2e:null 2f:["$","title",null,{"children":"Matrix Form of Sandwich Product | Steven De Keninck | Observable"}] 34:T1376,res = Algebra({p:p*1,q:q*1,r:r*1,graded:true}).inline((algebrite,p,q,r,target)=>{ // figure out the amount of bivectors. const fac = x=>x<=1?1:x*fac(x-1); const choose = (n,k)=>fac(n)/(fac(k)*fac(n-k)); // Figure out how much elements in vector and motor. var n = p+q+r; var vn = n; var v2n = choose(n,2); var v4n = choose(n,4); var origtarget = target if (target != 2 && target != 3) { target = target?n-1:1; } // Create a general vector. var vnames = "xyztu ".split('').slice(0,vn); if (r==1) { vnames = "wxyzt".split('').slice(0,vn); // if (origtarget == 1) vnames = //"wxyzt".split('').slice(5-vn,5); } else if (origtarget == 1) vnames = vnames.reverse(); var v = Element.nVector(target,...vnames); // Create a general motor. (do NOT use 'e' or 'i'!!) var names = "abcdjfghklmnopqrs".split(''); var m = Element.Scalar(...names.splice(0,1)) + Element.Bivector(...names.splice(0,v2n)) + Element.nVector(4,...names.splice(0,v4n)); names = "abcdjfghklmnopqrs".split(''); var mc = Element.Scalar(...names.splice(0,1)) + Element.Bivector(...names.splice(0,v2n).map(x=>'(-'+x+')')) + Element.nVector(4,...names.splice(0,v4n)); if (origtarget==2) { names = "klmnopqrstuvwxyz".split(''); v = Element.Scalar(...names.splice(0,1)) + Element.Bivector(...names.splice(0,v2n)) + Element.nVector(4,...names.splice(0,v4n)); vnames = "klmnopqrstuvwxyz".split('').slice(0,1+v2n+v4n); } if (origtarget==3) { names = "klmnopqrstuvwxyz".split(''); v = 0; for (var i=0; i<=n; ++i) { v = v + Element.nVector(i, ...names.splice(0,choose(n,i))); } vnames = "klmnopqrstuvwxyz".split('').slice(0,2**n); } // Apply the sandwich. var r = m * v; r = r.map(g=>g.map(e=>'('+e+')')); // grrr extra wrapping needed. var r2 = r * mc; // Call the simplifyier var [rb,r2b] = [r,r2].map(r=>r.map(g=>g.map(e=>{ e = e.replace(new RegExp("\\+-","g"),"-").replace(new RegExp("\\*(-[a-z1]+)","g"),'*($1)') e = (algebrite.eval(e)).toString(); return e; }))); // Select out contributions from/to each vector coefficient. var res = r2b[target]; if (origtarget==2) res = [r2b[0],r2b[2],r2b[4]].flat().filter(x=>x); if (origtarget==3) res = [r2b[0],r2b[1],r2b[2],r2b[3],r2b[4],r2b[5]].flat().filter(x=>x); var res2 = res.map(x=>[]); var termcount = {},factors; for (var j=0; j{ if (res[j][0]=='(') { var pos = res[j].indexOf(')'); var before = res[j].slice(0,pos+3); var after = res[j].slice(pos+3); var terms = [before, ...after.replace(new RegExp("-","g"),"+-").split('+')].filter(x=>x.indexOf(token)!=-1); } else var terms = res[j].replace(new RegExp("-","g"),"+-").split('+').filter(x=>x.indexOf(token)!=-1); terms = terms.map(x=>x.replace("*"+token,"")); // for this set, do all the terms share a factor? if (terms[0] && terms[0][0]=='(') { terms = terms[0].slice(1,terms[0].length-1).replace(new RegExp("-","g"),"+-").split('+').filter(x=>x); } factors = {}; terms.forEach(x=>x.replace('-','').split('*').forEach(f=>factors[f]=(factors[f]||0)+1)); factors = Object.entries(factors).filter(x=>x[1]==terms.length).map(x=>x[0]); terms=terms.map(x=>{factors.forEach(f=>x=x.replace(f+'*','')); return x;}); terms.forEach(x=>termcount[x.replace('-','')]=(termcount[x.replace('-','')]||0)+1); if (terms.length==1) res2[j][i]=terms[0]; else res2[j][i] = terms.length?(factors.join(''))+'('+terms.join('+').replace(new RegExp("\\+-","g"),"-").replace(new RegExp("\\*","g"),"")+')':'0'; }) // Create some code var all = (origtarget==2) ? [r2b[0],r2b[2],r2b[4]].flat().filter(x=>x) : r2b[target]; var code = ` var [${(q==1)?'x,y,z,t':'w,x,y,z'},a,b,c,d,j,f,g,h] = [1,2,3,4, 1, 2**.5,2**.5,1, 0,0,0, 0]; var res=[]; ${all.map((x,i)=>`res[${i}] = ${x.replace(new RegExp("([a-z]+)\\^2","g"),"$1*$1")};`).join('\n')} return res; `; var ret = {termcount,res2,r2b,r2, v:(v+'').replace(new RegExp("e(\\d+)","g"),"\\mathbf e_{$1}"), m:(m+'').replace(new RegExp("e(\\d+)","g"),"\\mathbf e_{$1}"), code:code.split('\n').slice(0,3+all.length)} // Test only for STA.. if ((p==3) && (q==1) && (n==4) && (target!=2)) { var fu = new Function(code); ret.check = fu(); ret.check2 = (2**.5*(1e12+1e13)+1e14)>>>Element.nVector(target,...(target==1?[1,2,3,4]:[4,3,2,1])) + ''; } // Test for PGA if ((p==3) && (q==0) && (n==4) && (target!=2) && (target!=3)) { var fu = new Function(code); ret.check = fu(); ret.check2 = (1+2**.5*(1e01+1e02)+1e03)>>>Element.nVector(target,...(target!=1?[1,2,3,4]:[4,3,2,1])) + ''; } // Return results return ret; })(algebrite,p*1,q*1,r*1,target*1);35:[] 32:{"id":"0cd172e6d765198a","slug":"matrix-form-of-sandwich-product","title":"Matrix Form of Sandwich Product","thumbnail":"17a3e0a8ecd6710e81395f14872e507b34f369a8b76aae0e985f33d19baa8053","access_level":"public","create_time":"$D2021-02-23T14:24:10.266Z","update_time":"$D2021-06-22T09:49:04.839Z","view_time":null,"first_public_time":"$D2021-02-25T09:27:06.956Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"679a6b4e77863c60","login":"enkimute","name":"Steven De Keninck","avatar_url":"https://avatars-next.observableusercontent.com/75edf08e8548e6687838709326b27b1ad20835bbb277aca3370b72df3a466bd8","tier":"starter_2024"},"version":679,"publish_version":679,"role":null,"authored":false,"authors":[{"type":"individual","id":"f7e54611864321b4","login":"enkimute","name":"Steven De Keninck","avatar_url":"https://avatars-next.observableusercontent.com/75edf08e8548e6687838709326b27b1ad20835bbb277aca3370b72df3a466bd8","tier":null}],"members":[{"type":"individual","id":"f7e54611864321b4","login":"enkimute","name":"Steven De Keninck","avatar_url":"https://avatars-next.observableusercontent.com/75edf08e8548e6687838709326b27b1ad20835bbb277aca3370b72df3a466bd8","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":2,"collection":null,"type":"notebook","body":{"title":"Matrix Form of Sandwich Product","cells":[{"id":0,"value":"md`# Matrix Form of Sandwich Product\n\nResponding to a question on our [Geometric Algebra discord](https://discord.gg/SJpVCYvx), here's a short notebook that produces the matrix for conjugation of a **vector**,**n-1 vector** or **rotor** with a **rotor** in an arbitrary Geometric Algebra. Set the metric of your algebra ${tex`\\mathbb R_{p,q,r}`} below:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"viewof p = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"viewof q = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"viewof r = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":628,"value":"md`select what you want to conjugate:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":537,"value":"viewof target = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":321,"value":"{\nvar r2=tex`\n\\Huge\n\\text{Algebra : }\\mathbb R_{${p},${q},${r}}\n\\normalsize\n\\\\~\\\\\n\\text{for a motor: }\n\\\\~\\\\\n${res.m}\n\\\\~\\\\\n\\text{the matrix form of the sandwich (with a ${target==1?'n-1 vector':(target==2)?'rotor':'vector'}) is: }\n\\\\~\\\\\n\\begin{bmatrix}\n${res.res2.map(x=>x.map(x=>(res.res2.length>=5?'\\\\scriptscriptstyle{':'\\\\scriptstyle{')+x+'}').join('&')).join('\\\\\\\\')}\n\\end{bmatrix}\n\\\\~\\\\\n\n`\nvar d = document.createElement('div');\n \nd.style.overflowX='auto';\nd.appendChild(r2);\n return d\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":420,"value":"md`For ease of implementation, it sometimes makes more sense to implement the matrix product directly, without ever explicitely constructing the above matrix. Here's an easy-to-port version of that using the same symbol names:\n\n(and assuming the element ${tex`${res.v}`})\n\n

\n${res.code.slice(2).join('
')}\n\n

\n\nAlthough you may want some CSE on that :).\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":370,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":31,"value":"algebrite = {\n return import('https://cdn.jsdelivr.net/gh/davidedc/Algebrite/dist/algebrite.bundle-for-browser.js').then((m)=>window.Algebrite)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"Algebra = require('ganja.js');","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 33:"$32:body"