Overlays: fit to the outer upward triangle's slanted-side midpoints (ρ ≈ 0.256)

overlay/index.html

@@ -66,7 +66,9 @@ function seg(a,b,n=500){const o=[];for(let i=0;i<=n;i++){const t=i/n;o.push([a[0
 const sides=[];T.forEach(([lx,by,ay,rx,ty])=>{const A=P(lx,by),B=P(150,ay),C=P(rx,by);[[A,B],[B,C],[C,A]].forEach(([a,b])=>sides.push({up:!!ty,pts:seg(a,b)}))});
 // Fit: send the outer downward triangle's vertices (where two outer lobes meet) onto the
 // side midpoints of the outer upward triangle, which lie on the same three rays: rho^2 = |V||M|.
-const V0=P(150,160.66014976539617), M1=P(150,174.24660560764943);
+const V0=P(150,160.66014976539617);
+const _A=P(52.984011026495736,174.24660560764943),_B=P(150,50),_C=P(247.01598897350425,174.24660560764943);
+const MIDS=[[(_A[0]+_B[0])/2,(_A[1]+_B[1])/2],[(_C[0]+_B[0])/2,(_C[1]+_B[1])/2]], M1=MIDS[0];
 // outer yantra: lotus rings, petals, bhupura (square with four gates), in yantra units (rim = 1)
 const circ=(r,n=360)=>Array.from({length:n+1},(_,i)=>{const a=i/n*Math.PI*2;return[r*Math.cos(a),r*Math.sin(a)]});
 function petals(r0,r1,k){const o=[];for(let i=0;i<k;i++){const a0=i/k*Math.PI*2,a1=(i+1)/k*Math.PI*2,am=(a0+a1)/2,pts=[];for(let j=0;j<=40;j++){const t=j/40;const a=a0+(am-a0)*t;const r=r0+(r1-r0)*Math.sin(t*Math.PI/2);pts.push([r*Math.cos(a),r*Math.sin(a)])}for(let j=0;j<=40;j++){const t=j/40;const a=am+(a1-am)*t;const r=r1-(r1-r0)*(1-Math.cos(t*Math.PI/2));pts.push([r*Math.cos(a),r*Math.sin(a)])}o.push(pts)}return o}
@@ -87,12 +89,12 @@ function draw(){
   if(L.unit){ring(1,col.ring);sides.forEach(sd=>path(sd.pts.map(p=>inv(p,1)),col.ring,.7,1.1));O.forEach(o=>path(o.map(p=>inv(p,1)),col.ring,.4,1))}
   if(L.fit){ring(rho,col.fit);sides.forEach(sd=>path(sd.pts.map(p=>inv(p,rho*rho)),col.fit,.85,1.2));O.forEach(o=>path(o.map(p=>inv(p,rho*rho)),col.fit,.45,1))}
   if(L.orig){sides.forEach(sd=>path(sd.pts,sd.up?col.up:col.down,1,1.3));O.forEach(o=>path(o,col.fg,.45,1));path(circ(1),col.fg,.6,1)}
-  if(L.fit&&Math.abs(rho-RHO_FIT)<1e-3){ctx.fillStyle=col.fit;const A=Math.atan2(M1[1],M1[0]),m=Math.hypot(...M1);for(let k=0;k<3;k++){const a=A+k*2*Math.PI/3;ctx.beginPath();ctx.arc(X([m*Math.cos(a),0]),Y([0,m*Math.sin(a)]),3.5,0,7);ctx.fill()}}
+  if(L.fit&&Math.abs(rho-RHO_FIT)<1e-3){ctx.fillStyle=col.fit;MIDS.forEach(m=>{ctx.beginPath();ctx.arc(X(m),Y(m),3.5,0,7);ctx.fill()})}
   ctx.fillStyle=col.bindu;ctx.beginPath();ctx.arc(cx,cy,2.4,0,Math.PI*2);ctx.fill();
   document.getElementById('rv').textContent=rho.toFixed(4);
   document.getElementById('note').textContent=
    'Yantra in gold and pink; its inversion in the rim (ρ = 1) in violet; its inversion in a circle of radius ρ in teal. '+
-   (Math.abs(rho-RHO_FIT)<1e-3?`Fit: ρ = ${RHO_FIT.toFixed(4)}. Two big lobes and one inner lobe meet at the image of a triple point (the T4 apex on the T6 base); this ρ lands it on the side midpoints of the outer upward triangle (teal dots).`:'Fit trefoil snaps ρ so the lobe meeting points land on the outer triangle’s side midpoints.');
+   (Math.abs(rho-RHO_FIT)<1e-3?`Fit: ρ = ${RHO_FIT.toFixed(4)}. Two big lobes and one inner lobe meet at the image of a triple point (the T4 apex on the T6 base); this ρ puts those meeting points at the radius of the outer upward triangle’s slanted-side midpoints (teal dots), about 0.615 of the rim.`:'Fit trefoil snaps ρ so the lobe meeting points land on the outer triangle’s side midpoints.');
 }
 const $=id=>document.getElementById(id);
 ['orig','unit','fit','outer'].forEach(k=>$('l-'+k).onclick=e=>{L[k]=!L[k];e.currentTarget.setAttribute('aria-pressed',L[k]);draw()});