cubic-beziers-through-points/browser/index.module.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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function(n){const[[t,e],[r,o],[u,i],[c,s]]=n,l=c+3*r,f=t+3*u,a=l-f,h=Jn(3*r)+Jn(l)+(Jn(3*u)+Jn(f))+Jn(a);if(Jn(a)-Gn*h>0)return!1;const g=s+3*o,d=e+3*i,p=g-d,b=Jn(3*o)+Jn(g)+(Jn(3*i)+Jn(d))+Jn(p);return!(Jn(p)-Gn*b>0)&&0===Rn(Un(Hn([c],Yn(3,r)),Hn([t],Yn(3,u))))&&0===Rn(Un(Hn([s],Yn(3,o)),Hn([e],Yn(3,i))))}(n)?function(n){if(!Qn(n))return!1;const[t,e,r,o]=n,[u,i]=t,[c,s]=e,[l,f]=r,[a,h]=o,g=Xn(Wn(Zn(c/4,l/4),3),Zn(u/4,a/4)),d=Xn(Wn(Zn(s/4,f/4),3),Zn(i/4,h/4));return 0===nt(Xn(Zn(u/2,a/2),g))&&0===nt(Xn(Zn(i/2,h/2),d))}(n)?$n(n)?$t:zt:Qn(n)?jt:_t:Qn(n)?Ft:function(n){const{coeffs:[t,e,r],errBound:[o,u,i]}=function(n){const{coeffs:[[t,e,r],[o,u,i]],errorBound:[[c,s],[l,f]]}=function(n){const[[t,e],[r,o],[u,i],[c,s]]=n,l=c-t,f=r-u,a=l+3*f,h=u+t,g=h-2*r,d=r-t,p=s-e,b=o-i,m=p+3*b,N=i+e,T=N-2*o,O=o-e;return{coeffs:[[a,3*g,3*d,t],[m,3*T,3*O,e]],errorBound:[[tt(l)+6*tt(f)+tt(a),6*(tt(h)+tt(g)),6*tt(d),0],[tt(p)+6*tt(b)+tt(m),6*(tt(N)+tt(T)),6*tt(O),0]]}}(n),a=et(t),h=et(e),g=et(r),d=et(o),p=et(u),b=et(i),m=e*o,N=t*u,T=t*i,O=r*o,v=e*i,w=r*u,M=s*d+h*l+et(m),y=c*p+a*f+et(N),I=c*b+et(T),x=g*l+et(O),E=s*b+et(v),S=g*f+et(w),P=m-N,B=et(P),D=M+y+B,L=O-T,A=et(L),F=x+I+A,_=v-w,j=et(_),z=E+S+j,$=P*P,k=P*L,C=_*P,V=L*L,q=C+V;return{coeffs:[$,k,q],errBound:[2*D*B+et($),D*A+B*F+et(k),z*B+j*D+et(C)+(2*A*F+et(V))+et(q)].map((n=>rt*n))}}(n);if(vt(t)<=o){const[[t,e],[r,o],[u,i],[c,s]]=n,l=Tt(mt(c,t),bt(3,mt(r,u))),f=Ot(Nt(u,t),-2*r),a=Tt(mt(s,e),bt(3,mt(o,i))),h=Ot(Nt(i,e),-2*o),g=pt(f,a),d=pt(l,h);if(0===gt(g,d))return At}const c=e*e,s=4*t*r,l=c-s,f=2*u*vt(e)+wt*c+(4*(o*vt(r)+vt(t)*i)+wt*vt(s))+wt*vt(l);if(l<-f)return Dt;if(l>f)return Bt;let[a,h,g]=function(n){const[[t,e,r],[o,u,i]]=function(n){const[[t,e],[r,o],[u,i],[c,s]]=n;return[[st(ot(c,t),it(3,ot(r,u))),it(3,ct(ut(u,t),-2*r)),it(3,ot(r,t)),[t]],[st(ot(s,e),it(3,ot(o,i))),it(3,ct(ut(i,e),-2*o)),it(3,ot(o,e)),[e]]]}(n),c=lt(e,o),s=lt(t,u),l=lt(t,i),f=lt(r,o),a=lt(e,i),h=lt(r,u),g=at(c,s),d=at(f,l),p=at(a,h),b=lt(g,g),m=lt(g,d),N=lt(g,p),T=lt(d,d);return[b,m,ft(N,T)]}(n);const d=dt(pt(h,h),bt(4,pt(a,g))),p=ht(d);return p<0?Dt:p>0?Bt:Lt}(n);if(3===n.length)return Qn(n)?function(n){const[[t,e],[r,o],[u,i]]=n,c=t+u,s=c-2*r,l=qn(c)+qn(s);if(qn(s)-l>0)return!1;const f=e+i,a=f-2*o,h=qn(f)+qn(a);return!(qn(a)-h>0)&&0===Cn(kn(Vn(t,u),[2*r]))&&0===Cn(kn(Vn(e,i),[2*o]))}(n)?$n(n)?Pt:St:Et:xt;if(2===n.length)return $n(n)?It:yt;if(1===n.length)return Mt;throw new Error("The given bezier curve must be of order <= 3")}const Ct=function(n,t){const e=n*t,r=An*n,o=r-(r-n),u=n-o,i=An*t,c=i-(i-t),s=t-c;return[u*s-(e-o*c-u*c-o*s),e]},Vt=Fn,qt=_n,{atan2:Yt}=Math;function Ht(n,t){const e=n[0],r=n[1],o=t[0],u=t[1];let i=qt(Ct(u,e),Ct(o,r))[1],c=Vt(Ct(o,e),Ct(u,r))[1];return Yt(i,c)}const{abs:Rt}=Math;function Ut(n){const t=[];for(let e=0;e<n.length-1;e++){const r=[n[e+1][0]-n[e][0],n[e+1][1]-n[e][1]];0!==(0!==r[0]||r[1])&&t.push(r)}const e=t.length;let r=0;for(let n=0;n<e-1;n++)r+=Rt(Ht(t[n],t[n+1]));return r}function Gt(n,t,e){if(4===n.length)return function(n,t,e){return 0===t?1===e?n:function(n,t){const e=n[0],r=n[1],o=n[2],u=n[3],i=e[0],c=e[1],s=r[0],l=r[1],f=o[0],a=o[1],h=i-t*(i-s),g=s-t*(s-f),d=h-t*(h-g),p=c-t*(c-l),b=l-t*(l-a),m=p-t*(p-b);return[[i,c],[h,p],[d,m],[d-t*(d-(g-t*(g-(f-t*(f-u[0]))))),m-t*(m-(b-t*(b-(a-t*(a-u[1])))))]]}(n,e):1===e?function(n,t){const e=n[0],r=n[1],o=n[2],u=n[3],i=e[0],c=e[1],s=r[0],l=r[1],f=o[0],a=o[1],h=u[0],g=u[1],d=i-t*(i-s),p=s-t*(s-f),b=f-t*(f-h),m=d-t*(d-p),N=p-t*(p-b),T=c-t*(c-l),O=l-t*(l-a),v=a-t*(a-g),w=T-t*(T-O),M=O-t*(O-v);return[[m-t*(m-N),w-t*(w-M)],[N,M],[b,v],[h,g]]}(n,t):function(n,t,e){const r=n[0],o=n[1],u=n[2],i=n[3],c=r[0],s=r[1],l=o[0],f=o[1],a=u[0],h=u[1],g=t*t,d=t*g,p=e*e,b=e*p,m=t*e,N=c-l,T=a-l,O=N+T,v=t*N,w=e*N,M=i[0]-c-3*T,y=s-f,I=h-f,x=y+I,E=t*y,S=e*y,P=i[1]-s-3*I;return[[d*M+(3*t*(t*O-N)+c),d*P+(3*t*(t*x-y)+s)],[m*(t*M+2*O)+(g*O+c-(w+2*v)),m*(t*P+2*x)+(g*x+s-(S+2*E))],[m*(e*M+2*O)+(p*O+c-(2*w+v)),m*(e*P+2*x)+(p*x+s-(2*S+E))],[b*M+(3*e*(e*O-N)+c),b*P+(3*e*(e*x-y)+s)]]}(n,t,e)}(n,t,e);if(3===n.length)return function(n,t,e){return 0===t?1===e?n:function(n,t){const e=n[0],r=n[1],o=n[2],u=e[0],i=e[1],c=r[0],s=r[1],l=t*t,f=u-c,a=i-s;return[[u,i],[-t*f+u,-t*a+i],[l*(f+(o[0]-c))-(2*t*f-u),l*(a+(o[1]-s))-(2*t*a-i)]]}(n,e):1===e?function(n,t){const e=n[0],r=n[1],o=n[2],u=e[0],i=e[1],c=r[0],s=r[1],l=o[0],f=o[1],a=t*t,h=u-c,g=l-c,d=i-s,p=f-s;return[[a*(h+g)-(2*t*h-u),a*(d+p)-(2*t*d-i)],[t*g+c,t*p+s],[l,f]]}(n,t):function(n,t,e){const r=n[0],o=n[1],u=n[2],i=r[0],c=r[1],s=o[0],l=o[1],f=t*t,a=e*e,h=t*e,g=i-s,d=g+(u[0]-s),p=c-l,b=p+(u[1]-l);return[[f*d-(2*t*g-i),f*b-(2*t*p-c)],[h*d-(g*(e+t)-i),h*b-(p*(e+t)-c)],[a*d-(2*e*g-i),a*b-(2*e*p-c)]]}(n,t,e)}(n,t,e);if(2===n.length)return function(n,t,e){return 0===t?1===e?n:function(n,t){const e=n[0],r=n[1],o=e[0],u=e[1];return[[o,u],[t*(r[0]-o)+o,t*(r[1]-u)+u]]}(n,e):1===e?function(n,t){const e=n[0],r=n[1],o=e[0],u=e[1],i=r[0],c=r[1];return[[t*(i-o)+o,t*(c-u)+u],[i,c]]}(n,t):function(n,t,e){const r=n[0],o=n[1],u=r[0],i=r[1],c=o[0],s=o[1];return[[t*(c-u)+u,t*(s-i)+i],[e*(c-u)+u,e*(s-i)+i]]}(n,t,e)}(n,t,e);if(1===n.length)return n;throw new Error("The given bezier curve must be of order <= 3.")}function Jt(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:Kt}=Math;function Qt(n,t=1.125,e=4){const r=kt(n);return r.collinear?function(n){if(!Qn(n))return!1;const t=n.map((n=>n[0])),e=n.map((n=>n[1]));return!(Jt(t)&&Jt(e))}(n)?Number.POSITIVE_INFINITY:0:3===r.realOrder?"cusp"===r.nodeType?Number.POSITIVE_INFINITY:Wt(Xt,n,t,e):2===r.realOrder?Wt(Zt,n,t,e):0}function Wt(n,t,e,r){const o=function(n,t=.4,e=2**-16){let r={r:[0,1]},o=r;for(;void 0!==o;){const r=o.r;if(Ut(Gt(n,r[0],r[1]))<=t||r[1]-r[0]<=e){o=o.next;continue}const u=(r[0]+r[1])/2,i=[r[0],u],c=[u,r[1]];o.r=i,o.next={r:c,next:o.next}}o=r;const u=[];for(;void 0!==o;)u.push(o.r[0]),void 0===o.next&&u.push(o.r[1]),o=o.next;return u}(t,e);let u=0;for(let e=0;e<o.length-1;e++)u+=Dn(n(Gt(t,o[e],o[e+1])),[0,1],r);return u}function Xt(n){const t=n[0],e=n[1],r=n[2],o=n[3],u=t[0],i=t[1],c=e[0],s=e[1],l=r[0],f=r[1],a=o[0],h=o[1],g=a-u+3*(c-l),d=h-i+3*(s-f),p=l+u-2*c,b=f+i-2*s;return n=>{const t=n*n,e=t*g+2*n*p+(c-u),r=t*d+2*n*b+(s-i),o=e*(n*d+b)-r*(n*g+p),l=e*e+r*r;return 4/3*o*o/(l*l*Kt(l))}}function Zt(n){const t=n[0],e=n[1],r=n[2],o=t[0],u=t[1],i=e[0],c=e[1],s=r[0],l=r[1],f=s-2*i+o,a=l-2*c+u;return n=>{const t=f*n+(i-o),e=a*n+(c-u),r=t*a-e*f,s=t*t+e*e;return.5*r*r/(s*s*Kt(s))}}const{cos:ne,sin:te}=Math;function ee(n){const{α:t,β:e,s0:r,s1:o,L:u,rot:i,p:c}=n;return function(n){const t=n[0].length;if(4===t){const[[t,e,r,o],[u,i,c,s]]=n;return[[o,s],[o+r/3,s+c/3],[o+2*r/3+e/3,s+2*c/3+i/3],[o+r+e+t,s+c+i+u]]}if(3===t){const[[t,e,r],[o,u,i]]=n;return[[r,i],[r+e/2,i+u/2],[r+e+t,i+u+o]]}if(2===t){const[[t,e],[r,o]]=n;return[[e,o],[e+t,o+r]]}if(1===t){const[[t],[e]]=n;return[[t,e]]}throw new Error("The given bezier curve must be of order <= 3.")}([[u*(r*ne(t)-2+o*ne(e)),u*(3-2*r*ne(t)-o*ne(e)),u*(r*ne(t)),0*u],[u*(r*te(t)+o*te(e)),u*(-2*r*te(t)-o*te(e)),u*(r*te(t)),0*u]]).map(Mn(te(i),ne(i))).map(wn(c))}function re(n){return t=>{const[e,r]=oe(t,n);return Qt(r)+Qt(e)}}function oe(n,t){const{ϕ:e}=t,r=t._info,o=r.$ps,u=t.$ps,i=ee({...o,β:n-e,s0:1,s1:1}),c=ee({...u,α:n,s0:1,s1:1});{const n=r.ps,e=t.ps;return[[n[0],i[1],i[2],n[3]],[e[0],c[1],c[2],e[3]]]}}const{PI:ue}=Math,ie=ue,ce=.0625,[se,le]=[1,1],{min:fe,max:ae,PI:he,atan2:ge,sin:de,cos:pe}=Math;function be(n){const t=function(n){const t=n.length,e=[];for(let n=0;n<t;n++)e.push({});for(let u=0;u<t;u++){const i=(u-1+t)%t,c=(u+1)%t,s=e[i],l=e[u],f=e[c];l._info=s,l.info_=f;const a=n[i],h=n[u],g=n[c],d=(o=g,-Ht(xn(a,r=h),xn(r,o)));l.ϕ=d;const p=[ae(-ie,d-ie),fe(+ie,d+ie)];l.range=p;const b=wn(yn(h))(g),[m,N]=[b[0],b[1]],T=ge(N,m),O=Mn(de(-T),pe(-T))(b)[0];l.L=O,l.rot=T,l.p=h}var r,o;for(let n=0;n<t;n++){const t=e[n],{ϕ:r,info_:o,p:u,L:i,rot:c}=t,s=r/2,l={α:s,β:-o.ϕ/2,p:u,L:i,rot:c,s0:se,s1:le},f=ee(l);t.ps=[u,f[1],f[2],o.p],t.$ps=l,t.l=s-he*ce,t.m=s,t.r=s+he*ce}return e}(n),e=t.length;for(let n=0;n<4;n++)for(let n=0;n<e;n++){const e=t[n],r=e._info,o=re(e);let u,{l:i,m:c,r:s}=e,l=o(i),f=o(c),a=o(s);for(;;){if(({l:i,m:c,r:s}=e),u=(s-i)/2,l>=f&&f<=a){e.l=c-u/2,e.r=c+u/2;break}l<=f&&f<=a?(e.l=i-u,e.m=i,e.r=c,a=f,f=l,l=o(e.l)):l>=f&&f>=a?(e.l=c,e.m=s,e.r=s+u,l=f,f=a,a=o(e.r)):l<=f&&f>=a&&(e.l=i-u,e.m=i,e.r=c,a=f,f=l,l=o(e.l))}const h=e.m,[g,d]=oe(h,e);e.ps=d,e.$ps=Bn(d),r.ps=g,r.$ps=Bn(g)}return t.map((n=>n.ps))}var me=t.j;export{me as cubicBeziersThroughPoints};