歯車について勉強する2 の履歴(No.4)

歯車について勉強するシリーズ†

• 工作/歯車について勉強する 歯車の形状についての基礎
• 工作/歯車について勉強する2 仮想的なラックを使って実用的な歯車形状を切り出す

ラックを使って歯車を切り出す†

歯車について勉強する でやったようにインボリュート曲線やトロコイド曲線、歯先や歯底やフィレット用の円弧を繋げて歯車形状を作ろうとすると、特に歯元の形状をきれいに作るのがとても難しい。

実際の歯車が製作される手順と同様に、仮想的な切削用のラックを想定して、ラックの歯形により切り出される歯車形状を計算すると実際の歯車と同様の自然な歯形を生成できるようだ。

参考： https://involutegearsoft.hatenablog.com/entry/involute-trochoid-draw-tool2

下図では緑色が切削用のラックであり、赤色の歯車はこのラックの上を転がることによりラックと重なる部分が切削されて形状が定まる。下図ではラックと接している点に小丸が描かれるので、ラックのどの部分で切削されるかを感じ取れる。

ラックは通常の歯車歯形と比べて歯先に太い点線で示した曲線の分だけ高くなっていて、その部分で相手の歯元に必要な隙間を生じる。またこの部分にはフィレットを付けることが多い（そのため曲線になっている）。

LANG: p5js_live
var t;let e;const a=[];function s(t,e,s,h){let r=20*s+20,n=t.createSpan(e).position(20,r),i=t.createSpan(String(h.value));i.position(100,r);let l=t.createSlider(h.min,h.max,h.value,h.step);return l.position(150,r),l.size(300),l.elt.oninput=()=>i.html(String(l.value())),a.push(n,i,l),l}function h(t,e,s,r=0,n=!1){let i=100*r+20,l=20*s+20,o=t.createCheckbox();o.position(i,l),o.checked(n);let u=t.createSpan(e).position(i+20,l);return u.mouseClicked(()=>{o.checked(!o.checked()),o.elt.onchange&&o.elt.onchange()}),u.style("cursor","pointer "),a.push(u,o),o}class r{constructor(t,e){this.x=t,this.y=e}static polar(t,e){return new r(t*Math.cos(e),t*Math.sin(e))}angle(){return Math.atan2(this.y,this.x)}equal(t){return 1e-6>Math.abs(this.x-t.x)&&1e-6>Math.abs(this.y-t.y)}add(t){return new r(this.x+t.x,this.y+t.y)}sub(t){return new r(this.x-t.x,this.y-t.y)}mul(t){return new r(this.x*t,this.y*t)}rotate(t){let e=Math.cos(t),a=Math.sin(t);return new r(this.x*e-this.y*a,this.x*a+this.y*e)}addRotated(t,e){return this.add(e.rotate(t))}flipY(){return new r(this.x,-this.y)}flipX(){return new r(-this.x,this.y)}norm(){return Math.sqrt(this.x*this.x+this.y*this.y)}normalize(t=1){let e=this.norm()/t;return new r(this.x/e,this.y/e)}polar(){return[this.norm(),this.angle()]}inner(t){return this.x*t.x+this.y*t.y}outer(t){return this.x*t.y-this.y*t.x}}function n(t,e){return new r(t,e)}class i extends Array{constructor(t,...e){t instanceof r?(super(t,...e),this.option={}):(super(...e),this.option=t??{})}static circle(t,e=n(0,0),a={},s=100){let h=new i(a);for(let a=0;a<=s;a++){let r=2*Math.PI*a/s;h.push(n(t*Math.cos(r),t*Math.sin(r)).add(e))}return h}draw(t){if(void 0===this.option?.stroke?t.stroke("black"):t.stroke(this.option.stroke),void 0===this.option?.weight?t.strokeWeight(1):t.strokeWeight(this.option.weight),void 0===this.option?.dash?t.drawingContext.setLineDash([]):t.drawingContext.setLineDash(this.option.dash),void 0===this.option?.fill){t.noFill(),t.beginShape();for(let e=0;e<this.length;e++)t.vertex(this[e].x,this[e].y);t.endShape()}else{t.fill(this.option.fill),t.beginShape();for(let e=0;e<this.length;e++)t.vertex(this[e].x,this[e].y);t.endShape(t.CLOSE)}}apply(t,e=this.option){return new i(e,...this.filter(t=>t).map(t))}crossPoint(t){return l(this[0],this[1],t[0],t[1])}direction(t=0){return this[t].sub(this[t+1]).normalize()}}function l(t,e,a,s){let h=e.sub(t),r=s.sub(a),n=a.sub(t),i=h.outer(r);if(0==i)return;let l=n.outer(h)/i,o=n.outer(r)/i;if(!(l<0)&&!(l>1)&&!(o<0)&&!(o>1))return t.add(h.mul(o))}function o(t,e,a){let s=a.sub(e),h=t.sub(e);return Math.abs(s.outer(h)/s.norm())}function u(t,e,a,s=1e-6){for(;t+s<e;){let s=t+(e-t)/3,h=e-(e-t)/3;a(s)<a(h)?e=h:t=s}return(t+e)/2}const c={width:600,height:600,ox:0,oy:0};(t=p).setup=()=>{let{width:r,height:n}=c;c.ox=r/2,c.oy=n/2,t.createCanvas(r,n).style("position","relative"),e=function(t){let e=h(t,"ツールを非表示",27.5,0),r={m:s(t,"ﾓｼﾞｭｰﾙ",0,{min:50,max:800,value:75,step:1}),z:s(t,"歯数",1,{min:4,max:100,value:6,step:1}),alpha:s(t,"圧力角",2,{min:10,max:32,value:20,step:1}),shift:s(t,"転位",3,{min:-.5,max:2,value:0,step:.05}),fillet:s(t,"ﾌｨﾚｯﾄ",4,{min:0,max:.4,value:.4,step:.01}),backlash:s(t,"ﾊﾞｯｸﾗｯｼｭ",5,{min:0,max:2,value:0,step:.01}),theta:s(t,"回転角",6,{min:-90,max:90,value:0,step:.05}),play:h(t,"自動更新",7.5,0,!0),stop:!1};return a.forEach((t,e)=>{t.style("opacity","0.2"),t.mouseOver(()=>a.forEach(t=>t.style("opacity","1"))),t.mouseOut(()=>a.forEach(t=>t.style("opacity","0.2")))}),r.theta.mousePressed(()=>{r.stop=!0}),r.theta.mouseReleased(()=>{r.stop=!1}),e.elt.onchange=()=>{a.forEach((t,a)=>{a<2||(e.checked()?t.hide():t.show())})},{get m(){return Number(r.m.value())},get z(){return Number(r.z.value())},get alpha(){return Number(r.alpha.value())/180*Math.PI},get shift(){return Number(r.shift.value())*this.m},get theta(){return-(Number(r.theta.value())/180)*Math.PI},set theta(value){r.theta.value(-(180*value)/Math.PI),r.theta.elt.oninput()},get fillet(){return Number(r.fillet.value())*this.m},get mk(){return+this.m},get mf(){return 1.25*this.m},get rp(){return this.m*this.z/2},get rf(){return this.m*this.z/2-this.mf},get rk(){return this.m*this.z/2+this.mk},get backlash(){return Number(r.backlash.value())/100*2*Math.PI/this.z},get play(){return r.play.checked()&&!r.stop}}}(t)},t.draw=()=>{let a;let{ox:s,oy:h}=c;t.background(220);let{m:d,z:m,alpha:f,shift:y,theta:g,rp:w,rk:M,rf:k,backlash:x,play:b}=e,{sin:v,max:P,PI:I,cos:z,abs:G,min:S,ceil:R}=Math;if(b){let t=e.theta-.1*I/180;t<=-I/2&&(t=I/2),e.theta=t}let[E,N,C,B,q]=function(t){let e,{m:a,z:s,alpha:h,shift:r,theta:l,fillet:o,mf:u,mk:c,rp:d,rk:m}=t,f=[];o=function(t){let{m:e,alpha:a,fillet:s,mf:h,mk:r}=t;return Math.max(s=Math.min(s,(h-r)/(1-Math.sin(a)),(e*Math.PI/4-(e+h-r)*Math.tan(a))/Math.tan(Math.PI/4-a/2)),0)}(t);let y=Math.PI*a,g=new i({stroke:"Green",dash:[5,5],weight:1},n(-y-u*Math.tan(h),d+u+r),n(-y+c*Math.tan(h),d-c+r),n(-y/2-c*Math.tan(h),d-c+r),n(-y/2+u*Math.tan(h),d+u+r),n(-u*Math.tan(h),d+u+r),n(c*Math.tan(h),d-c+r),n(y/2-c*Math.tan(h),d-c+r),n(y/2+u*Math.tan(h),d+u+r),n(y-u*Math.tan(h),d+u+r),n(y+c*Math.tan(h),d-c+r),n(3*y/2-c*Math.tan(h),d-c+r),n(3*y/2+u*Math.tan(h),d+u+r));f.push(g);let w=new i({stroke:"Green",dash:[3,3],weight:2},n(-u*Math.tan(h),d+u+r),n(u*Math.tan(h),d-u+r));f.push(w);let M=new i({stroke:"Green",dash:[3,3],weight:2},n(u*Math.tan(h),d-u+r),n(a*Math.PI/4,d-u+r));f.push(M);let k=new i({stroke:"Green",dash:[10,6,4,6]},n(a*Math.PI/4,0),n(a*Math.PI/4,m+a+r));if(f.push(k),o>0){let t=w.direction().rotate(-Math.PI/2),a=w.apply(e=>e.add(t.mul(o))),s=M.apply(t=>t.add(n(0,o)));(e=a.crossPoint(s))||(e=a.crossPoint(k));let r=new i({stroke:"Green",dash:[3,3],weight:2});if(e){let t=h+Math.PI,a=3*Math.PI/2;for(let s=0;s<=20;s++){let h=t+(a-t)*s/20;r.push(n(e.x+o*Math.cos(h),e.y+o*Math.sin(h)))}f.push(r),w[1]=r[0],M[0]=r[r.length-1]}else e=w[1]}else{let t=w.crossPoint(k);t?(w[1]=t,M.splice(0),e=t):e=w[1]}f.push(i.circle(a/50,e,{stroke:"Green",fill:"Green"}));let x=new i({stroke:"Green",weight:2},n(c*Math.tan(h),d-c+r),n(a*Math.PI/4,d-c+r)),b=new i({stroke:"Green",weight:2},w[0],x[0]);return f.push(x,b),w[1].equal(x[0])?w.splice(0):w[0]=x[0],f.push(new i({stroke:"Green",dash:[10,10]},n(-a*Math.PI,d+r),n(a*Math.PI,d+r))),[f,b[0],b[1],e,o]}(e),L=[],O=(t,e)=>e.add(n(w*t,0)).rotate(t),Y=(t,e)=>e.rotate(t+I/2).add(n(w*(z(t)-t*v(t)),w*(v(t)+t*z(t))));E=E.map(t=>t.apply(t=>O(g,t)));let D=t=>{let e=O(t,N),a=O(t,C).sub(e),s=Y(t,N),h=Y(t,C).sub(s),r=s.outer(a)/h.outer(a);return[e.sub(a.mul(r)),r]},W,X,A=[];W=u(0,90-f,t=>G(D(t)[0].norm()-M-y)),D(W)[0].angle()>I*(1+1/m)/2-x&&(W=u(0,W,t=>G(D(t)[0].angle()-(I*(1+1/m)/2-x)))),X=u(0,-2*f,t=>D(t)[0].norm());let F=new i({stroke:"Red",weight:2});for(let t=0;t<100;t++){let e=W+(X-W)*t/100;F.push(D(e)[0]),A.push(e)}let j=new i({stroke:"Red",weight:2});if(F[0].angle()-I*(1+1/m)/2+x<-.001){let t=F[0].angle(),e=I*(1+1/m)/2-x;for(let a=0;a<=20;a++){let s=t+(e-t)*a/20;j.push(n((M+y)*z(s),(M+y)*v(s)))}}L.push(j);let H=new i({stroke:"Green",dash:[2,2]}),J,K,Q=[];for(J=0;;J-=.01){let t=B.add(n(w*J,0)).rotate(J);if(H.unshift(t),t.norm()>w+y+d/4)break}for(K=.01;;K+=.01){let t=B.add(n(w*K,0)).rotate(K);if(H.push(t),t.norm()>w+y+d/4)break}E.push(H);let T=new i({stroke:"Red",weight:2}),U=t=>{let e=O(t,B),a=Y(t,B).normalize(q),s=e.addRotated(-I/2,a),h=e.addRotated(I/2,a);return s.x+s.y<h.x+h.y?s:h},V=u(J,K,t=>U(t).norm());O(V,B).norm()>w&&([J,K]=[K,J]),K=V;let Z=!1,$=1,_=J;for(let t=0;t<=100;t+=$){let e=J+(K-J)*t/100,s=U(e);Z||s.norm()>C.norm()&&s.sub(C).angle()<I/2+f||(Z=!0,_=e,T.push(a),Q.push(e-(K-J)*$/100)),Z&&(T.push(s),Q.push(e));let h=O(e,B);if(t>0&&h.sub(O(V,B)).norm()<d/1e3){let t=-G(s.sub(O(V,B)).angle()),e=-I*(1+1/m)/2+x,a=R(G(e-t)/I*180);for(let s=1;s<=a;s++){let h=t+(e-t)*s/a;T.push(O(V,B).add(r.polar(q,h)))}break}t>0&&s.sub(a).norm()>d/100&&($/=2),a=s}J=_,(()=>{F=F.apply(t=>t),T=T.apply(t=>t);let[t,e,a]=function(t,e){let a=0,s=0;for(;a<t.length-1&&s<e.length-1;){if(t[a+1].y>e[s].y){a++;continue}if(t[a].y<e[s+1].y){s++;continue}let h=l(t[a],t[a+1],e[s],e[s+1]);if(h)return[h,a,s];t[a+1].y>e[s+1].y?a++:s++}return[]}(F,T);if(t)F.splice(e+1,1/0,t)

ラックの形状†

ラックは歯数を無限大にした歯車と考えられる。

• 歯の角度 = 圧力角 $\alpha$
• 基準高さで歯と隙間とが１：１になる
• 歯末の高さ $m$ = 基準高さより上の長さ
• 歯元の高さ $1.25m$ = 基準高さより下の長さ

歯車切削用のラックの形状†

歯車切削用のラックの形状は通常のラックと書きの点で異なる：

• 歯末の高さは $m$ ではなく相手の歯末の高さと等しく $1.25m$ になっている
• 必要に応じて歯末にフィレットを付ける
  • フィレットは本来の歯末の高さまでの部分に影響しないようその上の部分に付ける

計算上での切削†

LANG:js
   // 直進＆回転
   const involute= (t: number, p: Vector) =>
     p.add(vec(0, -rp * t)).rotate(t);

   // 直進＆回転の微分
   const dInvolute = (t: number, p: Vector) =>
     p.rotate(t + PI / 2).add(
       vec(
         rp * (sin(t) + t * cos(t)), //
         rp * (-cos(t) + t * sin(t))
       )
     );

LANG:js
   // ラックの直線部分 (点 r1 と点 r2 を通る) の包絡線
   const lineTrace = (t: number) => {
     const p1 = shiftRotate(t, r1);
     const Dp = shiftRotate(t, r2).sub(p1);
     const dp1 = dShiftRotate(t, r1);
     const dDp = dShiftRotate(t, r2).sub(dp1);
     const s = dp1.outer(Dp) / dDp.outer(Dp);
     return [p1.sub(Dp.mul(s)), -s] as [Vector, number];
   };

歯車を描く†

LANG: p5js_live
const t=[];function e(e,n,i,r){let l=20*i+20,s=e.createSpan(n).position(20,l),a=e.createSpan(String(r.value));a.position(100,l);let h=e.createSlider(r.min,r.max,r.value,r.step);return h.position(150,l),h.size(300),h.elt.oninput=()=>a.html(String(h.value())),t.push(s,a,h),h}function n(e,i,r,l=0,s=!1){let a=100*l+20,h=20*r+20,o=e.createCheckbox();o.position(a,h),o.checked(s);let u=e.createSpan(i).position(a+20,h);return u.mouseClicked(()=>{o.checked(!o.checked()),o.elt.onchange&&o.elt.onchange()}),u.style("cursor","pointer "),t.push(u,o),o}class i{constructor(t,e){this.x=t,this.y=e}static polar(t,e){return new i(t*Math.cos(e),t*Math.sin(e))}angle(){return Math.atan2(this.y,this.x)}equal(t){return 1e-6>Math.abs(this.x-t.x)&&1e-6>Math.abs(this.y-t.y)}add(t){return new i(this.x+t.x,this.y+t.y)}sub(t){return new i(this.x-t.x,this.y-t.y)}mul(t){return new i(this.x*t,this.y*t)}rotate(t){let e=Math.cos(t),n=Math.sin(t);return new i(this.x*e-this.y*n,this.x*n+this.y*e)}addRotated(t,e){return this.add(e.rotate(t))}flipY(){return new i(this.x,-this.y)}flipX(){return new i(-this.x,this.y)}norm(){return Math.sqrt(this.x*this.x+this.y*this.y)}normalize(t=1){let e=this.norm()/t;return new i(this.x/e,this.y/e)}polar(){return[this.norm(),this.angle()]}inner(t){return this.x*t.x+this.y*t.y}outer(t){return this.x*t.y-this.y*t.x}}function r(t,e){return new i(t,e)}class l extends Array{constructor(t,...e){t instanceof i?(super(t,...e),this.option={}):(super(...e),this.option=t??{})}static extractPoints(t,e){if(t.length<=e)return t;let n=t.totalLength(),i=new l(t.option),r=0,s=0;for(let l=0;l<t.length-1;l++)s>=r&&(i.push(t[l]),r=n*i.length/e),s+=t[l].sub(t[l+1]).norm();return i.push(t[t.length-1]),i}static circle(t,e=r(0,0),n={},i=100){let s=new l(n);for(let n=0;n<=i;n++){let l=2*Math.PI*n/i;s.push(r(t*Math.cos(l),t*Math.sin(l)).add(e))}return s}totalLength(){let t=0;for(let e=0;e<this.length-1;e++)t+=this[e].sub(this[e+1]).norm();return t}draw(t,e=!1){if(void 0===this.option?.stroke?t.stroke("black"):t.stroke(this.option.stroke),void 0===this.option?.weight?t.strokeWeight(1):t.strokeWeight(this.option.weight),void 0===this.option?.dash?t.drawingContext.setLineDash([]):t.drawingContext.setLineDash(this.option.dash),void 0===this.option?.fill){t.noFill(),t.beginShape();for(let e=0;e<this.length;e++)t.vertex(this[e].x,this[e].y);t.endShape()}else{t.fill(this.option.fill),t.beginShape();for(let e=0;e<this.length;e++)t.vertex(this[e].x,this[e].y);t.endShape(t.CLOSE)}e&&this.forEach(e=>t.circle(e.x,e.y,1.5))}apply(t,e=this.option){return new l(e,...this.map(t))}crossPoint(t){return s(this[0],this[1],t[0],t[1])}direction(t=0){return this[t].sub(this[t+1]).normalize()}}function s(t,e,n,i){let r=e.sub(t),l=i.sub(n),s=n.sub(t),a=r.outer(l);if(0==a)return;let h=s.outer(r)/a,o=s.outer(l)/a;if(!(h<0)&&!(h>1)&&!(o<0)&&!(o>1))return t.add(r.mul(o))}function a(t,e,n){let i=n.sub(e),r=t.sub(e);return Math.abs(i.outer(r)/i.norm())}function h(t,e,n,i=1e-6){for(;t+i<e;){let i=t+(e-t)/3,r=e-(e-t)/3;n(i)<n(r)?e=r:t=i}return(t+e)/2}function o(t){let{m:e,z:n}=t={...t};t.inner&&([t.mk,t.mf]=[t.mf,t.mk],t.fillet=0);let o=e*n/2,{r1:c,r2:f,filletCenter:m,filletRadius:d}=function(t){let e,{m:n,z:i,alpha:l,shift:s,fillet:a,mf:h,mk:o}=t,u=i*n/2,{sin:c,cos:f,tan:m,min:d,max:g,PI:x}=Math;a=d(a,(h-o)/(1-c(l)));let b=g(0,(n*x/4-(n+h-o)*m(l))/m(x/4-l/2));return e=a>b?r(u-h+s+(a=b),-n*x/4):r(u-h+s+a,(-h+a)*m(l)-a/f(l)),{r1:r(u+s+h,h*m(l)),r2:r(u-o+s,-o*m(l)),filletCenter:e,filletRadius:a}}(t),g=(t,e)=>e.add(r(0,-o*t)).rotate(t),x=(t,e)=>{let{sin:n,cos:i,PI:l}=Math;return e.rotate(t+l/2).add(r(o*(n(t)+t*i(t)),o*(-i(t)+t*n(t))))},b=t=>{let e=g(t,c),n=g(t,f).sub(e),i=x(t,c),r=x(t,f).sub(i),l=i.outer(n)/r.outer(n);return[e.sub(n.mul(l)),l]},v=t=>{let e=g(t,m),n=x(t,m).normalize(d),i=e.addRotated(-Math.PI/2,n),r=e.addRotated(Math.PI/2,n);return i.x-i.y<r.x-r.y?i:r},{cInvolute:y,involuteT:k}=function(t,e){let{alpha:n,mk:i,mf:r,shift:s,z:a,m:o,backlash:u}=t,c=o*a/2,{abs:f,PI:m}=Math,d,g,x=[];d=h(0,90-n,t=>f(e(t)[0].norm()-c-i-s)),e(d)[0].angle()>m/a/2-u&&(d=h(0,d,t=>f(e(t)[0].angle()-(m/a/2-u)))),g=h(0,-2*n,t=>e(t)[0].norm()),e(g)[0].norm()&&(g=function(t,e,n,i=1e-6){let r=n(t),l=n(e);if(r*l>0)return Math.abs(r)<Math.abs(l)?t:e;for(r>l&&([t,e]=[e,t]);Math.abs(t-e)>i;){let i=(t+e)/2;0>n(i)?t=i:e=i}return(t+e)/2}(0,g,t=>e(t)[0].norm()-(c-r+s)));let b=new l;for(let t=0;t<300;t++){let n=d+(g-d)*t/300;b.push(e(n)[0]),x.push(n)}return{cInvolute:b,involuteT:x}}(t,b),M=new l,z=[];return t.inner||([M,z]=function(t,e,n,s,a,o){let u,c;let{abs:f,PI:m,ceil:d}=Math,{shift:g,alpha:x,m:b,z:v,mk:y,backlash:k}=t,M=b*v/2,z=M+y;for(u=0;!(s.add(r(0,-M*u)).rotate(u).norm()>z+g);u-=.05);for(c=.05;!(s.add(r(0,-M*c)).rotate(c).norm()>z+g);c+=.05);let w=h(u,c,t=>n(t).norm());e(w,s).norm()>M&&([u,c]=[c,u]),c=w;let P=!1,I=null,S=1,C=u,N=[],E=new l;for(let t=0;t<=60;t+=S){let r=u+(c-u)*t/60,l=n(r);P||l.norm()>o.norm()&&l.sub(o).angle()<x||(P=!0,C=r,I&&(E.push(I),N.push(r-(c-u)*S/60))),P&&(E.push(l),N.push(r));let h=e(r,s);if(t>0&&h.sub(e(w,s)).norm()<b/1e3){let t=f(l.sub(e(w,s)).angle()),n=m-m/v/2+k,r=d(f(n-t)/m*180);for(let l=1;l<=r;l++){let h=t+(n-t)*l/r;E.push(e(w,s).add(i.polar(a,h)))}break}t>0&&l.sub(I).norm()>b/20&&(S/=2),I=l}return u=C,[E,N]}(t,g,v,m,d,f),{cInvolute:y,involuteT:k,cFillet:M,filletT:z}=function(t,e,n,i,r,l){let[o,u,c]=function(t,e){let n=0,i=0;for(;n<t.length-1&&i<e.length-1;){if(t[n+1].x>e[i].x){n++;continue}if(t[n].x<e[i+1].x){i++;continue}let r=s(t[n],t[n+1],e[i],e[i+1]);if(r)return[r,n,i];t[n+1].x>e[i+1].x?n++:i++}return[]}(t,i);if(o)t.splice(u+1,1/0,o),e.splice(u+1,1/0,h(e[u],e[u+1],t=>n(t)[0].sub(o).norm())),i.splice(0,c+1,o),r.splice(0,c+1,h(r[c],r[c+1],t=>l(t).sub(o).norm()));else{let[s,o,u,c]=function(t,e){let n=0,i=0,r=[1/0];for(;n<t.length-1&&i<e.length-1;){if(t[n+1].x>e[i].x){n++;continue}if(t[n].x<e[i+1].x){i++;continue}if(t[n].x<e[i].x){let l=a(t[n],e[i],e[i+1]);l<r[0]&&(r=[l,n,i,0])}else{let l=a(e[i],t[n],t[n+1]);l<r[0]&&(r=[l,n,i,1])}t[n+1].x>e[i+1].x?n++:i++}return r}(t,i);c?(t.splice(o+1,1/0,i[u]),i.splice(0,u),e.splice(o+1,1/0,h(e[o],e[o+1],t=>n(t)[0].sub(i[u]).norm())),r.splice(0,u)):(t.splice(o+1,1/0),i.splice(0,u+1,t[o]),e.splice(o+1,1/0),r.splice(0,u+1,h(r[u],r[u+1],e=>l(e).sub(t[o]).norm())))}return{cInvolute:t,involuteT:e,cFillet:i,filletT:r}}(y,k,b,M,z,v)),function(t,e,n){let{m:i,mk:r,shift:s,z:a,backlash:h,mf:o}=t,{PI:c}=Math,f=[],m=u(i*a/2+r+s,c/a/2-h,e[0].angle());f.push(m);let d=l.extractPoints(e,30).filter((t,e,n)=>e<2||e>n.length-3||e%2==1);f.push(new l(e.option,...d));let g=l.extractPoints(n,30).filter((t,e,n)=>e<2||e>n.length-3||e%2==1);if(f.push(new l(n.option,...g)),t.inner){let t=u(e.at(-1).norm(),e.at(-1).angle(),-c/a/2-h);f.push(t)}else if(n.length>0){let t=u(i*a/2-o+s,n[n.length-1].angle(),-c/a/2-h);f.push(t)}f=f.map(t=>t.apply(t=>t.rotate(h-c/a/2)));let x=[];x.push(e[0].rotate(h-c/a/2).polar()),x.push(e.at(-1).rotate(h-c/a/2).polar()),n.length>0&&x.push(n.at(-1).rotate(h-c/a/2).polar());let b=f.reduce((t,e)=>t.slice(0,-1).concat(e.map(t=>t.polar())),[]),v={curve:[...b=b.map(t=>[t[0],-t[1]]).reverse().concat(b)],connections:[...x=x.map(t=>[t[0],-t[1]]).reverse().concat(x)]};for(let t=1;t<a;t++)v.curve=v.curve.concat(b.map(e=>[e[0],e[1]-2*t*c/a])),v.connections=v.connections.concat(x.map(e=>[e[0],e[1]-2*t*c/a]));return v}(t,y,M)}function u(t,e,n,s=r(0,0),a=.02,h=1e-4){let{abs:o,ceil:c,max:f}=Math,m=new l;if(t*o(n-e)>h){let r=f(2,c(o(n-e)/a));for(let l=0;l<=r;l++){let a=e+(n-e)*l/r;m.push(s.add(i.polar(t,a)))}}return m}let c=null,f=null,m=null;(i=>{let r;let l={width:600,height:600,ox:0,oy:0};i.setup=()=>{let{width:s,height:a}=l;l.ox=s/2,l.oy=a/2,i.createCanvas(s,a).style("position","relative"),r=function(i){let r=n(i,"ツールを非表示",27.5,0),l={m:e(i,"ﾓｼﾞｭｰﾙ",0,{min:10,max:200,value:40,step:1}),alpha:e(i,"圧力角",1,{min:10,max:32,value:20,step:1}),z:e(i,"歯数A",2,{min:4,max:200,value:19,step:1}),shift:e(i,"転位A",3,{min:-.5,max:2,value:0,step:.05}),z2:e(i,"歯数B",4,{min:4,max:200,value:6,step:1}),shift2:e(i,"転位B",5,{min:-.5,max:2,value:0,step:.05}),fillet:e(i,"ﾌｨﾚｯﾄ",6,{min:0,max:.4,value:.4,step:.01}),backlash:e(i,"ﾊﾞｯ

ｸﾗｯｼｭ",7,{min:0,max:2,value:0,step:.01}),velocity:e(i,"速度",8,{min:-1,max:10,value:1,step:.1}),theta:e(i,"回転角",9,{min:-180,max:180,value:0,step:.05}),df:n(i,"歯底円",10.5,0,!1),db:n(i,"基礎円",10.5,1,!1),dp:n(i,"基準円",10.5,2,!0),dk:n(i,"歯先円",10.5,3,!1),play:n(i,"自動更新",11.5,0,!0),connection:n(i,"接続点",11.5,1,!1),inner:n(i,"内歯車",11.5,2,!1),stop:!1};return t.forEach*1e,n)=>{e.style("opacity","0.2"),e.mouseOver*2)=>t.forEach(t=>t.style("opacity","1"(#notefoot_2)),e.mouseOut*3)=>t.forEach(t=>t.style("opacity","0.2"(#notefoot_3))}),l.theta.mousePressed*4)=>{l.stop=!0}),l.theta.mouseReleased*5)=>{l.stop=!1}),r.elt.onchange=()=>{t.forEach*6t,e)=>{e<2||(r.checked()?t.hide():t.show((#notefoot_6)})},{get m(){return Number(l.m.value((#notefoot_5)},get alpha(){return Number(l.alpha.value((#notefoot_4)/180*Math.PI},get z(){return Number(l.z.value(},get shift(){return Number(l.shift.value())*this.m},get z2(){return Number(l.z2.value())},get shift2(){return Number(l.shift2.value())*this.m},get theta(){return-(Number(l.theta.value())/180)*Math.PI},set theta(value){l.theta.value(-(180*value)/Math.PI),l.theta.elt.oninput()},get fillet(){return Number(l.fillet.value())*this.m},get mk(){return+this.m},get mf(){return 1.25*this.m},get backlash(){return Number(l.backlash.value())/100*2*Math.PI/this.z},get play(){return l.play.checked()&&!l.stop},get velocity(){return Number(l.velocity.value())/180*Math.PI},get df(){return l.df.checked()},get db(){return l.db.checked()},get dp(){return l.dp.checked()},get dk(){return l.dk.checked()},get connection(){return l.connection.checked()},get inner(){return l.inner.checked()}}}(i)},i.draw=()=>{var t,e,n;let h;let{ox:u,oy:d}=l;i.background(220);let{gear1:g,gear2:x}=(t=r,h=null!==c?Object.keys(e=c).filter(n=>t[n]!==e[n]):[],(!f||h.filter(t=>!["m2","shift2"].includes(t)).length>0)&&(f=o(t)),(!m||h.filter(t=>!["m","shift","inner"].includes(t)).length>0)&&(m=o({...t,z:t.z2,shift:t.shift2,inner:!1})),n=t,c=["m","z","alpha","shift","z2","shift2","fillet","backlash","inner"].reduce*7t,e)=>(t[e]=n[e],t),{}),{gear1:f,gear2:m});r.play&&(r.theta-r.velocity<-Math.PI?r.theta+=-r.velocity+2*Math.PI:r.theta+=-r.velocity),s(i,g,r.inner?u+r.m*r.z/2-r.shift:u-r.m*r.z/2-r.shift,d,r.inner?r.theta/r.z+Math.PI+Math.PI/r.z:-r.theta/r.z,r.connection,r.inner?r.m*(5+r.z)/2+r.shift:0),s(i,x,u+r.m*r.z2/2+r.shift2,d,r.theta/r.z2+Math.PI+Math.PI/r.z2,r.connection),a(i,r.inner?u+r.m*r.z/2-r.shift:u-r.m*r.z/2-r.shift,d,r),a(i,u+r.m*r.z2/2+r.shift2,d,{...r,z:r.z2,shift:r.shift2})};let s=(t,e,n,i,r=0,l=!1,s=0)=>{t.strokeWeight(1),t.stroke("black"),t.fill("white"),s&&(t.circle(n,i,2*s),t.fill(220,t.beginShape(),e.curve.forEach(e=>{t.vertex(n+e[0]*Math.cos(e[1]+r),i+e[0]*Math.sin(e[1]+r))}),t.endShape("close"),l&&e.connections.forEach(e=>{t.circle(n+e[0]*Math.cos(e[1]+r),i+e[0]*Math.sin(e[1]+r),2)})},a=(t,e,n,i)=>{let{df:r,db:l,dp:s,dk:a,m:h,z:o,mk:u,mf:c,alpha:f,shift:m}=i;t.strokeWeight(1),t.stroke("Blue"),t.noFill(),t.drawingContext.setLineDash([2,6]),r&&t.circle(e,n,(h*o/2-c+m)*2),l&&t.circle(e,n,h*o*Math.cos(f)),s&&t.circle(e,n,h*o),a&&t.circle(e,n,(h*o/2+u+m)*2),t.drawingContext.setLineDash([])}})(p);

内歯車について：

• インボリュート曲線部分のみで歯形としている
• フィレットは付けていない
• 内歯車の転位には対応していない ← TODO

コメント・質問†