">
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<a href="https://1.bp.blogspot.com/-uEmTiBl_FN8/V1qi3k6qUxI/AAAAAAAAD9Q/5-2FiHlqHIImbap21C02Zz8tf16XC04rwCLcB/s1600/lion_man_02.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-uEmTiBl_FN8/V1qi3k6qUxI/AAAAAAAAD9Q/5-2FiHlqHIImbap21C02Zz8tf16XC04rwCLcB/s400/lion_man_02.png" width="400" /></a></div>
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<br />
<br />
<h3>
Equations:</h3>
<h3>
(x² + y² - 3) sqrt(x² + y²)</h3>
<h3>
= a(1 + sign(a)by/sqrt(x² + y²))^ln(c + abs(x)/sqrt(x² + y²))</h3>
<br />
<br />
<h3>
Examples:</h3>
<h3>
<a href="http://forum.padowan.dk/viewtopic.php?id=1024" target="_blank">Download page</a></h3>
<br />benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-4689338563795220512016-06-05T23:28:00.000-07:002016-06-05T23:34:49.938-07:00Weird Face - a family of implicit polar equations<div class="separator" style="clear: both; text-align: center;">
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( Softwares used: gnuplot, PhotoScape )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0tag:blogger.com,1999:blog-2261985361614499342.post-45451098757560303732016-06-05T23:15:00.000-07:002016-06-05T23:15:40.466-07:00Face Slimming<div class="separator" style="clear: both; text-align: center;">
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The main equations are:</h3>
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r³ - 3r + 1.95 = 4 abs(sinθ)^(c + sinθ)</h3>
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(c = 1, 2, 3, ..., 11)</h3>
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( Softwares used: gnuplot, PhotoScape )benicehttp://www.blogger.com/profile/09841851785144902423noreply@blogger.com0