29:null 2a:["$","title",null,{"children":"3D Projective Geometry starter page | Jim Hinds | Observable"}] 2f:T80c, plot=PGA3d.inline`()=>{ // In dual projectivized space, the origin is represented by the e12 bivector. // in 3D PGA, grade-1 elements or vectors (e0,e1,e2) represent // reflections AND planes (the invariant of a reflection) var plane = (a,b,c,d)=>a*1e1 + b*1e2 + c*1e3 + d*1e0; // grade-2 elements or bivectors (e01,e02,e03,e12,e13,e23) represent // rotations/translations AND lines/infinite lines (the invariant // of a rotation/translation). var line = (...plucker)=>plucker*[1e01,1e02,1e03,1e12,1e13,1e23]; // grade-3 elements or trivectors (e012,e013,e023,e123) represent // point/glide/roto reflections AND points (the invariants ..) // Again, we use dualisation for basis independent definition. var point = (x,y,z)=>!(1e0 + x*1e1 + y*1e2 + z*1e3); // Lets define some points to demonstrate basic incidence and rendering. var A=point(0,.8,0), B=point(.8,-1,-.8), C=point(-.8,-1,-.8), D=point(.8,-1,.8), E=point(-.8,-1,.8); var ec = ()=>E & C, p = ()=>A & B & C; // Sum points to get the average position. var avg = A+B+E bc = B+C; // Join (&) points into lines var l = ()=>avg & bc; // Intersect a line and a plane into a point. var intersect = ()=>l ^ (A & E & D); // Sum lines to get average lines. var l2 = ()=>l.Normalized + ec.Normalized; return [ "Drag A,B,C,D,E", // title. 0xD0FFE1, [A,B,D], // polygons 0x00AA88, [A,B],[A,C],[A,D],[A,E],[B,C],[C,E],[E,D],[D,B], // edges 0x224488, A,"A",B,"B",C,"C",D,"D",E,"E", // points 0x884488, ec,"E&C", ()=>p*0.1,"A&B&C", // join of points 0x884488, ()=>B+C, "B+C", ()=>A+B+E, "A+B+E", l, // sum of points 0x00AA88, intersect, "line ^ plane", // meets 0xFF8844, l2, "sum of lines", // sum of lines. ] return plot; }`(); 30:Tf8b,## Ganja starterkit : PGA3D P(R*_3,0,1) This example implements the table on page 15 of [Gunn's Geometric Algebra for Computer Graphics](http://page.math.tu-berlin.de/~gunn/Documents/Papers/GAforCGTRaw.pdf). We apply the same strategy from above and start from a Clifford Algebra in R_3,0,1. ```javascript // Create a Clifford Algebra with 3 positive and one zero generator. Algebra(3,0,1,()=>{ // Output algebra info to the console. this.describe(); // The default basis is 1,e0,e1,e2,e3,e01,e02,e03,e12,e13,e23,e012,e013,e023,e123,e0123 // The metric for vectors is 0, 1, 1, 1 - Vectors will represent planes. // The metric for bivectors is 0, 0, 0, -1, -1, -1 - Bivectors will represent lines. // The metric for trivectors is 0, 0, 0, -1 - Trivectors will represent points. // The pseudoscalar is degenerate, so use the built-in duality operator instead. // The bivectors consist of three motor elements and three rotation element - // exactly what is needed to represent translations and rotations in euclidean space. // In dual projectivized space, the origin is represented by the e12 bivector. var origin=1e123, EX=-1e023, EY=1e013, EZ=-1e012; // Points and planes can be specified directly. var point = (x,y,z)=>origin+x*EX+y*EY+z*EZ, plane = (a,b,c,d)=>a*1e1+b*1e2+c*1e3+d*1e0; // Table from "Geometric Algebra for Computer Graphics" p.15 var LineFromPoints = (P,Q)=>P&Q, LineFromPlanes = (a,b)=>a^b, PointFromPlanes = (a,b,c)=>a^b^c, PlaneFromPoints = (P,Q,R)=>P&Q&R, DistPointToPlane = (a,P)=>a&P, DistPoints = (P,Q)=>(P&Q).Length, AnglePlanes = (a,b)=>Math.acos((a<P<P<P<PI^a, PlaneThroughPointPerpLine = (PI,P)=>P<P<P<(P<PI&EP, AngleLines = (PI,EP)=>Math.acos((PI<a*X*a, Rotor = (PI,alpha)=>Math.cos(alpha/2) + Math.sin(alpha/2)*PI, RotationAroundLine = (X,PI,alpha)=>Rotor(PI,alpha)*X*~Rotor(PI,alpha), Translator = (x,y,z)=>1+0.5*(x*EX+y*EY+z*EZ); // To demonstrate graphing, we create some points and lines. // Users can drag points in the graph, lambda expressions can be // used to create dynamic updating items. var A = point(-1,-1, 0), B = point(1,-1, 0), C = point(-1,1,0); // Ganja.js can directly graph 3D PGA elements. Pass in an array of // items to render. (numbers are colors, strings are labels, PGA points // and lines are rendered automatically and arrays can be used for line // segments and polygons). The graph function returns a HTML SVG element. document.body.appendChild(this.graph([ // use numbers to set the current color. 0x444444, // strings label the items they follow, first string is a title. "title", // render points (user can drag these) A, B, C, "Label for point", // render lines ()=>(B&C&A)<<(B&C)<<-A,"Label for line", // line segments ()=>[A,B], "Label for segment", // polygons 0xffeeee, ()=>[A,B,C], 0xff7777, "Label for polygon" ],{grid:false, animate:false})); // When using the animation mode, all lambda's will be evaluated every frame. // Use Date.now() or similar. (many examples in the coffeeshop.) }); ```31:Tb04,PGA3D = notTheThumb&&quick_sheet({ title : "3D Euclidean Projective GA", titlesym : "", altname : "mv", sample : "1e0+1e1+1e2+1e3", sig : [3,0,1], evaluate : "1e1*1e12", cards : [{ description : "Plane Based Geometry", origin : "e_{123}", points : "e_{123} + xe_{012} + ye_{013} + ze_{023}", "ideal points" : "xe_{012} + ye_{013} + ze_{023}", lines : "p_1 \\vee p_2", planes : "p_1 \\vee p_2 \\vee p_3", join : "p_1 \\vee p_2", meet : "l_1 \\wedge l_2", rotors : "e^{\\frac\\alpha 2B}", motors : "e^{\\frac\\delta 2B}" },{ description:"Sub-Algebras", text:md`

Scalars	1
Complex numbers	e.g. 1,e₁₂
Hyperbolic numbers	e.g. 1,e₁
Dual numbers	e.g. 1,e₀
Quaternions	1,e₁₂, e₁₃, e₂₃
Dual Quaternions	1,e₀₁, e₀₂, e₀₃,e₁₂, e₁₃, e₂₃, e₀₁₂₃

` },{ description : "Matrices", text : md`\`\`\`javascript var M = [[1e1,1e2],[0,1]]; // 2x2 PGA matrix. var V = [1,1e123]; // 1x2 PGA vector. var R = M*V; // matrix-vector product. var N = M*M; // matrix-matrix product. var s = V*V; // vector-vector dot product. var O = !M; // conjugate-transpose matrix. \`\`\` ` },{ description : "Graphing", text : md`SVG and webGL output can be used for 3D PGA with support for points, lines, line segments, polygons, labels and colors. Shown below is a euclidean screw motion. ${ Algebra(3,0,1,()=>{ var bv = 1e02+1e13, screw=[]; for (var j=0; j<10; j++) screw = screw.concat( [...Array(40)] .map((x,i)=>Math.E**((i/16-1.2)*bv)* (Math.cos(j/10*Math.PI)+Math.sin(j/10*Math.PI)*1e13)>>> (!(1e0+1e1))) .map((x,i,a)=>[x,i?a[i-1]:x]) ); return this.graph(screw,{grid:1,width:window.innerWidth-100}); }) } ${md` \`\`\`javascript Algebra(3,0,1,()=>{ var bv = 1e02+1e13, screw=[]; for (var j=0; j<10; j++) screw = screw.concat( [...Array(40)] .map((x,i)=>Math.E**((i/16-1.2)*bv)>>> (Math.cos(j/10*Math.PI)+Math.sin(j/10*Math.PI)*1e13)>>> (!(1e0+1e1))) .map((x,i,a)=>[x,i?a[i-1]:x]) ); return this.graph(screw); }); \`\`\``} ` }] })32:Tfba,quick_sheet = (data)=>{ var al = Algebra(data.sig[0],data.sig[1],data.sig[2]); var len = data.sig[0]+data.sig[1]+data.sig[2]; if (len > 6) debugger var a=al.describe(); if (len<6) { var A=new al(), B=new al(); for (var i=0; i2?"A":("("+A+")").replace(/_(\d+)/g,"_{$1}"); var BS = len>2?"B":("("+B+")").replace(/_(\d+)/g,"_{$1}"); var evalex = (x)=>{ try { return al.inline(new Function("return "+x))().toString().replace(/_(\d+)/g,"_{$1}"); } catch (e) { return e.message; } } var updateex = (x,el)=>{ while (el.firstChild) el.removeChild(el.firstChild); el.appendChild(md`= ${tex`${evalex(x)}`}`) }} console.log('here'); return Object.assign(md`# ${data.title} ${tex`\mathbb {${data.titlesym}} \cong \mathbb{R}_{${data.sig}}`}

create

${md`\`\`\`javascript var ${data.titlesym||"A"} = Algebra(${data.sig.join(',')}); var ${data.altname||"x"} = Algebra(${data.sig.join(',')},()=>${data.sample}); \`\`\``}

basis and metric

${(len<6?a.basis:a.basis[1]).map(x=>"${(len<6?a.metric:a.metric[1]).map((x,i)=>("

"+(""+x).replace(/e(\d+)/,"e_$1")).join('')}

")+((x==1)?"+1":x)).join('')}

${a.mulTable&&md`

Cayley Table

${a.mulTable.map(x=>x.map(x=>'')}

'+x.replace(/e(\d+)/,"e_$1")).join('')).join('

`||""} ${data.cards&&data.cards.map(x=>md`

${x.description}

${ Object.keys(x).length>2? html`${Object.keys(x).slice(1).map((y,i)=>html`

${y}	${tex`${x[y]}`}`)}

` :md`${x.text}`}

`)||""} ${len in{"1":1,"2":1,"3":1}&&md`

Operators

${len>2?"

	A	":""}${len>2?md`${tex`${("("+A+")").replace(/_(\d+)/g,"_{$1}")}`}`:""} ${len>2?"
	B	":""}${len>2?md`${tex`${("("+B+")").replace(/_(\d+)/g,"_{$1}")}`}`:""}
Addition	+	${tex`${AS}+${BS}=${(""+A.Add(B)).replace(/_(\d+)/g,"_{$1}")}`}
Subtraction	-	${tex`${AS}-${BS}=${(""+A.Sub(B)).replace(/_(\d+)/g,"_{$1}")}`}
Geometric Product	*	${tex`${AS}*${BS}=${(""+A.Mul(B)).replace(/_(\d+)/g,"_{$1}")}`}
Outer Product	^	${tex`${AS} \wedge ${BS}=${(""+A.Wedge(B)).replace(/_(\d+)/g,"_{$1}")}`}
Left Contraction	<<	${tex`${AS}\cdot ${BS}=${(""+A.Dot(B)).replace(/_(\d+)/g,"_{$1}")}`}
Regressive Product	&	${tex`${AS} \vee ${BS}=${(""+A.Vee(B)).replace(/_(\d+)/g,"_{$1}")}`}
Conjugation	~	${tex`\overline{${AS}} =${(""+A.Conjugate).replace(/_(\d+)/g,"_{$1}")}`}
Dual	!	${tex`!{${AS}} =${(""+A.Dual).replace(/_(\d+)/g,"_{$1}")}`}

`||""} ${data.evaluate&&((()=>{ var r=md`

Expression Evaluator

= ${tex`${evalex(data.evaluate)}`}

`; var i = r.querySelector('input'); var j = r.querySelector('div.out') i.onkeyup = ()=>updateex(i.value,j); return r; })())||""} `,{className:"sheet",id:data.title})}33:[] 2d:{"id":"1bc768c7be23bbc4","slug":"3d-projective-geometry-starter-page","title":"3D Projective Geometry starter page","thumbnail":"370e953b165ff7585f584b2743e130214be358300954024be65c74a98c7ba821","access_level":"public","create_time":"$D2021-07-13T02:57:33.697Z","update_time":"$D2021-09-09T02:57:00.635Z","view_time":null,"first_public_time":"$D2021-07-17T04:05:59.073Z","has_unpublished_edits":null,"fork":false,"stars":4,"starred":false,"owner":{"type":"team","id":"7a8f10f72a2ba64d","login":"jahbini","name":"Jim Hinds","avatar_url":"https://avatars-next.observableusercontent.com/01f298326a526f2c1b0e97719d94a5dab57fce98fa343bd6326a24fa51dedbcc","tier":"starter_2024"},"version":329,"publish_version":329,"role":null,"authored":false,"authors":[{"type":"individual","id":"a89bf008a705c3be","login":"jahbini","name":"Jim Hinds","avatar_url":"https://avatars-next.observableusercontent.com/01f298326a526f2c1b0e97719d94a5dab57fce98fa343bd6326a24fa51dedbcc","tier":null}],"members":[{"type":"individual","id":"a89bf008a705c3be","login":"jahbini","name":"Jim Hinds","avatar_url":"https://avatars-next.observableusercontent.com/01f298326a526f2c1b0e97719d94a5dab57fce98fa343bd6326a24fa51dedbcc","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":5,"collection":null,"type":"notebook","body":{"title":"3D Projective Geometry starter page","cells":[{"id":0,"value":"# 3D Projective Geometry starter page\n## PGA3D -- taken from ganja.js/examples/example_pga3d_points_and_lines.html\n\nCode is broken up in three parts -\n- initialization (bottom of page)\n- display plot (adjacent below\n- scene definition (below plot)\n- referance of PGA3d follows","pinned":true,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":137,"value":"PGA3d.inline`()=>{\n var camera=1+0e0, t=.20;\n return this.graph(()=>{\n t = t+ 0.001;\n camera.set(Math.cos(t)+Math.sin(t)*1e23);\n return ${plot}\n }\n ,{grid:true,animate:false,camera,labels:true,gridSize:1,scale:1.5,width:window.innerWidth-100})\n}`();","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":322,"value":"## Plot Definition\n","pinned":true,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":91,"value":"$30","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":32,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"Algebra = require(\"ganja.js\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"Algebra(3,0,1,()=>{\n var gridX = Array.apply([],{length:21}).map((x,i)=>[1e123+2e012+1e013+(i-10)*.2e023,1e123-2e012+1e013+(i-10)*.2e023]);\n var gridY = Array.apply([],{length:21}).map((x,i)=>[1e123+2e023+1e013+(i-10)*.2e012,1e123-2e023+1e013+(i-10)*.2e012]);\n var orig = 1e123-1e013;\n debugger;\n //var p = Algebra.Point(1,1);\n var p2 = 1e123+1e013-1.35e023+.8e012;\n var p3 = 1e123+1e013+1.35e023+1.1e012;\n //var p4 = p.slice();\n var camera=1+0e0, t=0;\n debugger;\n console.log('line : ',(orig&p2&p3)^(p2&p3));\n var el=this.graph(()=>{\n t = t+ 0.001;\n camera.set(Math.cos(t)+Math.sin(t)*1e13);\n return [orig,\"origin in 3D\",0x123456,gridX,\"gridx\",gridY,\"GRIDY\",0x4444cc, p2,\"point in 2D\",orig&p2,\"vector in 3D\",0xffcccc,[orig,p2,p3],0xCC4444,\"bivector in 3D\",p2&p3,\"line in 2D\"]},{gl:true, animate:true, camera}); //\n el.style.width='100%',el.style.height='500px'; return el;\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":38,"value":"notTheThumb=true","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":81,"value":"// Create a Clifford Algebra with 3 positive and one zero generator.\nPGA3d=Algebra(3,0,1);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2e:"$2d:body"