cubic-beziers-through-points/browser/index.min.js

A function to fit fair (bending energy minimizing) cubic bezier curves through a set of given ordered points in the plane.

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0===t?1===r?n:function(n,t){const r=n[0],e=n[1],o=n[2],c=r[0],s=r[1],u=e[0],i=e[1],f=t*t,l=c-u,a=s-i;return[[c,s],[-t*l+c,-t*a+s],[f*(l+(o[0]-u))-(2*t*l-c),f*(a+(o[1]-i))-(2*t*a-s)]]}(n,r):1===r?function(n,t){const r=n[0],e=n[1],o=n[2],c=r[0],s=r[1],u=e[0],i=e[1],f=o[0],l=o[1],a=t*t,h=c-u,p=f-u,d=s-i,g=l-i;return[[a*(h+p)-(2*t*h-c),a*(d+g)-(2*t*d-s)],[t*p+u,t*g+i],[f,l]]}(n,t):function(n,t,r){const e=n[0],o=n[1],c=n[2],s=e[0],u=e[1],i=o[0],f=o[1],l=t*t,a=r*r,h=t*r,p=s-i,d=p+(c[0]-i),g=u-f,b=g+(c[1]-f);return[[l*d-(2*t*p-s),l*b-(2*t*g-u)],[h*d-(p*(r+t)-s),h*b-(g*(r+t)-u)],[a*d-(2*r*p-s),a*b-(2*r*g-u)]]}(n,t,r)},kn=function(n,t,r){return 0===t?1===r?n:function(n,t){const r=n[0],e=n[1],o=r[0],c=r[1];return[[o,c],[t*(e[0]-o)+o,t*(e[1]-c)+c]]}(n,r):1===r?function(n,t){const r=n[0],e=n[1],o=r[0],c=r[1],s=e[0],u=e[1];return[[t*(s-o)+o,t*(u-c)+c],[s,u]]}(n,t):function(n,t,r){const e=n[0],o=n[1],c=e[0],s=e[1],u=o[0],i=o[1];return[[t*(u-c)+c,t*(i-s)+s],[r*(u-c)+c,r*(i-s)+s]]}(n,t,r)};function Cn(n,t,r){if(4===n.length)return Xn(n,t,r);if(3===n.length)return Yn(n,t,r);if(2===n.length)return kn(n,t,r);if(1===n.length)return n;throw new Error("The given bezier curve must be of order <= 3.")}function Dn(n){return function(n){for(let t=1;t<n.length;t++)if(n[t-1]>n[t])return!1;return!0}(n)||function(n){for(let t=1;t<n.length;t++)if(n[t-1]<n[t])return!1;return!0}(n)}const{sqrt:Gn}=Math;function Hn(n,t=1.125,r=4){const e=Ln(n);return e.collinear?function(n){if(!Y(n))return!1;const t=n.map((n=>n[0])),r=n.map((n=>n[1]));return!(Dn(t)&&Dn(r))}(n)?Number.POSITIVE_INFINITY:0:3===e.realOrder?"cusp"===e.nodeType?Number.POSITIVE_INFINITY:Jn(Kn,n,t,r):2===e.realOrder?Jn(Qn,n,t,r):0}function Jn(n,t,r,e){const o=function(n,t=.4,r=2**-16){const e={r:[0,1]};let o=e;for(;void 0!==o;){const e=o.r;if(Rn(Cn(n,e[0],e[1]))<=t||e[1]-e[0]<=r){o=o.next;continue}const c=(e[0]+e[1])/2,s=[e[0],c],u=[c,e[1]];o.r=s,o.next={r:u,next:o.next}}o=e;const c=[];for(;void 0!==o;)c.push(o.r[0]),void 0===o.next&&c.push(o.r[1]),o=o.next;return c}(t,r);let c=0;for(let r=0;r<o.length-1;r++)c+=a(n(Cn(t,o[r],o[r+1])),[0,1],e);return c}function Kn(n){const t=n[0],r=n[1],e=n[2],o=n[3],c=t[0],s=t[1],u=r[0],i=r[1],f=e[0],l=e[1],a=o[0],h=o[1],p=a-c+3*(u-f),d=h-s+3*(i-l),g=f+c-2*u,b=l+s-2*i;return n=>{const t=n*n,r=t*p+2*n*g+(u-c),e=t*d+2*n*b+(i-s),o=r*(n*d+b)-e*(n*p+g),f=r*r+e*e;return 4/3*o*o/(f*f*Gn(f))}}function Qn(n){const t=n[0],r=n[1],e=n[2],o=t[0],c=t[1],s=r[0],u=r[1],i=e[0],f=e[1],l=i-2*s+o,a=f-2*u+c;return n=>{const t=l*n+(s-o),r=a*n+(u-c),e=t*a-r*l,i=t*t+r*r;return.5*e*e/(i*i*Gn(i))}}const{cos:Un,sin:Wn}=Math;function Zn(n){const{α:t,β:e,s0:c,s1:s,L:u,rot:i,p:f}=n;return function(n){const t=n[0].length;if(4===t){const[[t,r,e,o],[c,s,u,i]]=n;return[[o,i],[o+e/3,i+u/3],[o+2*e/3+r/3,i+2*u/3+s/3],[o+e+r+t,i+u+s+c]]}if(3===t){const[[t,r,e],[o,c,s]]=n;return[[e,s],[e+r/2,s+c/2],[e+r+t,s+c+o]]}if(2===t){const[[t,r],[e,o]]=n;return[[r,o],[r+t,o+e]]}if(1===t){const[[t],[r]]=n;return[[t,r]]}throw new Error("The given bezier curve must be of order <= 3.")}([[u*(c*Un(t)-2+s*Un(e)),u*(3-2*c*Un(t)-s*Un(e)),u*(c*Un(t)),0*u],[u*(c*Wn(t)+s*Wn(e)),u*(-2*c*Wn(t)-s*Wn(e)),u*(c*Wn(t)),0*u]]).map(o(Wn(i),Un(i))).map(r(f))}function nt(n){return t=>{const[r,e]=tt(t,n);return Hn(e)+Hn(r)}}function tt(n,t){const{ϕ:r}=t,e=t._info,o=e.$ps,c=t.$ps,s=Zn({...o,β:n-r,s0:1,s1:1}),u=Zn({...c,α:n,s0:1,s1:1});{const n=e.ps,r=t.ps;return[[n[0],s[1],s[2],n[3]],[r[0],u[1],u[2],r[3]]]}}const{PI:rt}=Math,et=rt,ot=.0625,[ct,st]=[1,1],ut=4;function it(n,t){return[t[0]-n[0],t[1]-n[1]]}const{min:ft,max:lt,PI:at,atan2:ht,sin:pt,cos:dt}=Math;function gt(n){const t=n.length,c=[];for(let n=0;n<t;n++)c.push({});for(let i=0;i<t;i++){const f=(i-1+t)%t,l=(i+1)%t,a=c[f],h=c[i],p=c[l];h._info=a,h.info_=p;const d=n[f],g=n[i],b=n[l],m=(u=b,-qn(it(d,s=g),it(s,u)));h.ϕ=m;const T=[lt(-et,m-et),ft(+et,m+et)];h.range=T;const y=r(e(g))(b),[O,v]=[y[0],y[1]],M=ht(v,O),w=o(pt(-M),dt(-M))(y)[0];h.L=w,h.rot=M,h.p=g}var s,u;for(let n=0;n<t;n++){const t=c[n],{ϕ:r,info_:e,p:o,L:s,rot:u}=t,i=r/2,f={α:i,β:-e.ϕ/2,p:o,L:s,rot:u,s0:ct,s1:st},l=Zn(f);t.ps=[o,l[1],l[2],e.p],t.$ps=f,t.l=i-at*ot,t.m=i,t.r=i+at*ot}return c}const bt={has:function(n,t,r){const e=n.get(t);return void 0!==e&&e.has(r)},add:function(n,t,r){let e=n.get(t);void 0===e&&(e=new Set,n.set(t,e)),e.add(r)},remove:function(n,t,r){let e=n.get(t);return void 0!==e&&!!e.delete(r)&&(0===e.size&&n.delete(t),!0)}};function mt(n){const t=new Map;for(const r of n)bt.add(t,r[0],r[1]);const r=[];for(const e of n)bt.has(t,e[0],e[1])&&(r.push(e),bt.remove(t,e[0],e[1]));return r}function Tt(n){const t=gt(mt(n)),r=t.length;let e=0;for(let n=0;n<ut;n++){e=0;for(let n=0;n<r;n++){const r=t[n],o=r._info,c=nt(r);let s,{l:u,m:i,r:f}=r,a=c(u),h=c(i),p=c(f);for(;;){if(({l:u,m:i,r:f}=r),s=(f-u)/2,a>=h&&h<=p){r.l=i-s/2,r.r=i+s/2,e+=h;break}a<=h&&h<=p?(r.l=u-s,r.m=u,r.r=i,p=h,h=a,a=c(r.l)):a>=h&&h>=p?(r.l=i,r.m=f,r.r=f+s,a=h,h=p,p=c(r.r)):a<=h&&h>=p&&(r.l=u-s,r.m=u,r.r=i,p=h,h=a,a=c(r.l))}const d=r.m,[g,b]=tt(d,r);r.ps=b,r.$ps=l(b),o.ps=g,o.$ps=l(g)}}return{cubics:t.map((n=>n.ps)),Energy:e}}function yt(n){const t=gt(mt(n)),r=t.length;let e=0;for(let n=0;n<r;n++){const r=t[n];e+=nt(r)(r.m)}return e}function Ot(n){return Tt(n).cubics}function vt(n,t,r,e){const o=t.slice(),c=e.slice(),s=n.length,u=s-1,i=s-2,f=s-3,l=[];l[0]=-n[0];for(let t=1;t<u;t++){if(0===o[t-1])return;const e=n[t]/o[t-1];o[t]-=e*r[t-1],c[t]-=e*c[t-1],l[t]=-e*l[t-1]}if(l[i]-=r[i],0===o[i])return;c[i]/=o[i],l[i]/=o[i];for(let n=f;n>=0;n--){if(0===o[n])return;c[n]=(c[n]-r[n]*c[n+1])/o[n],l[n]=(l[n]-r[n]*l[n+1])/o[n]}c[u]-=r[u]*c[0]+n[u]*c[i];const a=o[u]+r[u]*l[0]+n[u]*l[i];if(0!==a){c[u]/=a;for(let n=0;n<u;n++)c[n]+=l[n]*c[u];return c}}function Mt(n){const t=n.length,e=[...Array(t)].map((n=>1)),o=[...Array(t)].map((n=>4)),s=[...Array(t)].map((n=>1)),u=[];for(let e=0;e<t;e++){const o=e+1===t?0:e+1;u.push(c(r(c(n[e],2),n[o]),2))}const i=vt(e,o,s,u.map((n=>n[0]))),f=vt(e,o,s,u.map((n=>n[1]))),l=[];for(let e=0;e<t;e++){const o=e+1===t?0:e+1;l.push(r(c(n[o],2),[-i[o],-f[o]]))}const a=[];for(let r=0;r<t;r++){const e=r+1===t?0:r+1;a.push([n[r],[i[r],f[r]],l[r],n[e]])}return a}var wt=t.r,It=t.B6,Pt=t.RX,Et=t.nV;export{wt as cubicBeziersThroughPoints,It as cubicBeziersThroughPoints_C2,Pt as cubicsAndEnergyThroughPoints,Et as energyThroughPoints};