2b:null 2c:["$","title",null,{"children":"Conformal 3D geometric algebra? | Kelley van Evert | Observable"}] 31:T429,md`- Javascript library used: [ganja.js](https://github.com/enkimute/ganja.js) - Nicely written: [Geometric operations implemented by conformal geometric algebra neural nodes](https://arxiv.org/abs/1306.1358) - Good overview: [Geometric Computing in Computer Graphics — using Conformal Geometric Algebra](http://www.gaalop.de/dhilden_data/CLUScripts/CandG_PrePrint.pdf) - Tutorial: [Geometric algebra: a computational framework for geometrical applications (part I: algebra)](https://staff.fnwi.uva.nl/l.dorst/clifford/dorst-mann-I.pdf) - Lots and lots of beautiful imagery: [Articulating Space: Geometric Algebra for Parametric Design – Symmetry, Kinematics, and Curvature](http://versor.mat.ucsb.edu/ArticulatingSpace.pdf) - ${html`*`} = geometric product, ^ = wedge, & = vee, << = dot, ! = dual, >>> = sandwich`32:T4e2,md`

| construct | representation | dual representation | |---| | point | ${tex`\large P = \bm{p} + \frac{1}{2}\,p^2\,e_\infty + e_0`} || | sphere | ${tex`\large s = P - \frac{1}{2}\,r^2\,e_\infty`} | ${tex`\large \mathop{!}s = P_1 \land P_2 \land P_3 \land P_4`} | plane
(*${tex`\large d`} = distance to origin*) | ${tex`\large \pi = \bm n + d\,e_\infty`} | ${tex`\large \mathop{!}\pi = P_1 \land P_2 \land P_3 \land e_\infty`} | circle | ${tex`\large z = s_1 \land s_2`} | ${tex`\large \mathop{!}z = P_1 \land P_2 \land P_3`} | line | ${tex`\large \ell = \pi_1 \land \pi_2`} | ${tex`\large \mathop{!}\ell = P_1 \land P_2 \land e_\infty`} | point pair | ${tex`\large R = s_1 \land s_2 \land s_3`} | ${tex`\large \mathop{!}R = P_1 \land P_2`}

`33:T54b,intersection = CGA3D.inline((extract_detailed) => { const ei = 1e5 + 1e4; // e_infty const eo = .5 * (1e5 - 1e4); // e_0 // (and apparently, swapping the +/- in both e0 and ei is OK) const E = ei^eo; const eq = (a,b) => a-b == 0; // can't use == for array equality const pt = (x,y,z) => x*1e1 + y*1e2 + z*1e3 + .5*(x*x+y*y+z*z)*ei + eo; // var a = pt(4,5,6); // var b = pt(7,8,9); // var d = Math.hypot(7-4, 8-5, 9-6); // return eq(a< { const r = Math.sqrt(Math.abs((pp << pp)[0])); const le = pp << (ei^pp) - (pp << pp)*ei; const ri = r*(ei << pp); const denom = (ei << pp)**2; return [ -(le - ri) / denom, -(le + ri) / denom, ]; }; var sphere1 = pt(3,0,-.5)^pt(-1,0,-.5)^pt(1,2,-.5)^pt(1,0,1.5); var sphere2 = pt(3,0,1)^pt(-1,0,1)^pt(1,2,1)^pt(1,0,3); var line = pt(0,0,0)^pt(1,0,0)^ei; var intersection1 = (sphere1&line); // return [extract(intersection1), extract_detailed(intersection1)]; return extract(intersection1); // <-- WORKS var sphere3 = pt(4,1,0)^pt(0,1,0)^pt(2,3,0)^pt(2,1,2); var intersection2 = (sphere3&sphere2&sphere1); return extract(intersection2); //