︠1399bbdf-b25a-40e7-9c76-bc904be5ca09i︠
%html
<font size=4>
<p><span style="font-size: xx-small;"><span style="font-weight: normal;"><span style="font-size: x-small;"><span style="font-weight: normal;"><span style="font-size: medium;">This work is licensed under a&nbsp;<a href="http://creativecommons.org/licenses/by-sa/3.0/us/" rel="license">Creative Commons Attribution-Share Alike 3.0 United States License</a>.</span></span></span></span></span></p>
<p><a href="http://creativecommons.org/licenses/by-sa/3.0/us/" rel="license"><img style="border: 0px initial initial;" src="http://i.creativecommons.org/l/by-sa/3.0/us/88x31.png" alt="Creative Commons License" /></a></p><br>
<h3>
    Further Plotting Techiques in Sage
</h3>
    Thanks to Sage's integration of projects like <a href="http://matplotlib.org/">matplotlib</a> and <a href="http://jmol.sourceforge.net/">Jmol</a>, Sage has comprehensive plotting capabilities. This worksheet aims to go somewhat in depth with them. In particular, we will be looking at <a href="http://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot.html">2D plotting</a> functionality (also see the <a href="http://doc.sagemath.org/html/en/reference/plotting/index.html">2D Graphics</a> documentation page).</font>
︡0be37d1a-d8c3-42b6-b0c7-2e46ef7ef4be︡{"done":true,"html":"<font size=4>\n<p><span style=\"font-size: xx-small;\"><span style=\"font-weight: normal;\"><span style=\"font-size: x-small;\"><span style=\"font-weight: normal;\"><span style=\"font-size: medium;\">This work is licensed under a&nbsp;<a href=\"http://creativecommons.org/licenses/by-sa/3.0/us/\" rel=\"license\">Creative Commons Attribution-Share Alike 3.0 United States License</a>.</span></span></span></span></span></p>\n<p><a href=\"http://creativecommons.org/licenses/by-sa/3.0/us/\" rel=\"license\"><img style=\"border: 0px initial initial;\" src=\"http://i.creativecommons.org/l/by-sa/3.0/us/88x31.png\" alt=\"Creative Commons License\" /></a></p><br>\n<h3>\n    Further Plotting Techiques in Sage\n</h3>\n    Thanks to Sage's integration of projects like <a href=\"http://matplotlib.org/\">matplotlib</a> and <a href=\"http://jmol.sourceforge.net/\">Jmol</a>, Sage has comprehensive plotting capabilities. This worksheet aims to go somewhat in depth with them. In particular, we will be looking at <a href=\"http://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot.html\">2D plotting</a> functionality (also see the <a href=\"http://doc.sagemath.org/html/en/reference/plotting/index.html\">2D Graphics</a> documentation page).</font>"}
︠8eb91146-3730-4b7d-8cf6-c10c1f6c67dfi︠
%html
<font size=4>
<h4>
    Cartesian Plots
</h4>
    A simple quadratic is easy.Note that alternatively, you can also use <code>plot(x^2,-2,2)</code> or <code>plot(x^2,xmin=-2,xmax=2)</code>.</font>
︡980a1956-e3c4-47d7-9eaf-925159c3cffe︡{"done":true,"html":"<font size=4>\n<h4>\n    Cartesian Plots\n</h4>\n    A simple quadratic is easy.Note that alternatively, you can also use <code>plot(x^2,-2,2)</code> or <code>plot(x^2,xmin=-2,xmax=2)</code>.</font>"}
︠7114bdd4-6beb-4d39-84f7-52d1885649b4s︠
plot(x^2,(x,-2,2))
︡5cc0ed96-6dcb-4260-97ac-3c22b336d6cb︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_1ALN8I.svg","show":true,"text":null,"uuid":"4629ee03-e3d8-496c-b1e5-f0205b655ca5"},"once":false}︡{"done":true}︡
︠891cf39b-2acb-41cd-8dd3-8e9484403d7ci︠
%html
<font size=4>
You can combine graphics object by adding them.</font>
︡8808cb04-f33c-4396-8e70-9c47513fe5e2︡{"done":true,"html":"<font size=4>\nYou can combine graphics object by adding them.</font>"}
︠6ef3a1fc-202d-404b-9a8c-26f03f49c909s︠
regular = plot(x^2,(x,-2,2),color="purple")
skinny = plot(4*x^2,(x,-2,2),color="darkgreen")
regular + skinny
︡aa785756-16a9-4145-896a-4e7864790623︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_lRqYA0.svg","show":true,"text":null,"uuid":"b9f5b4e2-c67c-4148-b99a-5165a185657d"},"once":false}︡{"done":true}︡
︠f40f181f-32e1-48ab-bc62-a94aea77d135i︠
%html
<font size=4>
<b>Problem:</b> Plot a green $y=\sin(x)$ together with a red $y=2\cos(x)$. Note that you can use <code>pi</code> as part of your range.
</font>
︡c376e496-dbea-4095-951b-f03598a23d80︡{"done":true,"html":"<font size=4>\n<b>Problem:</b> Plot a green $y=\\sin(x)$ together with a red $y=2\\cos(x)$. Note that you can use <code>pi</code> as part of your range.\n</font>"}
︠144ecb5a-e292-4752-8410-15dea4390f66︠

︡cd809d4c-98a1-4d86-9e86-35230a5c1692︡
︠7c62d995-9a29-4b88-ab19-c124afe89066i︠
%html
<font size=4>
Boundaries of a plot can be specified, in addition to the overall size.</font>

︡697e5e3a-5a20-4372-a965-060c593a14f4︡{"done":true,"html":"<font size=4>\nBoundaries of a plot can be specified, in addition to the overall size.</font>"}
︠2c2511ba-94ce-430e-8968-4751e602cba8s︠
plot(1+e^(-x^2),xmin=-2,xmax=2,ymin=0,ymax=2.5,figsize=10)
︡d814de56-a6c0-4699-8797-38b1bc76a4e9︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_lYkRcC.svg","show":true,"text":null,"uuid":"ada46515-ec1e-4d6a-89d4-2e1db76f8100"},"once":false}︡{"done":true}︡{"done":true}︡
︠1fa628b3-5a08-4039-aa91-150ebf99b09ci︠
%html
<font size=4>
<b>Problem:</b> Plot $y=5+3 \sin(4x)$ with suitable boundaries.</font>
︡93bbd135-8fa6-4d19-bc02-b51eb49ed703︡{"done":true,"html":"<font size=4>\n<b>Problem:</b> Plot $y=5+3 \\sin(4x)$ with suitable boundaries.</font>"}
︠6cc22a80-25d0-49da-a17d-a1adab859a9d︠

︡a4e9113b-433d-4197-8517-bcaa384d831b︡
︠795073bf-22fd-4636-9fec-96743cb4020ai︠
%html
<font size=4>
You can add lots of extra information.</font>
︡ff8150f9-d3a7-46ef-8103-4de626f4624c︡{"done":true,"html":"<font size=4>\nYou can add lots of extra information.</font>"}
︠6e5b258f-9905-49dd-8a27-0a3d7ac2cc4cs︠
exponential = plot(1+e^(-x^2),xmin=-2,xmax=2,ymin=0,ymax=2.5)
max_line = plot(2,xmin=-2,xmax=2,ymin=0,ymax=2.5, linestyle="-.", color="red")
min_line = plot(1,xmin=-2,xmax=2,ymin=0,ymax=2.5, linestyle=":", color="magenta")
exponential + max_line + min_line

︡fe18462b-e628-4e71-953a-5f420bd0958e︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_b3dPZ3.svg","show":true,"text":null,"uuid":"defa4be3-6d57-49c9-95bb-ff978fcc10c6"},"once":false}︡{"done":true}︡
︠45ac074b-6525-4baf-be51-0ba2e123efcei︠
%html
<font size=4>
You can fill regions with transparent color, and thicken the curve. This example uses several such options.</font>

︡b7263974-267b-4b11-ae1f-3af907914768︡{"done":true,"html":"<font size=4>\nYou can fill regions with transparent color, and thicken the curve. This example uses several such options.</font>"}
︠77ab76a6-24f4-4f63-b74d-2f4dd7767173s︠
exponential = plot(1+e^(-x^2),xmin=-2,xmax=2,ymin=0,ymax=2.5,fill=.5,fillcolor='grey',fillalpha=.3)
min_line = plot(1,xmin=-2,xmax=2,ymin=0,ymax=2.5, linestyle="-", thickness=6,color="magenta")
exponential + min_line
︡d424061d-52cf-42b0-b6ff-44552d08bfd7︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_EyYjNC.svg","show":true,"text":null,"uuid":"7a17a0cc-1b66-4e5e-9ae4-6f9cc357dbed"},"once":false}︡{"done":true}︡
︠dbf248d9-e7ff-4948-9119-ef21141fd45ai︠
%html
<font size=4>
You can also combine plots using the <code>sum</code> function and list comprehension.</font>

︡eb41cd6b-6247-44f4-bc43-1b4a1259fb91︡{"done":true,"html":"<font size=4>\nYou can also combine plots using the <code>sum</code> function and list comprehension.</font>"}
︠35f3aa72-694e-4a06-80be-17215f1812c7s︠
sum([plot(x^n,(x,0,1), color=rainbow(5)[n]) for n in range(5)])
︡4fd0ee34-f1ef-45b7-a71a-d7463de0c3cc︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_xC_IAZ.svg","show":true,"text":null,"uuid":"471b9799-3df7-4cf3-b6c9-d1b87e45f85a"},"once":false}︡{"done":true}︡
︠0782a8bd-53bb-4a62-b20a-b5ebbd2d067bi︠
%html
<font size=4>
<b>Problem:</b>
Create a plot showing the cross-section area for the following solid of revolution problem: <br>
    Consider the area bounded by $y=x^2-3x+6$ and the line $y=4$. Find the volume created byrotating this area around the line $y=1$.<br>
(Note: You don't have to calculate this volume, all we want here is the appropriate picture of the cross-section.)</font>
︡9648f934-2659-4500-95b9-0bf348db7197︡{"done":true,"html":"<font size=4>\n<b>Problem:</b>\nCreate a plot showing the cross-section area for the following solid of revolution problem: <br>\n    Consider the area bounded by $y=x^2-3x+6$ and the line $y=4$. Find the volume created byrotating this area around the line $y=1$.<br>\n(Note: You don't have to calculate this volume, all we want here is the appropriate picture of the cross-section.)</font>"}
︠6fc61bfb-5fd4-4810-bf83-5d94fc82c62d︠

︡0a42f86a-501d-44d9-987f-756577b4510a︡
︠dae1a3e4-8a44-431b-a3e4-ed41e0e55445i︠
%html
<font size=4>
You can also label the axes and add labels to your graphs.</font>

︡1276c28c-4cb0-475a-8dbc-6878b731a5f1︡{"done":true,"html":"<font size=4>\nYou can also label the axes and add labels to your graphs.</font>"}
︠ea2e55dc-e77b-49fe-8f01-463f3225e84fs︠
p = plot(sin(pi*x), -pi, pi, legend_label='f(x) = sin(pi x)', color="red", linestyle="--")
p.axes_labels(['x','y=f(x)'])
p.show()
︡ccaafe12-299c-464d-b37e-32918bcea22a︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_n2f2m8.svg","show":true,"text":null,"uuid":"20c9e70c-be8d-4b87-8c5d-819a3dec1ad4"},"once":false}︡{"done":true}︡
︠c1fd9f46-865a-4b7b-bd80-8c9d6defce7ai︠
%html
<font size=4>
Here is an example where we combine two such plots, using $\LaTeX$ typesetting for the labels:
</font>
︡5d062b99-13eb-4938-ac48-2ff60c0cd247︡{"done":true,"html":"<font size=4>\nHere is an example where we combine two such plots, using $\\LaTeX$ typesetting for the labels:\n</font>"}
︠b19b5505-8da9-4e8e-9995-5e40f5379c01s︠
p1 = plot(sin(pi*x), -pi, pi, legend_label='$f(x) = \sin(\pi x)$', color="red", linestyle="--",axes_labels=['$x$','$y$'])
p2 = plot(cos(pi*x), -pi, pi, legend_label='$g(x) = \cos(\pi x)$', color="blue", linestyle=":",axes_labels=['$x$','$y$'])
p1 + p2
︡7fcc0941-1bfc-4b97-b07f-d1cba97bf08a︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_7kHSzy.svg","show":true,"text":null,"uuid":"bf9ded99-48cd-46ba-9827-d47f9f12c39b"},"once":false}︡{"done":true}︡
︠f783e356-7579-42ad-b127-02d29cf0fe08i︠
%html
<font size=4>
<b>Problem:</b>
In econmics, a demand curve $q(p) = \frac{1000}{p}-10$ relates the quantity $q$ demanded by the market of some product to its price $p$. Plot this function with appropriate informative labels.</font>

︡82569fa9-9540-4694-a180-7f690876fbe6︡{"done":true,"html":"<font size=4>\n<b>Problem:</b>\nIn econmics, a demand curve $q(p) = \\frac{1000}{p}-10$ relates the quantity $q$ demanded by the market of some product to its price $p$. Plot this function with appropriate informative labels.</font>"}
︠f5990ef4-0ac3-4c50-ba3d-92fbe5d0ecca︠

︡8c61dc03-8b6d-4d50-95ea-f9d22704ec21︡
︠c9a42a3b-dda4-46ad-a355-ff812c124d76i︠
%html
<font size=4>
You can also add grid lines to a plot, or frame it. Have a look at the following examples.</font>

︡8d83124d-eeb0-468d-8ed2-6b6e2b0fb9a7︡{"done":true,"html":"<font size=4>\nYou can also add grid lines to a plot, or frame it. Have a look at the following examples.</font>"}
︠cae9784b-b077-479a-8577-faa045522d4cs︠
plot(x^3-x,-2,2,gridlines=True,frame=False)
plot(x^3-x,-2,2,gridlines='minor',frame=True)
plot(x^3-x,-2,2,gridlines=True,frame=True)
plot(x^3-x,-2,2,gridlines='minor',frame=False)
plot(x^3-x,-2,2,gridlines=False,frame=False)
plot(x^3-x,-2,2,gridlines=False,frame=False)

︡e326e00e-d351-4650-a362-74cd63f9b949︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_S6pKap.svg","show":true,"text":null,"uuid":"d06b6307-ead9-4cd2-a5ae-63d6c43b6313"},"once":false}︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_1mAYdR.svg","show":true,"text":null,"uuid":"e6cbcb85-7125-4619-bace-bb1562bda894"},"once":false}︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_zzCXDS.svg","show":true,"text":null,"uuid":"ef7a3908-341d-4a34-8420-da154051a329"},"once":false}︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_z7ArZ3.svg","show":true,"text":null,"uuid":"08c8a545-50be-4ff9-aa93-1d76ca9b18f4"},"once":false}︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_kegV4j.svg","show":true,"text":null,"uuid":"e7def3ad-22a7-4dc0-9fcc-8afbf090fd39"},"once":false}︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_P6ipWv.svg","show":true,"text":null,"uuid":"491a0122-8982-4a50-a1e8-8d40ba925d15"},"once":false}︡{"done":true}︡
︠88df7328-226e-4982-b132-e4c57f0b8f87i︠
%html
<font size=4>
Here is an example where we add an arrow and some text to a plot.</font>

︡876af715-125f-4dc9-8789-5edc600bcc07︡{"done":true,"html":"<font size=4>\nHere is an example where we add an arrow and some text to a plot.</font>"}
︠eea5c177-7f19-4da7-a3a2-ef9373c3b189s︠
tangent_line=plot(2*x,-2,2,gridlines='minor',color="red")
curve = plot(x^2+1,-2,2)
big_dot= point((1,2), size=60)
the_arrow = arrow((0.2,2.8),(0.9,2.1))
the_words = text('f(x) = x^2+1 and tangent line at (1,2)', (0.5,-2.75),fontsize=20,color="black")
big_image=tangent_line+curve+big_dot+the_arrow+the_words
big_image.show()
︡ea5a1579-071f-464a-929b-643086ce2ce6︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_w88Z8B.svg","show":true,"text":null,"uuid":"234fdb5e-f430-422a-add8-d1da67c8adaf"},"once":false}︡{"done":true}︡
︠677b5271-5ef6-47e1-a76f-d3a9b1e25dcbi︠
    
%html
<font size=4>
We can add text (using $\LaTeX$ typesetting) to graph an integral.</font>

︡ddb7bf8a-6bdf-4bf0-8c5e-06f4e2071f8b︡{"done":true,"html":"<font size=4>\nWe can add text (using $\\LaTeX$ typesetting) to graph an integral.</font>"}
︠d5e8ba4f-bbf3-410e-a06c-cd861bdd0219s︠
p1= plot(sqrt(x),0,2,fill=True)
p2=text('$\int_0^2 \sqrt{x}\,dx$',(0.5,1.1),fontsize=20,color='black')
p1 + p2
︡77f7f05b-529b-46b5-b8e1-66b7ea5ded3a︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_uG_mHh.svg","show":true,"text":null,"uuid":"45e53364-bb22-4262-b8c6-42fed72a5790"},"once":false}︡{"done":true}︡
︠654b78e1-8575-48df-ae6f-e86e4444486ci︠
%html
<font size=4>
<b>Problem:</b>
Graph the integral for the function $\sqrt{x}-1$ for $x$ between $0$ and $2$.
</font>

︡7eb8ee28-58f9-4a70-94de-db7527a1e5f7︡{"done":true,"html":"<font size=4>\n<b>Problem:</b>\nGraph the integral for the function $\\sqrt{x}-1$ for $x$ between $0$ and $2$.\n</font>"}
︠51776797-ae08-4342-9f0c-d9348cb15cbc︠

︡706a8fc1-f176-4d18-9f00-cde0ef18cb7a︡
︠cddef212-70c7-4b4d-812c-dadc543ca1f3i︠
%html
<font size=4>
<h4>
    Parametric Plots
</h4>
    A parametric plot needs a list of two functions of the parameter, i.e., it is a function $t\mapsto (x(t), y(t))$ and thus from the one-dimensional space $\mathbb{R}$ to the two-dimensional space $\mathbb{R}^2$. Notice that we must declare $t$ as a variable first (the variable $x$ is automatically declared as such in Sage). Because the following graphic is slightly wider than it is tall, we use the <code>aspect_ratio</code>option (such options are also called <em>keywords</em>) to ensure the axes are correct for how we want to view this object.</font>

︡efca1cf0-050b-48b3-ae12-44d3f4fefccc︡{"done":true,"html":"<font size=4>\n<h4>\n    Parametric Plots\n</h4>\n    A parametric plot needs a list of two functions of the parameter, i.e., it is a function $t\\mapsto (x(t), y(t))$ and thus from the one-dimensional space $\\mathbb{R}$ to the two-dimensional space $\\mathbb{R}^2$. Notice that we must declare $t$ as a variable first (the variable $x$ is automatically declared as such in Sage). Because the following graphic is slightly wider than it is tall, we use the <code>aspect_ratio</code>option (such options are also called <em>keywords</em>) to ensure the axes are correct for how we want to view this object.</font>"}
︠4a5d1bca-7702-4bca-aaaf-1bbb2eb52294s︠
t=var('t')
parametric_plot([cos(t)+3*cos(t/9),sin(t)-3*sin(t/9)],(t,0,18*pi),fill=True,aspect_ratio=1)
︡3f68e36e-982a-49df-9fd2-935f1bcc7853︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_tcL_kO.svg","show":true,"text":null,"uuid":"43997129-20c9-4732-838a-455a4f4bf8d9"},"once":false}︡{"done":true}︡
︠ff7d4362-8bd1-4dee-a9e8-7df91c661991i︠
%html
<font size=4>
<b>Problem:</b>
The following parametric equations will create a so-called <em>hypocycloid</em>:
    $$x(t) = 17*\cos(t)+3*\cos(17t/3)$$
    $$y(t) = 17*\sin(t)-3*\sin(17t/3)$$
    Create this as a parametric plot.</font>

︡f6b49a55-1952-45f1-b6fc-f38bacd7a941︡{"done":true,"html":"<font size=4>\n<b>Problem:</b>\nThe following parametric equations will create a so-called <em>hypocycloid</em>:\n    $$x(t) = 17*\\cos(t)+3*\\cos(17t/3)$$\n    $$y(t) = 17*\\sin(t)-3*\\sin(17t/3)$$\n    Create this as a parametric plot.</font>"}
︠342aadee-1a85-4ada-9f9c-9b0e0a3e010e︠

︡79165055-d9b5-47d2-acf7-f4ba32b24475︡
︠a31fc253-b44b-4140-a637-8d1b3cc8b87di︠
%html
<font size=4>
Sage automatically plots a 2D or 3D plot, a curve or a surface, depending on how many variables and coordinates you specify. Have a look at the following examples (move the 3D examples around).</font>

︡d419009c-bd96-4cf4-8605-3d1bbcaf5a19︡{"done":true,"html":"<font size=4>\nSage automatically plots a 2D or 3D plot, a curve or a surface, depending on how many variables and coordinates you specify. Have a look at the following examples (move the 3D examples around).</font>"}
︠a7f4c601-216c-40c6-9566-c618a192a802s︠
t=var('t')
parametric_plot([log(1+t^2),sin(t)],(t,0,pi))
︡7dddd809-78f8-4fdf-b5cb-01811bd72c48︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_ai34MQ.svg","show":true,"text":null,"uuid":"e68e5a43-a9b2-481f-83ed-435cd8797d1f"},"once":false}︡{"done":true}︡
︠a508148b-e239-4109-bcbc-02dfda928544s︠
t=var('t')
parametric_plot3d([log(1+t^2),sin(t),cos(t)],(t,0,6*pi))
︡c049d77a-0473-4c83-8640-ba2a9012d7e6︡{"file":{"filename":"e213be43-e476-4d93-85a3-8614d1a92098.sage3d","uuid":"e213be43-e476-4d93-85a3-8614d1a92098"}}︡{"done":true}︡
︠515bbc2a-bff7-404b-9847-195b9033c922s︠
t,r=var('t,r')
parametric_plot([log(1+t^2),sin(r*t),cos(r*t)],(t,0,pi),(r,-1,1))
︡1c4f7545-b9b4-403b-b550-74fe251ac506︡{"file":{"filename":"b7d0b313-8407-47ff-be4e-65e40850996f.sage3d","uuid":"b7d0b313-8407-47ff-be4e-65e40850996f"}}︡{"done":true}︡
︠c9f609cb-b6ca-4c28-af22-9f20f7b79e30i︠
%html
<font size=4>
<b>Problem:</b> A neat example where we can use parametric plots are <em>Lissajous Curves</em>. These are curves of the form 
    $$x(t) = \sin(a t) \qquad y(t)=\sin(b t)$$
where $t\in [0,2\pi]$ and $a$ and $b$ are some relatively prime natural numbers. Draw a few such Lissajous Curves, changing $a$ and $b$ and observing how the curve changes accordingly.
</font>

︡9115db8e-d542-4cc4-af1f-6a408473bda0︡{"done":true,"html":"<font size=4>\n<b>Problem:</b> A neat example where we can use parametric plots are <em>Lissajous Curves</em>. These are curves of the form \n    $$x(t) = \\sin(a t) \\qquad y(t)=\\sin(b t)$$\nwhere $t\\in [0,2\\pi]$ and $a$ and $b$ are some relatively prime natural numbers. Draw a few such Lissajous Curves, changing $a$ and $b$ and observing how the curve changes accordingly.\n</font>"}
︠7e8d0a8f-650b-42ce-976c-f98aff2f39dc︠

︡9c08ec1a-e2ee-4ff2-852e-10921d783ea2︡
︠49bf4d64-b807-48b5-87aa-e049e80a11cd︠

︡17e09b22-5863-4b0f-a3b4-ca2c8927a35d︡
︠52d3f570-1c54-4cc6-8dd4-6268e1af2df3i︠
%html
<font size=4>
<h4>
    Polar Plots
</h4>
    Sage can do polar plots - from your knowledge of complex numbers (and their polar form) can you figure out how they work? So, here a curve called a <em>cardioid</em>:</font>

︡497970f4-65cf-4c47-bdac-c03831998ed5︡{"done":true,"html":"<font size=4>\n<h4>\n    Polar Plots\n</h4>\n    Sage can do polar plots - from your knowledge of complex numbers (and their polar form) can you figure out how they work? So, here a curve called a <em>cardioid</em>:</font>"}
︠f2331925-dd95-407c-b6db-e84b0737d911s︠
polar_plot(2+2*cos(x),(x,0,2*pi),color=hue(0.8),thickness=4)
︡df45220f-747a-4e8b-b346-2c583eccda84︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_RhGA4s.svg","show":true,"text":null,"uuid":"24a3c5d7-0698-41c4-9aba-9752b2dbd455"},"once":false}︡{"done":true}︡
︠998e1c60-8992-43f4-8936-e32848edf741i︠
%html
<font size=4>
More than one polar curve can be specified in a list.  Notice the automatic graded shading of the fill color.</font>


︡143ab63c-c40d-4448-a7e4-f06c621a6bab︡{"done":true,"html":"<font size=4>\nMore than one polar curve can be specified in a list.  Notice the automatic graded shading of the fill color.</font>"}
︠949b07ce-4f71-495f-857b-6bfd3f814acfs︠
t=var('t')
polar_plot([cos(4*t)+1.5, 0.5*cos(4*t)+2.5],(t,0,2*pi),color="darkgreen",thickness=2, fill=True,fillcolor='orange')

︡7f4e5acf-622b-484e-bf3c-6e2a6accd2fe︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_0trVp9.svg","show":true,"text":null,"uuid":"89510f03-ff8e-4d32-a24f-e23045bf4eb9"},"once":false}︡{"done":true}︡
︠3c1830fa-199a-4c7f-8fe3-5f8cbac29ee1i︠
%html
<font size=4>
<b>Problem:</b> Create a plot for the following problem: <br>
    Find the area that is inside the circle $r=2$, but outside the cardioid $2+2 \cos(\theta)$.</font>

︡58e6d0e7-7f5e-453a-a2dc-b38610f0199b︡{"done":true,"html":"<font size=4>\n<b>Problem:</b> Create a plot for the following problem: <br>\n    Find the area that is inside the circle $r=2$, but outside the cardioid $2+2 \\cos(\\theta)$.</font>"}
︠05322369-d905-43be-af68-4bfc1e50e4f3︠

︡c290b6fd-5ea6-43b8-aae4-9d56e7798cec︡
︠682c84d0-d295-490e-9550-9e42d11a70c3i︠
%html
<font size=4>
Sometimes the default value of points Sage uses to generate a plot is too low, and the plot looks too "crinkly". Have a look at the following example:</font>

︡62f03400-b818-41ec-8908-1a4095acd786︡{"done":true,"html":"<font size=4>\nSometimes the default value of points Sage uses to generate a plot is too low, and the plot looks too \"crinkly\". Have a look at the following example:</font>"}
︠fda094ba-326d-4451-b230-3d9a30c8aa65s︠
polar_plot(exp(x/10),(x,-12*pi,12*pi))
polar_plot(exp(x/10),(x,-12*pi,12*pi), plot_points=1000)
︡18ce8b52-ba8b-4ea9-bcd0-c4e9f1edc75a︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_lhDIWG.svg","show":true,"text":null,"uuid":"98d3bb43-8e62-4bd2-bafd-a5ccc7007791"},"once":false}︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_iY_mnn.svg","show":true,"text":null,"uuid":"5b691f2a-66e6-417f-ab45-207c04918e68"},"once":false}︡{"done":true}︡
︠c7554cfe-1caf-4b0c-8e8a-d28c51629468︠
%html
<font size=4>
<h4>
    Plotting Data Points
</h4>
    Sometimes one wishes to simply plot some data points. Here we demonstrate several ways of plotting points and data via the simple approximation to the Fibonacci numbers given by
$$F_n \approx \frac1{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n,$$
which is quite good after about $n=5$.<br>
    First, we notice that the Fibonacci numbers are build in.
</font>

︡40fcea11-ed99-4d4c-8108-61c66689eddc︡{"done":true,"html":"<font size=4>\n<h4>\n    Plotting Data Points\n</h4>\n    Sometimes one wishes to simply plot some data points. Here we demonstrate several ways of plotting points and data via the simple approximation to the Fibonacci numbers given by\n$$F_n \\approx \\frac1{\\sqrt{5}}\\left(\\frac{1+\\sqrt{5}}{2}\\right)^n,$$\nwhich is quite good after about $n=5$.<br>\n    First, we notice that the Fibonacci numbers are build in.\n</font>"}
︠a772d8e1-fee1-4ab5-b808-c836dfd1cd06s︠
fibonacci_sequence(6)
︡151e759f-b063-4e01-8158-b48417a689da︡{"stdout":"<generator object fibonacci_sequence at 0x7f6c9a8930f0>\n"}︡{"done":true}︡
︠5ff5293e-4e1a-4413-a2bd-c6a9f71931f0s︠
list(fibonacci_sequence(6))
︡40e39c99-0a3f-4830-97ff-decb573ef391︡{"stdout":"[0, 1, 1, 2, 3, 5]\n"}︡{"done":true}︡
︠b72e7868-30e3-4025-ab2c-656b637605aas︠
list(enumerate(fibonacci_sequence(6)))
︡5d8f2a31-104e-44ef-b9aa-b3f933f6c9c3︡{"stdout":"[(0, 0), (1, 1), (2, 1), (3, 2), (4, 3), (5, 5)]\n"}︡{"done":true}︡
︠b5c8e44f-3b9a-450e-a0df-36217873cc91i︠
%html
<font size=4>
In this cell, we just define the numbers and coordinate pairs we are about to plot.</font>

︡e61f8aab-843d-4022-b62b-0c5cc849edb9︡{"done":true,"html":"<font size=4>\nIn this cell, we just define the numbers and coordinate pairs we are about to plot.</font>"}
︠069a8a4f-f368-4848-b0d1-cfdb59051240s︠
n = var('n')
fibonacci = list(enumerate(fibonacci_sequence(6)))
f(n)= (1/sqrt(5))*((1+sqrt(5))/2)^n
asymptotic = [ (i, f(i)) for i in range(6)]
︡1395fe40-7f5f-4dfb-99c6-95908954f465︡{"done":true}︡
︠56ddba49-7cd4-4a4c-a56f-1761c44b4107s︠
fibonacci
︡cff4e094-0cce-4c16-a78c-4c97faaf4ba3︡{"stdout":"[(0, 0), (1, 1), (2, 1), (3, 2), (4, 3), (5, 5)]\n"}︡{"done":true}︡
︠a6cb074a-ee1c-47b5-8d77-2a5715bd08cds︠
asymptotic
︡0748eb89-feee-4c5b-9e74-106749d5f5a5︡{"stdout":"[(0, 1/5*sqrt(5)), (1, 1/10*sqrt(5)*(sqrt(5) + 1)), (2, 1/20*sqrt(5)*(sqrt(5) + 1)^2), (3, 1/40*sqrt(5)*(sqrt(5) + 1)^3), (4, 1/80*sqrt(5)*(sqrt(5) + 1)^4), (5, 1/160*sqrt(5)*(sqrt(5) + 1)^5)]\n"}︡{"done":true}︡
︠587e4971-bddb-48cd-bc24-d655259966fes︠
show(asymptotic)
︡2a372d71-c7fd-49c3-9b19-8ced8bb74ee1︡{"html":"<div align='center'>[($\\displaystyle 0$, $\\displaystyle \\frac{1}{5} \\, \\sqrt{5}$), ($\\displaystyle 1$, $\\displaystyle \\frac{1}{10} \\, \\sqrt{5} {\\left(\\sqrt{5} + 1\\right)}$), ($\\displaystyle 2$, $\\displaystyle \\frac{1}{20} \\, \\sqrt{5} {\\left(\\sqrt{5} + 1\\right)}^{2}$), ($\\displaystyle 3$, $\\displaystyle \\frac{1}{40} \\, \\sqrt{5} {\\left(\\sqrt{5} + 1\\right)}^{3}$), ($\\displaystyle 4$, $\\displaystyle \\frac{1}{80} \\, \\sqrt{5} {\\left(\\sqrt{5} + 1\\right)}^{4}$), ($\\displaystyle 5$, $\\displaystyle \\frac{1}{160} \\, \\sqrt{5} {\\left(\\sqrt{5} + 1\\right)}^{5}$)]</div>"}︡{"done":true}︡
︠e1145414-a073-483b-be3d-b7877ae39fa9i︠
%html
<font size=4>
And here we plot not just the two sets of points, but also use several of the documented options for plotting points; the choice of markers for different data is particularly helpful for generating publishable graphics.</font>


︡575f7377-936b-45db-83ff-8186202d3d1b︡{"done":true,"html":"<font size=4>\nAnd here we plot not just the two sets of points, but also use several of the documented options for plotting points; the choice of markers for different data is particularly helpful for generating publishable graphics.</font>"}
︠6da60030-4349-444a-8a5b-9313df65414ds︠
fib_plot=list_plot(fibonacci, color="red",marker="o",size=40)
asy_plot=line(asymptotic,marker="D",color="black",thickness=2)
show(fib_plot+asy_plot,aspect_ratio=1,figsize=12)
︡8756ad9a-6b83-4820-83e7-53c0fe6b9c30︡{"file":{"filename":"/projects/a717257b-7b34-4f0a-bccc-47996d6ed607/.sage/temp/compute7-us/17920/tmp_NgGaOo.svg","show":true,"text":null,"uuid":"04562848-273d-4202-b7ec-a6cbe2cefa84"},"once":false}︡{"done":true}︡
︠bfebeed4-9ffc-45fd-b2c5-1f38b0709601︠

︠eefb3dc8-926c-4b05-9a29-121426a84489︠