Drawing Planar Diagrams - Revision history

ScottDataRobot at 22:20, 21 February 2013

2013-02-21T22:20:07Z

← Older revision		Revision as of 18:20, 21 February 2013
Line 8:		Line 8:
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpAndAbout\|		{{HelpAndAbout\|
	n = 1 \|		n = 2 \|
	n1 = 2 \|		n1 = 3 \|
	in = <nowiki>DrawPD</nowiki> \|		in = <nowiki>DrawPD</nowiki> \|
	out= <nowiki>DrawPD[pd] takes the planar diagram description pd and creates a graphics object containing a picture of the knot. DrawPD[pd,options], where options is a list of rules, allows the user to control some of the parameters. OuterFace->n sets the face at infinity to the face numbered n. OuterFace->{e_1,e_2,...,e_n} sets the face at infinity to a face which has edges e_1, e_2, ..., e_n in the planar diagram description. Gap->g sets the size of the gap around a crossing to length g.</nowiki> \|		out= <nowiki>DrawPD[pd] takes the planar diagram description pd and creates a graphics object containing a picture of the knot.
			DrawPD[pd,options], where options is a list of rules, allows the user to control some of the parameters. OuterFace->n sets the face at infinity to the face numbered n.
			OuterFace->{e_1,e_2,...,e_n} sets the face at infinity to a face which has edges e_1, e_2, ..., e_n in the planar diagram description. Gap->g sets the size of the gap around a crossing to length g.</nowiki> \|
	about= <nowiki>DrawPD was written by Emily Redelmeier at the University of Toronto in the summers of 2003 and 2004.</nowiki>}}		about= <nowiki>DrawPD was written by Emily Redelmeier at the University of Toronto in the summers of 2003 and 2004.</nowiki>}}
	<!--END-->		<!--END-->
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	n = 4 \|		n = 4 \|
	in = <nowiki>Show[DrawPD[TorusKnot[4, 3]]]</nowiki> \|		in = <nowiki>Show[DrawPD[TorusKnot[4, 3]]]</nowiki> \|
	img= ~~Drawing_Planar_Diagrams_Out_3~~.gif \|		img= Drawing_Planar_Diagrams_Out_4.gif \|
	out= <nowiki>-Graphics-</nowiki>}}		out= <nowiki>-Graphics-</nowiki>}}
	<!--END-->		<!--END-->
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{Graphics\|		{{Graphics\|
	n = 7 \|		n = 6 \|
	in = <nowiki>Show[DrawPD[MillettUnknot]]</nowiki> \|		in = <nowiki>Show[DrawPD[MillettUnknot]]</nowiki> \|
	img= Drawing_Planar_Diagrams_Out_6.gif \|		img= Drawing_Planar_Diagrams_Out_6.gif \|
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{Graphics\|		{{Graphics\|
	n = 9 \|		n = 7 \|
	in = <nowiki>Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]</nowiki> \|		in = <nowiki>Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]</nowiki> \|
	img= ~~Drawing_Planar_Diagrams_Out_8~~.gif \|		img= Drawing_Planar_Diagrams_Out_7.gif \|
	out= <nowiki>-Graphics-</nowiki>}}		out= <nowiki>-Graphics-</nowiki>}}
	<!--END-->		<!--END-->

DrorsRobot at 17:04, 30 August 2005

2005-08-30T17:04:38Z

← Older revision		Revision as of 13:04, 30 August 2005
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	{{In\|		{{In\|
	n = 5 \|		n = 5 \|
			in = <nowiki>MillettUnknot = PD[
	in = <nowiki>MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];</nowiki>}}
			X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4],
			X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19],
			X[19,12,20,13], X[13,20,14,1]
			];</nowiki>}}
	<!--END-->		<!--END-->

Drorbn at 17:02, 30 August 2005

2005-08-30T17:02:52Z

← Older revision		Revision as of 13:02, 30 August 2005
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	</div>		</div>

	[[Image:Circles.gif\|thumb~~\|none~~\|250px]][[Image:CirclesAndKnot.gif\|thumb~~\|none~~\|250px]]		[[Image:Circles.gif\|thumb\|250px]][[Image:CirclesAndKnot.gif\|thumb\|250px]]

	<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first picture on the right (circle 13 is the unit disk itself).		<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first picture on the right (circle 13 is the unit disk itself).

Drorbn at 17:02, 30 August 2005

2005-08-30T17:02:31Z

← Older revision		Revision as of 13:02, 30 August 2005
Line 63:		Line 63:
	</div>		</div>

		⚫	[[Image:Circles.gif\|thumb\|none\|250px]][[Image:CirclesAndKnot.gif\|thumb\|none\|250px]]
⚫	<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first ~~half~~ of the ~~picture below~~ (circle 13 is the unit disk itself).

		⚫	<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first picture on the right (circle 13 is the unit disk itself).
⚫	~~<center>~~[[Image:Circles.gif\|thumb\|none\|~~300px~~]][[Image:CirclesAndKnot.gif\|thumb\|none\|~~300px~~]]~~</center>~~

	But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the second ~~half~~ of the ~~picture above~~. Removing all the circles, what remains is the desired clean planar picture of [[4_1]].		But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the second picture on the right. Removing all the circles, what remains is the desired clean planar picture of [[4_1]].

	{{note\|Andreev1}} A. Andreev, ''On convex polyhedra in Lobacevskii spaces'' (in Russian), Math. Sbornik USSR, Nov. Ser. '''81''' (1970) 445-478.		{{note\|Andreev1}} A. Andreev, ''On convex polyhedra in Lobacevskii spaces'' (in Russian), Math. Sbornik USSR, Nov. Ser. '''81''' (1970) 445-478.

Drorbn at 17:00, 30 August 2005

2005-08-30T17:00:58Z

← Older revision		Revision as of 13:00, 30 August 2005
Line 65:		Line 65:
	<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first half of the picture below (circle 13 is the unit disk itself).		<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first half of the picture below (circle 13 is the unit disk itself).

	<center>[[Image:Circles.gif\|thumb\|300px]][[Image:CirclesAndKnot.gif\|thumb\|300px]]</center>		<center>[[Image:Circles.gif\|thumb\|none\|300px]][[Image:CirclesAndKnot.gif\|thumb\|none\|300px]]</center>

	But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the second half of the picture above. Removing all the circles, what remains is the desired clean planar picture of [[4_1]].		But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the second half of the picture above. Removing all the circles, what remains is the desired clean planar picture of [[4_1]].

Drorbn at 16:59, 30 August 2005

2005-08-30T16:59:51Z

← Older revision		Revision as of 12:59, 30 August 2005
Line 28:		Line 28:
	One problem we currently have is that crossings come out at non-uniform sizes, hence in the picture below you may need magnifying glasses to decide who's over and who's under:		One problem we currently have is that crossings come out at non-uniform sizes, hence in the picture below you may need magnifying glasses to decide who's over and who's under:

			<!--$$MillettUnknot = PD[
	<!--$$MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];$$-->
			X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4],
			X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19],
			X[19,12,20,13], X[13,20,14,1]
			];$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{In\|		{{In\|
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	<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first half of the picture below (circle 13 is the unit disk itself).		<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first half of the picture below (circle 13 is the unit disk itself).

	<center>[[Image:Circles.gif]][[Image:CirclesAndKnot.gif]]</center>		<center>[[Image:Circles.gif\|thumb\|300px]][[Image:CirclesAndKnot.gif\|thumb\|300px]]</center>

	But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the second half of the picture above. Removing all the circles, what remains is the desired clean planar picture of [[4_1]].		But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the second half of the picture above. Removing all the circles, what remains is the desired clean planar picture of [[4_1]].

DrorsRobot at 16:58, 30 August 2005

2005-08-30T16:58:22Z

← Older revision		Revision as of 12:58, 30 August 2005
Line 7:		Line 7:
	<!--$$?DrawPD$$-->		<!--$$?DrawPD$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
			{{HelpAndAbout\|
	{{HelpAndAbout1\|n=1\|s=DrawPD}}
			n = 1 \|
⚫	DrawPD[pd] takes the planar diagram description pd and creates a graphics object containing a picture of the knot. DrawPD[pd,options], where options is a list of rules, allows the user to control some of the parameters. OuterFace->n sets the face at infinity to the face numbered n. OuterFace->{e_1,e_2,...,e_n} sets the face at infinity to a face which has edges e_1, e_2, ..., e_n in the planar diagram description. Gap->g sets the size of the gap around a crossing to length g.
			n1 = 2 \|
	{{HelpAndAbout2\|n=2\|s=DrawPD}}
			in = <nowiki>DrawPD</nowiki> \|
⚫	DrawPD was written by Emily Redelmeier at the University of Toronto in the summers of 2003 and 2004.
		⚫	out= <nowiki>DrawPD[pd] takes the planar diagram description pd and creates a graphics object containing a picture of the knot. DrawPD[pd,options], where options is a list of rules, allows the user to control some of the parameters. OuterFace->n sets the face at infinity to the face numbered n. OuterFace->{e_1,e_2,...,e_n} sets the face at infinity to a face which has edges e_1, e_2, ..., e_n in the planar diagram description. Gap->g sets the size of the gap around a crossing to length g.</nowiki> \|
	{{HelpAndAbout3}}
		⚫	about= <nowiki>DrawPD was written by Emily Redelmeier at the University of Toronto in the summers of 2003 and 2004.</nowiki>}}
	<!--END-->		<!--END-->

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	<!--$$Show[DrawPD[TorusKnot[4, 3]]]$$-->		<!--$$Show[DrawPD[TorusKnot[4, 3]]]$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
			{{Graphics\|
	{{Graphics1\|n=3}}
			n = 4 \|
	Show[DrawPD[TorusKnot[4, 3]]]		in = <nowiki>Show[DrawPD[TorusKnot[4, 3]]]</nowiki> \|
	~~{{Graphics2\|n=3\|imagename~~=Drawing_Planar_Diagrams_Out_3.gif}}		img= Drawing_Planar_Diagrams_Out_3.gif \|
			out= <nowiki>-Graphics-</nowiki>}}
	<!--END-->		<!--END-->

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	<!--$$MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];$$-->		<!--$$MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{~~In1~~\|~~n=4}}~~		{{In\|
			n = 5 \|
	~~<pre~~ ~~style~~=~~"color:~~ ~~red; border: 0px; padding: 0em">~~<nowiki>MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];</nowiki>~~</pre>~~		in = <nowiki>MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];</nowiki>}}
	{{In2}}
	<!--END-->		<!--END-->

	<!--$$Show[DrawPD[MillettUnknot]]$$-->		<!--$$Show[DrawPD[MillettUnknot]]$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
			{{Graphics\|
	{{Graphics1\|n=5}}
			n = 7 \|
	Show[DrawPD[MillettUnknot]]		in = <nowiki>Show[DrawPD[MillettUnknot]]</nowiki> \|
	{{Graphics2\|n=5\|imagename=Drawing_Planar_Diagrams_Out_5.gif}}
		⚫	img= Drawing_Planar_Diagrams_Out_6.gif \|
			out= <nowiki>-Graphics-</nowiki>}}
	<!--END-->		<!--END-->

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	<!--$$Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]$$-->		<!--$$Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
			{{Graphics\|
	{{Graphics1\|n=6}}
			n = 9 \|
	Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]		in = <nowiki>Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]</nowiki> \|
⚫	~~{{Graphics2\|n=6\|imagename~~=Drawing_Planar_Diagrams_Out_6.gif}}
			img= Drawing_Planar_Diagrams_Out_8.gif \|
			out= <nowiki>-Graphics-</nowiki>}}
	<!--END-->		<!--END-->

Drorbn: /* How does it work? */

2005-08-30T14:36:44Z

How does it work?

← Older revision		Revision as of 10:36, 30 August 2005
Line 49:		Line 49:

	<div id="How does it work?">		<div id="How does it work?">
	== How does it work? ==		==== How does it work? ====
	</div>		</div>

69.156.15.171 at 00:42, 28 August 2005

2005-08-28T00:42:30Z

← Older revision		Revision as of 20:42, 27 August 2005
Line 6:		Line 6:

	<!--$$?DrawPD$$-->		<!--$$?DrawPD$$-->
			<!--Robot Land, no human edits to "END"-->
	<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
	{{HelpAndAbout1\|n=2\|s=DrawPD}}		{{HelpAndAbout1\|n=1\|s=DrawPD}}
	DrawPD[pd] takes the planar diagram description pd and creates a graphics object containing a picture of the knot. DrawPD[pd,options], where options is a list of rules, allows the user to control some of the parameters. OuterFace->n sets the face at infinity to the face numbered n. OuterFace->{e_1,e_2,...,e_n} sets the face at infinity to a face which has edges e_1, e_2, ..., e_n in the planar diagram description. Gap->g sets the size of the gap around a crossing to length g.		DrawPD[pd] takes the planar diagram description pd and creates a graphics object containing a picture of the knot. DrawPD[pd,options], where options is a list of rules, allows the user to control some of the parameters. OuterFace->n sets the face at infinity to the face numbered n. OuterFace->{e_1,e_2,...,e_n} sets the face at infinity to a face which has edges e_1, e_2, ..., e_n in the planar diagram description. Gap->g sets the size of the gap around a crossing to length g.
	{{HelpAndAbout2\|n=3\|s=DrawPD}}		{{HelpAndAbout2\|n=2\|s=DrawPD}}
	DrawPD was written by Emily Redelmeier at the University of Toronto in the summers of 2003 and 2004.		DrawPD was written by Emily Redelmeier at the University of Toronto in the summers of 2003 and 2004.
	{{HelpAndAbout3}}		{{HelpAndAbout3}}
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	<!--$$Show[DrawPD[TorusKnot[4, 3]]]$$-->		<!--$$Show[DrawPD[TorusKnot[4, 3]]]$$-->
			<!--Robot Land, no human edits to "END"-->
	<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
	{{Graphics1\|n=4}}		{{Graphics1\|n=3}}
	Show[DrawPD[TorusKnot[4, 3]]]		Show[DrawPD[TorusKnot[4, 3]]]
	{{Graphics2\|n=4\|imagename=~~Drawing_Planar_Diagrams_Out_4~~.gif}}		{{Graphics2\|n=3\|imagename=Drawing_Planar_Diagrams_Out_3.gif}}
	<!--END-->		<!--END-->

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	<!--$$MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];$$-->		<!--$$MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];$$-->
			<!--Robot Land, no human edits to "END"-->
	<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
	{{In1\|n=5}}		{{In1\|n=4}}
	MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];		<pre style="color: red; border: 0px; padding: 0em"><nowiki>MillettUnknot = PD[X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4], X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19], X[19,12,20,13], X[13,20,14,1]];</nowiki></pre>
	{{In2}}		{{In2}}
	<!--END-->		<!--END-->

	<!--$$Show[DrawPD[MillettUnknot]]$$-->		<!--$$Show[DrawPD[MillettUnknot]]$$-->
			<!--Robot Land, no human edits to "END"-->
	<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
	{{Graphics1\|n=6}}		{{Graphics1\|n=5}}
	Show[DrawPD[MillettUnknot]]		Show[DrawPD[MillettUnknot]]
	{{Graphics2\|n=6\|imagename=~~Drawing_Planar_Diagrams_Out_6~~.gif}}		{{Graphics2\|n=5\|imagename=Drawing_Planar_Diagrams_Out_5.gif}}
	<!--END-->		<!--END-->

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	<!--$$Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]$$-->		<!--$$Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]$$-->
			<!--Robot Land, no human edits to "END"-->
	<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
	{{Graphics1\|n=7}}		{{Graphics1\|n=6}}
	Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]		Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]
	{{Graphics2\|n=7\|imagename=~~Drawing_Planar_Diagrams_Out_7~~.gif}}		{{Graphics2\|n=6\|imagename=Drawing_Planar_Diagrams_Out_6.gif}}
	<!--END-->		<!--END-->

69.156.15.171 at 22:16, 24 August 2005

2005-08-24T22:16:22Z

← Older revision		Revision as of 18:16, 24 August 2005
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	</div>		</div>

	<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the ~~left~~ half of the picture below (circle 13 is the unit disk itself).		<code>DrawPD</code> uses Andreev's theorem {{ref\|Andreev1}}, {{ref\|Andreev2}}, which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of <math>G</math> one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot [[4_1]] is the first half of the picture below (circle 13 is the unit disk itself).

	<center>[[Image:Circles.gif]][[Image:CirclesAndKnot.gif]]</center>		<center>[[Image:Circles.gif]][[Image:CirclesAndKnot.gif]]</center>

	But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the ~~right~~ half of the picture above. Removing all the circles, what remains is the desired clean planar picture of [[4_1]].		But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the second half of the picture above. Removing all the circles, what remains is the desired clean planar picture of [[4_1]].

	{{note\|Andreev1}} A. Andreev, ''On convex polyhedra in Lobacevskii spaces'' (in Russian), Math. Sbornik USSR, Nov. Ser. '''81''' (1970) 445-478.		{{note\|Andreev1}} A. Andreev, ''On convex polyhedra in Lobacevskii spaces'' (in Russian), Math. Sbornik USSR, Nov. Ser. '''81''' (1970) 445-478.

← Older revision		Revision as of 18:20, 21 February 2013
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpAndAbout\|		{{HelpAndAbout\|
	n = 1 \|		n = 2 \|
	n1 = 2 \|		n1 = 3 \|
	in = <nowiki>DrawPD</nowiki> \|		in = <nowiki>DrawPD</nowiki> \|
	out= <nowiki>DrawPD[pd] takes the planar diagram description pd and creates a graphics object containing a picture of the knot. DrawPD[pd,options], where options is a list of rules, allows the user to control some of the parameters. OuterFace->n sets the face at infinity to the face numbered n. OuterFace->{e_1,e_2,...,e_n} sets the face at infinity to a face which has edges e_1, e_2, ..., e_n in the planar diagram description. Gap->g sets the size of the gap around a crossing to length g.</nowiki> \|		out= <nowiki>DrawPD[pd] takes the planar diagram description pd and creates a graphics object containing a picture of the knot.
			DrawPD[pd,options], where options is a list of rules, allows the user to control some of the parameters. OuterFace->n sets the face at infinity to the face numbered n.
			OuterFace->{e_1,e_2,...,e_n} sets the face at infinity to a face which has edges e_1, e_2, ..., e_n in the planar diagram description. Gap->g sets the size of the gap around a crossing to length g.</nowiki> \|
	about= <nowiki>DrawPD was written by Emily Redelmeier at the University of Toronto in the summers of 2003 and 2004.</nowiki>}}		about= <nowiki>DrawPD was written by Emily Redelmeier at the University of Toronto in the summers of 2003 and 2004.</nowiki>}}
	<!--END-->		<!--END-->
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	n = 4 \|		n = 4 \|
	in = <nowiki>Show[DrawPD[TorusKnot[4, 3]]]</nowiki> \|		in = <nowiki>Show[DrawPD[TorusKnot[4, 3]]]</nowiki> \|
	img= ~~Drawing_Planar_Diagrams_Out_3~~.gif \|		img= Drawing_Planar_Diagrams_Out_4.gif \|
	out= <nowiki>-Graphics-</nowiki>}}		out= <nowiki>-Graphics-</nowiki>}}
	<!--END-->		<!--END-->
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	{{Graphics\|		{{Graphics\|
	n = 7 \|		n = 6 \|
	in = <nowiki>Show[DrawPD[MillettUnknot]]</nowiki> \|		in = <nowiki>Show[DrawPD[MillettUnknot]]</nowiki> \|
	img= Drawing_Planar_Diagrams_Out_6.gif \|		img= Drawing_Planar_Diagrams_Out_6.gif \|
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	{{Graphics\|		{{Graphics\|
	n = 9 \|		n = 7 \|
	in = <nowiki>Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]</nowiki> \|		in = <nowiki>Show[DrawPD[MillettUnknot, {Gap -> 0.03}]]</nowiki> \|
	img= ~~Drawing_Planar_Diagrams_Out_8~~.gif \|		img= Drawing_Planar_Diagrams_Out_7.gif \|
	out= <nowiki>-Graphics-</nowiki>}}		out= <nowiki>-Graphics-</nowiki>}}
	<!--END-->		<!--END-->