Special Curves in Polar Coordinates: r=C and θ=θ0

2019-10-13 09:08

This applet demonstrates two special curves in the polar coordinate system. When r is a constant C, the graph is a circle centered at the pole with radius C. When θ is a constant θ0, the graph is a ray from the pole at angle θ0. It visually reveals the geometric meanings of polar equations r=C and θ=θ0, aiding the understanding of polar coordinates.

#coordinate-geometry #function #graphing

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.560.0" app="classic" platform="d" id="8550128d-63e5-4c5c-8882-b2d8c18d79a4"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="1376" height="738" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.16787790697674418" orientation="1" />
	</panes>
	<views>
		<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 ,  20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
		<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
		<view id="8" toolbar="1001 | 1002 | 1003  || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
		<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="385" window="100,100,700,550" />
		<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1119" window="100,100,600,400" />
		<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="231" window="100,100,600,400" />
		<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
		<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="64" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
		<view id="70" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
	</views>
	<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71  14  68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
	<input show="true" cmd="true" top="algebra" />
	<dockBar show="true" east="true" />
</perspective>
	</perspectives>
	<labelingStyle  val="0"/>
	<font  size="12"/>
</gui>
<euclidianView>
	<viewNumber viewNo="1"/>
	<size  width="1119" height="577"/>
	<coordSystem xZero="221.3831860954347" yZero="312.767628906516" scale="48.25596008321457" yscale="48.255960083214546"/>
	<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
	<bgColor r="255" g="255" b="255"/>
	<axesColor r="0" g="0" b="0"/>
	<gridColor r="192" g="192" b="192"/>
	<lineStyle axes="1" grid="0"/>
	<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
	<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
	<mode val="3"/>
</algebraView>
<kernel>
	<uses3D val="true"/>
	<continuous val="false"/>
	<usePathAndRegionParameters val="true"/>
	<decimals val="2"/>
	<angleUnit val="degree"/>
	<algebraStyle val="0" spreadsheet="0"/>
	<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<command name="CurveCartesian">
	<input a0="cos(u)" a1="sin(u)" a2="u" a3="0" a4="(2 * π)"/>
	<output a0="a"/>
</command>
<element type="curvecartesian" label="a">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="text1" exp="&quot;ρ=1&quot;"/>
<element type="text" label="text1">
	<show object="true" label="true" ev="40"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="1.0" size="0" style="0"/>
	<startPoint  x="-1.5200000000000011" y="1.4400000000000004" z="1.0"/>
</element>
<expression label="text2" exp="&quot;curve[cos(θ),sin(θ),θ,0,2pi]&quot;"/>
<element type="text" label="text2">
	<show object="true" label="true" ev="40"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="1.0" size="0" style="0"/>
	<startPoint  x="0.8800000000000003" y="-0.9400000000000003" z="1.0"/>
</element>
<expression label="f" exp="y = x" type="line"/>
<element type="line" label="f">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<fixed val="true"/>
	<coords x="-1.0" y="1.0" z="0.0"/>
	<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
	<eqnStyle style="explicit"/>
</element>
<expression label="text3" exp="&quot;θ=π/4&quot;"/>
<element type="text" label="text3">
	<show object="true" label="true" ev="40"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="1.0" size="0" style="0"/>
	<startPoint  x="3.237108684748946" y="2.798065511778506" z="1.0"/>
</element>
</construction>
</geogebra>

Special Curves in Polar Coordinates: r=C and θ=θ0

Loading the GeoGebra board and drawing, please wait…

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.560.0" app="classic" platform="d" id="8550128d-63e5-4c5c-8882-b2d8c18d79a4"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="1376" height="738" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.16787790697674418" orientation="1" />
	</panes>
	<views>
		<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 ,  20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
		<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
		<view id="8" toolbar="1001 | 1002 | 1003  || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
		<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="385" window="100,100,700,550" />
		<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1119" window="100,100,600,400" />
		<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="231" window="100,100,600,400" />
		<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
		<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="64" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
		<view id="70" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
	</views>
	<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71  14  68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
	<input show="true" cmd="true" top="algebra" />
	<dockBar show="true" east="true" />
</perspective>
	</perspectives>
	<labelingStyle  val="0"/>
	<font  size="12"/>
</gui>
<euclidianView>
	<viewNumber viewNo="1"/>
	<size  width="1119" height="577"/>
	<coordSystem xZero="221.3831860954347" yZero="312.767628906516" scale="48.25596008321457" yscale="48.255960083214546"/>
	<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
	<bgColor r="255" g="255" b="255"/>
	<axesColor r="0" g="0" b="0"/>
	<gridColor r="192" g="192" b="192"/>
	<lineStyle axes="1" grid="0"/>
	<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
	<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
	<mode val="3"/>
</algebraView>
<kernel>
	<uses3D val="true"/>
	<continuous val="false"/>
	<usePathAndRegionParameters val="true"/>
	<decimals val="2"/>
	<angleUnit val="degree"/>
	<algebraStyle val="0" spreadsheet="0"/>
	<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<command name="CurveCartesian">
	<input a0="cos(u)" a1="sin(u)" a2="u" a3="0" a4="(2 * π)"/>
	<output a0="a"/>
</command>
<element type="curvecartesian" label="a">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="text1" exp="&quot;ρ=1&quot;"/>
<element type="text" label="text1">
	<show object="true" label="true" ev="40"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="1.0" size="0" style="0"/>
	<startPoint  x="-1.5200000000000011" y="1.4400000000000004" z="1.0"/>
</element>
<expression label="text2" exp="&quot;curve[cos(θ),sin(θ),θ,0,2pi]&quot;"/>
<element type="text" label="text2">
	<show object="true" label="true" ev="40"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="1.0" size="0" style="0"/>
	<startPoint  x="0.8800000000000003" y="-0.9400000000000003" z="1.0"/>
</element>
<expression label="f" exp="y = x" type="line"/>
<element type="line" label="f">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<fixed val="true"/>
	<coords x="-1.0" y="1.0" z="0.0"/>
	<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
	<eqnStyle style="explicit"/>
</element>
<expression label="text3" exp="&quot;θ=π/4&quot;"/>
<element type="text" label="text3">
	<show object="true" label="true" ev="40"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="1.0" size="0" style="0"/>
	<startPoint  x="3.237108684748946" y="2.798065511778506" z="1.0"/>
</element>
</construction>
</geogebra>