2e:null 2f:["$","title",null,{"children":"Gears | Roeland Schoukens | Observable"}] 34:T6a2,function update_gear_canvas(gear_canvas, settings) { const {zoom_to, tooth_count_1, tooth_count_2, tooth_shape} = settings; const {chart_size} = gear_canvas; const tooth_count_1_corrected = tooth_count_1 + 2 * get_involute_extra_dist(settings); var x, y, zoom; if (zoom_to == "gear_both") { zoom = 1 / (tooth_count_1_corrected + tooth_count_2 + 4); x = 0.5 * (tooth_count_1_corrected + 4) * zoom; } else if (zoom_to == "gear_1") { zoom = .995 / (tooth_count_1_corrected + 4); x = 0.5; } else if (zoom_to == "gear_2") { zoom = .995 / (tooth_count_2 + 4); x = 0.5 * (5 - tooth_count_1_corrected) * zoom; } else if (zoom_to == "teeth_1"){ zoom = 0.040; x = 0.5 - zoom * 0.5 * tooth_count_1_corrected; } else { var extra_zoom = 1 + 2 / Math.min(tooth_count_1, tooth_count_2) + 1 / (6 + Math.max(tooth_count_1, tooth_count_2)) zoom = 0.10 * extra_zoom; x = 0.5 - zoom * 0.5 * tooth_count_1_corrected; } gear_canvas.tooth_shape = tooth_shape; gear_canvas.gear_1 = undefined; gear_canvas.gear_2 = undefined; gear_canvas.zoom = zoom; gear_canvas.x1 = x; try { gear_canvas.gear_1 = new Gear( {...settings, tooth_shape: opposite_gear_shape(tooth_shape), color: "#04f", tooth_count: tooth_count_1, opposite_tooth_count: tooth_count_2}); } catch {} try { gear_canvas.gear_2 = new Gear({...settings, color: "#f62", modify_involute: 0, tooth_count: tooth_count_2, opposite_tooth_count: tooth_count_1}); } catch {} }35:T769,function draw_gears_on_canvas(gear_canvas, settings, rotate) { const ctx = gear_canvas.ctx; var {x1, zoom, chart_size, gear_1, gear_2} = gear_canvas; const {tooth_count_1, tooth_count_2, tooth_shape} = settings; x1 *= chart_size; const y = 0.5 * chart_size; zoom *= chart_size; var show = settings.show; if (tooth_shape == "involute") { show = show.filter(x => x != "loa") } // erase ctx.fillStyle = "white"; ctx.clearRect(0, 0, ctx.canvas.width / gear_canvas.dpr, ctx.canvas.height / gear_canvas.dpr); if (gear_1) { var rot_rad = rotate * 2 * Math.PI / tooth_count_1; gear_1.draw(ctx, x1, y, zoom, show, rot_rad, 0); } var between_axes = 0.5 * (tooth_count_1 + tooth_count_2) + get_involute_extra_dist(settings); const x2 = x1 + zoom * between_axes; var rot_extra = 0; // calculate geometry quircks for involute meshing var rot_extra = 0, sinPA, cosPA; if (gear_1 && gear_2 && tooth_shape == "involute") { ({rot_extra, sinPA, cosPA} = get_involute_extra_rotation(gear_1, gear_2, between_axes)); } if (gear_2) { var rotate_2 = tooth_count_2 / 2 - rotate - (gear_1.tooth_width + gear_2.tooth_width) / 2; var rot_rad = rotate_2 * 2 * Math.PI / tooth_count_2 - rot_extra; gear_2.draw(ctx, x2, y, zoom, show, rot_rad, Math.PI); } if (tooth_shape == "involute" && gear_1 && gear_2 && settings.show.includes("loa")) { var p1 = {x: x1 + zoom * cosPA * gear_1.base, y: y - zoom * sinPA * gear_1.base}; var p2 = {x: x2 - zoom * cosPA * gear_2.base, y: y + zoom * sinPA * gear_2.base}; const line_width = 2 / ((1 + pixel_ratio)); ctx.lineWidth = 0.7 * line_width; ctx.beginPath(); ctx.moveTo(p1.x, p1.y); ctx.lineTo(p2.x, p2.y); ctx.strokeStyle = "#095"; ctx.lineWidth = line_width; ctx.setLineDash([]); ctx.stroke(); } }36:T426,function get_involute_extra_rotation(gear_1, gear_2, between_axes) { if (gear_1.tooth_shape != "involute") return undefined; // loa depends on distance between axes, it is the line which touches both base circles // right triangle math var total_base = gear_1.base + gear_2.base; var cosPA = total_base / between_axes; var sinPA = Math.sqrt(1 - cosPA * cosPA); var modified_r = 1 / cosPA; // rotation from involute root to touch poing on LOA: // rotatet forward by actual pressure angle, rotate backwards by // angle between involute root and pitch point var actual_pa = Math.acos(cosPA); var rot_to_modified_r = actual_pa - Math.sqrt(modified_r * modified_r - 1); // to actually touch, the gears need slightly more rotation as they move further apart var rot_1_extra = -gear_2.involute_root_angle + rot_to_modified_r; var rot_2_extra = -gear_1.involute_root_angle + rot_to_modified_r; return {rot_extra: rot_1_extra + gear_1.tooth_count / gear_2.tooth_count * rot_2_extra, cosPA: cosPA, sinPA: sinPA}; }37:T292a,Gear = { // This creates a gear, the module of the gear is one unit. function _Gear(settings) { const {tooth_shape = "involute", hole_shape = "", tooth_count, pressure_angle = 20} = settings; const tw = .5; const tooth_angle = 2 * Math.PI / tooth_count; const radius = tooth_count * 0.5; Object.assign(this, settings, {pitch: tooth_angle, tooth_count:tooth_count, tooth_width: tw}); if (tooth_shape == "lantern") { // lantern gear, this is a simple series of circles if (tooth_count < 4) throw RangeError("must have at least 4 teeth"); const l_radius = 0.25 * Math.PI; // 1/4 of the tooth pitch const l_c = r_th_to_p(radius, -tooth_angle / 4); const dd = 2.2 * l_radius; const N = 50; this.tooth_profile = []; // we draw a solid circle inside the lantern bars with lines // connecting them. { const N2 = 5; const r = radius - dd; const gap = Math.PI / N * l_radius / r; const th_max = -tooth_angle / 4 + gap; for (var i = -N2; i <= N2; ++i) { this.tooth_profile.push(r_th_to_p(r, i / N2 * th_max)); } } { for (var i = 0.5; i < N/2; ++i) { var p1 = r_th_to_p(l_radius, i / N * 2*Math.PI + tooth_angle / 4); this.tooth_profile.push({x: l_c.x - p1.x, y: l_c.y + p1.y}); } } Object.assign(this, {ad:l_radius, dd:dd, undercut_radius: radius - l_radius}); } else if (tooth_shape == "lantern-cycloid") { if (tooth_count < 4) throw RangeError("must have at least 4 teeth"); const l_radius = 0.25 * Math.PI; // 1/4 of the tooth pitch const dd = (settings.cycloid_round_bottom ? 1.05 : 1.2) * l_radius; this.tooth_profile = []; // undercut { const N = 30; const r1 = radius - dd + l_radius; const l_c = r_th_to_p(r1, tooth_angle / 4); const th_max = Math.acos(0.5 * l_radius / r1); if (settings.cycloid_round_bottom) { for (var i = .5; i < N; ++i) { var p1 = r_th_to_p(l_radius, -i/N * th_max - tooth_angle / 4); this.tooth_profile.push({x: l_c.x - p1.x, y: l_c.y + p1.y}); } } else { var p1 = r_th_to_p(l_radius / Math.cos(0.5 * th_max), -0.5 * th_max - tooth_angle / 4); this.tooth_profile.push({x: l_c.x - p1.x, y: l_c.y + p1.y}); } } const r_other = settings.opposite_tooth_count / tooth_count * radius; const lc = make_lantern_cycloid(radius, r_other, l_radius, tooth_angle, -Math.tan(tooth_angle / 4)) lc.points.forEach(p => this.tooth_profile.push(p)); Object.assign(this, {ad:2 * l_radius, dd:dd, undercut_radius: radius, loa:lc.loa}); } else if (tooth_shape == "cycloid") { if (tooth_count < 2) throw RangeError("must have at least 4 teeth"); // these look a bit better with a slightly larger ad/dd. // in practical gears these aren't fixed ratios. const dd = Math.min(1.40, radius * .85); const ad = 1.33; const param1 = 2 - (pressure_angle - 5) / 15; var cycloid = make_cycloid_tooth(radius, param1, dd, ad, tooth_count, settings.opposite_tooth_count, settings.cycloid_round_bottom); Object.assign(this, {ad:ad, dd:dd, r_hc: cycloid.r_hc, r_ec: cycloid.r_ec, loa_end_th:cycloid.loa_end_th, tooth_no_uc:cycloid.points_no_uc, tooth_profile: cycloid.points}); this.loa = [{x:radius, y:0}]; const r_hc = cycloid.r_hc; const th_max = cycloid.loa_end_th; for (var i = 0; i <= 20; ++i) { var th = th_max * (i + 0) / 20; var p1 = r_th_to_p(r_hc, th); this.loa.push({x:radius - r_hc + p1.x, y:-p1.y}); } var p1 = this.loa[this.loa.length - 1]; this.undercut_radius = Math.hypot(p1.x, p1.y); } else if (tooth_shape =="involute") { if (tooth_count < 3) throw RangeError("must have at least 4 teeth"); const modify_involute = settings.modify_involute || 0; // addendum and dedendum: // this is apparently a de facto standard: const ad = 1.00 + modify_involute; // equal to module const dd_unm = 1.16; // 16% over the addendum const dd = dd_unm - modify_involute; const pressure_angle_rad = Math.max(5, pressure_angle) * Math.PI/180; const involute = make_involute(radius, dd, ad, modify_involute, tooth_angle, pressure_angle_rad); const undercut = make_undercut_involute( radius, pressure_angle_rad, tooth_angle, dd, ad, dd_unm, modify_involute); const combined = min_curve(undercut, involute.points, tooth_angle * 1e-5); this.undercut_radius = combined.first_2 && Math.hypot(combined.first_2.x, combined.first_2.y); Object.assign(this, {ad:ad, dd:dd, base: involute.base, involute_root_angle: involute.root_angle, tooth_profile: combined.points, tooth_no_uc: involute.points, pressure_angle_rad: pressure_angle_rad}) this.loa = [ {x:radius, y:0}, r_th_to_p(this.base, -this.pressure_angle_rad)] } else { throw new RangeError("Not a known tooth shape: ", tooth_shape); } } _Gear.prototype.iterate_points = function(f, i_start = 0, i_end) { const {tooth_count} = this; i_end ??= tooth_count; const tooth_angle = 2 * Math.PI / tooth_count; const profile = this.tooth_profile; for (var i = i_start; i < i_end; ++i) { var m = make_rot(tooth_angle * (i - this.tooth_width / 2)); profile.forEach(p => { var p1 = tr(m, {x: p.x, y: -p.y}); f(p1); }); var m = make_rot(tooth_angle * (i + this.tooth_width / 2)); profile.slice().reverse().forEach(p => { var p1 = tr(m, p); f(p1); }); } } _Gear.prototype.draw_teeth = function(ctx, x, y, zoom, range, rot_rad) { const {tooth_count} = this; const [i_start, i_end] = range; const radius = tooth_count * 0.5; // set up transforms for this gear ctx.save(); ctx.translate(x, y); ctx.scale(zoom, zoom); ctx.rotate(rot_rad); // draw gear const angle_start = (i_start - .75) * Math.PI * 2 / tooth_count; const angle_end = (i_end - .75) * Math.PI * 2 / tooth_count; ctx.beginPath(); var p = r_th_to_p(radius - this.dd - 2, angle_start); ctx.moveTo(p.x, p.y); p = r_th_to_p(radius - this.dd, angle_start); ctx.lineTo(p.x, p.y); this.iterate_points(p => ctx.lineTo(p.x, p.y), i_start, i_end); p = r_th_to_p(radius - this.dd, angle_end); ctx.lineTo(p.x, p.y); p = r_th_to_p(radius - this.dd - 2, angle_end); ctx.lineTo(p.x, p.y); ctx.closePath(); const line_width = 2 / ((1 + pixel_ratio) * zoom); ctx.lineWidth = line_width; ctx.fillStyle = this.color; ctx.strokeStyle = this.color; ctx.globalAlpha = 0.2; ctx.fill(); ctx.globalAlpha = 1; ctx.stroke(); ctx.restore(); } _Gear.prototype.draw = function(ctx, x, y, zoom, show, rot_rad, rot_rad_guides) { const {tooth_count} = this; const radius = tooth_count * 0.5; const line_width = 2 / ((1 + pixel_ratio) * zoom); // set up transforms for this gear ctx.save(); ctx.translate(x, y); ctx.scale(zoom, zoom); // save before rotating ctx.save(); ctx.rotate(rot_rad); // draw gear ctx.beginPath(); this.iterate_points(p => ctx.lineTo(p.x, p.y)); ctx.closePath(); const max_hole_r = (radius - this.dd) * .85; switch (this.hole_shape) { case "": break; case "mark": draw_mark_hole(ctx, max_hole_r); break; case "round": draw_round_hole(ctx, max_hole_r); break; case "square": draw_square_hole(ctx, max_hole_r); break; case "keyed": draw_keyed_hole(ctx, max_hole_r); break; case "cross": draw_axle_hole(ctx, max_hole_r); break; } ctx.lineWidth = line_width; ctx.fillStyle = this.color; ctx.strokeStyle = this.color; ctx.globalAlpha = 0.2; ctx.fill(); ctx.globalAlpha = 1; ctx.stroke(); // draw other things if (show.includes("pitch")) { ctx.beginPath(); ctx.arc(0, 0, radius, 0, 2 * Math.PI, false); ctx.lineWidth = line_width; ctx.strokeStyle = "#000"; ctx.setLineDash([2 * line_width, 6 * line_width]); ctx.stroke(); } if (show.includes("axes")) { const x1 = radius + this.ad; ctx.beginPath(); ctx.moveTo(0, -x1); ctx.lineTo(0, x1); ctx.moveTo(-x1, 0); ctx.lineTo(x1, 0); ctx.strokeStyle = this.color; ctx.lineWidth = line_width * .4; ctx.setLineDash([10 * line_width, 2 * line_width]); ctx.stroke(); } // undo rotation ctx.restore(); ctx.save(); ctx.rotate(rot_rad_guides); if (this.base && show.includes("base")) { ctx.beginPath(); ctx.arc(0, 0, this.base, 0, 2 * Math.PI, false); ctx.strokeStyle = "#095"; ctx.lineWidth = line_width * .7; ctx.setLineDash([6 * line_width, 2 * line_width]); ctx.stroke(); } if (this.undercut_radius && show.includes("uc")) { ctx.beginPath(); ctx.arc(0, 0, this.undercut_radius, 0, 2 * Math.PI, false); ctx.strokeStyle = "#f0f"; ctx.lineWidth = line_width * .7; ctx.setLineDash([6 * line_width, 2 * line_width]); ctx.stroke(); } if (show.includes("ad_dd")) { ctx.beginPath(); ctx.arc(0, 0, radius - this.dd, 0, 2 * Math.PI, false); ctx.strokeStyle = "#f80"; ctx.lineWidth = line_width * .5; ctx.setLineDash([16 * line_width, 4 * line_width]); ctx.stroke(); ctx.beginPath(); ctx.arc(0, 0, radius + this.ad, 0, 2 * Math.PI, false); ctx.stroke(); } if (this.loa && show.includes("loa")) { const loa = this.loa; ctx.beginPath(); ctx.moveTo(loa[0].x, loa[0].y); for (var i = 1; i < loa.length; ++i) { ctx.lineTo(loa[i].x, loa[i].y); } ctx.strokeStyle = "#095"; ctx.lineWidth = line_width; ctx.setLineDash([]); ctx.stroke(); } // undo rotation ctx.restore(); // undo scale and offset ctx.restore(); } return _Gear; }38:Ta0b,function make_svg_gears(gear_canvas, settings) { const {gear_1, gear_2} = gear_canvas; const size_x = gear_1.tooth_count + gear_2.tooth_count + 4; const size_y = Math.max(gear_1.tooth_count, gear_2.tooth_count) + 4; const x1 = 2 + 0.5 * gear_1.tooth_count; const dx2 = (gear_1.tooth_count + gear_2.tooth_count) / 2 + get_involute_extra_dist(settings); const y = size_y / 2; var inv_rot_extra = get_involute_extra_rotation(gear_1, gear_2, dx2); var rot_extra = inv_rot_extra ? inv_rot_extra.rot_extra : 0; rot_extra *= 180 / Math.PI; rot_extra += 360 / gear_2.tooth_count * (gear_1.tooth_width + gear_2.tooth_width) / 2; const to_fixed = f => Number.parseFloat(f).toFixed(3); var path1 = ""; gear_canvas.gear_1.iterate_points(p => path1 += `L${to_fixed(p.x)} ${to_fixed(p.y)}`); path1 = 'M' + path1.substring(1) + "z"; var path2 = ""; gear_canvas.gear_2.iterate_points(p => path2 += `L${to_fixed(p.x)} ${to_fixed(p.y)}`); path2 = 'M' + path2.substring(1) + "z"; const zoom = Math.floor(Math.min(30, 2000 / size_x)); const stroke_w = 1 / zoom; const dash_array = `${to_fixed(8 / zoom)} ${to_fixed(2 / zoom)} ${to_fixed(2 / zoom)} ${to_fixed(2 / zoom)}`; return ``; }39:T6b9,cut_scheme = { if (!cut_graph_settings.draw) return "Click 'draw' to draw this scheme"; const {pressure_angle, teeth1, teeth2, modify_involute, frames, tooth_shape, time} = cut_graph_settings; const settings = {tooth_shape: opposite_gear_shape(tooth_shape), tooth_count: teeth1, opposite_tooth_count: teeth2, modify_involute: modify_involute, pressure_angle: pressure_angle, cycloid_round_bottom: true, color: "#f04"} const chart_size = width; const module = width * 0.1; const ctx = DOM.context2d(chart_size, chart_size * 0.5, window.devicePixelRatio); ctx.translate(0.5 * chart_size, 0.25 * chart_size + 0.5 * teeth1 * module); const g1 = new Gear(settings); const show = ["pitch", "base", "uc"]; const rot1 = 0; g1.draw(ctx, 0, 0, module, show, 1.5 * Math.PI - rot1, 0); const N = (frames - 1) / 2; const g2 = new Gear({...settings, modify_involute: 0, tooth_shape: tooth_shape, tooth_count: teeth2, opposite_tooth_count: teeth1, color: `rgba(0, 50, 250, ${Math.max(.2, 1 - Math.pow(0.01, 1 / frames))})`}); const d_th = 2 * Math.PI / teeth1; const i_scale = 1 / Math.max(1, N); for (var i = -N; i <= N; ++i) { const th1 = d_th * (i * i_scale + time); const th2 = .5 * Math.PI + th1 + (th1 + rot1) * teeth1 / teeth2 + Math.PI / teeth2; const R = module * 0.5 * (teeth1 + teeth2) + module * get_involute_extra_dist(settings); g2.draw_teeth(ctx, R * Math.sin(th1), -R * Math.cos(th1), module, [-2, 2], th2, 0) } return ctx.canvas; }3a:T9a2,// make an involute teeth curve. The involute crosses the pitch circle at Y = modify_involute. // and the "string" comes from the -Y direction. function make_involute(rp, dedendum, addendum, modify, tooth_dth, pa) { // get the angle we have to roll our straight line to arrive at a certain // radius function rToTheta(rb, r1) { // length of our tangent line var len = Math.sqrt(r1*r1 - rb*rb); // and apply radians definition return len / rb; } // base radius const rb = rp * Math.cos(pa); // start/end radius var r1 = Math.max(rp - dedendum, rb); const r2 = rp + addendum; // theta at pitch radius const thP = pa; // theta at start of involute curve // (may be below r1, should be th <= 0) const thAbs0 = pa - Math.sin(pa) * (rp + modify) / rb; if (Math.cos(pa) * rb < rp - dedendum) { // we may drop another part of the involute for large gears: // below the intersection between the pressure line and the addendum // line of a rack meshing with the gear. (actually, we assume for the // undercut that the rack addendum equals our dedendum). // If this intersect lies below the tangent point of the base circle and // the pressure line, we can drop the involute below this point. // That part will never touch an involute on any other gear, so instead // we may follow the the undercut curve. const intersect_y = dedendum * Math.tan(Math.PI * .5 - pa) r1 = Math.max(r1, Math.hypot(intersect_y, rp - dedendum)); } // range of theta we have to cover (relative to thAbs0) const th1 = rToTheta(rb, r1); const th2 = rToTheta(rb, r2); // angle where we intersect with the next tooth const angle_slope = Math.tan(0.25 * tooth_dth); var l = []; const steps = 5 + Math.ceil((th2 - th1) / .03); const dTh = (th2 - th1) / steps; for (var i = 0; i <= steps; ++i) { const th = th1 + i * dTh; const thAbs = thAbs0 + th // tangent point var p = { x: Math.cos(thAbs) * rb, y: -Math.sin(thAbs) * rb }; // arc length var len = rb * th; // and follow line segment p.x += len * Math.sin(thAbs); p.y += len * Math.cos(thAbs); // make sure the tooth doesn't intersect with the next tooth if (p.y < -angle_slope * p.x) { p = intersect_segment(l[l.length - 1], p, -angle_slope); l.push(p); break; } else { l.push(p); } } return {points: l, base: rb, root_angle: thAbs0}; } 3b:T52c,function make_undercut_involute(pitch_radius, pressure_angle_rad, tooth_pitch, dedendum, addendum, rack_addendum, rack_displace) { // A rack is the worst case for undercut, we calculate this curve by // tracing the upper corner of a tooth on the rack as it meshes with the gear. // start position is with a gear teeth perpendicular to a vertical rack. // pretend our dedendum is the addendum of the rack. // "modifying" a gear tooth shape refers to hobbing with the gear further from // the hob (usually) than normal. const tooth_angle_y = rack_addendum * Math.tan(pressure_angle_rad); const tooth_corner_0 = {x: pitch_radius - rack_addendum + rack_displace, y: tooth_angle_y}; var undercut = [] // angle we need to rotate until the rack is outside // our addendum circle. const rot_rad_unit = Math.acos((pitch_radius - rack_addendum + rack_displace) / (pitch_radius + addendum)); for (var t = -1.0; t < 1.0; t += 0.02) { // movement of rack. If the tooth corner crosses Y = 0 we're done. var tooth_corner = {...tooth_corner_0}; tooth_corner.y -= rot_rad_unit * -t * pitch_radius; if (tooth_corner.y > 0) break; // corner relative to rotating gear var M = make_rot(rot_rad_unit * -t); undercut.push(tr(M, tooth_corner)); } undercut.reverse(); return undercut; }3c:T112f,// make a cycloid teeth curve. The curve is a combination of two cycloids (one inside, one // outside the pitch circle). The curve crosses the pitch circle at Y = zero. The circles roll // towards -Y. // The undercut is is always assumed to start on the pitch circle, and it is shaped as a // circular arc matching the lantern bars of the other gear. // Strictly speaking, for small gears the undercut should be slightly higher, as there is a small // amount of interference with the next tooth, but we do not take this into account. function make_cycloid_tooth(rp, parameter, dedendum, addendum, teeth, other_teeth, round_bottom) { if (!teeth || !other_teeth) throw RangeError("must have non-zero amount of teeth"); const module = 2 * rp / teeth; const rp_other = rp * other_teeth / teeth; // choose radii for the cycloid circles function get_r_h(t1, t2, rp_inner) { // fixed points for parameter = 0 / 2 and 1: const a = 2 / rp_inner; const b = parameter < 1 ? (2 * t1) / rp_inner : 1.0001 / rp_inner; const t = Math.abs(parameter - 1); return 1 / ((1 - t) * a + t * b); } const r_hc = get_r_h(teeth, other_teeth, rp); const r_ec = get_r_h(other_teeth, teeth, rp_other); // Calculate where the addendum of the other gear intersects // our line of action for the hypocycloid var x_intersect = intersect_circles(rp_other + r_hc, r_hc, rp_other + addendum); const hc_loa_end_th = Math.acos(Math.max(-1, x_intersect / r_hc)); // tooth curve var tooth_angle = Math.PI * 0.5 / teeth; const angle_slope = Math.tan(tooth_angle); // if the dedendum has a flat bottom spanning the tooth angle, // and it touches the dedendum circle, this is the distance from the // centre to the corner points. const r_corner_dedendum = (rp - dedendum) / Math.cos(tooth_angle); var l = []; var has_undercut = false; if (Math.abs(parameter - 1) < 1e-4) { // parameter == 1 case: // The two circles we roll over our pitch circle // are half of the radius of the two gears. This results in straight lines // for the dedendum. if (!round_bottom) l.push({x: r_corner_dedendum, y:0}); has_undercut = round_bottom; } else { const dd_cos = rp - r_corner_dedendum; // we stop when p reaches the addendum circle. // First calculate the angles in our end state with the law of cosines. // A: centre of rolling circle, B: centre of gear, C: end point of cycloid var alpha = triangle_angle(rp - dd_cos, r_hc, rp - r_hc); // rolling angle is outside this triangle at vertex A var th_max_ad = r_hc / rp * (Math.PI - alpha) || Math.PI; var th_max = Math.min(r_hc / rp * hc_loa_end_th, th_max_ad); if (th_max != th_max_ad || round_bottom) { has_undercut = true; } const hc = make_cycloid(rp, -r_hc, th_max, angle_slope); if (!hc.has_point && !round_bottom) { var p_crease = intersect_segment(hc.points[hc.points.length - 2], hc.points[hc.points.length - 1], angle_slope); if (p_crease && p_crease.x > r_corner_dedendum && p_crease.x < rp) { hc.points.push(p_crease); } else { hc.points.push({x: r_corner_dedendum, y: hc.points.length ? hc.points[hc.points.length - 1].y : 0}); } } hc.points.slice().reverse().forEach(p => l.push(p)); } // we stop when p reaches the addendum circle. // First calculate the angles in our end state with the law of cosines. var alpha = triangle_angle(rp + addendum, r_ec, rp + r_ec) || Math.PI; var th_max = r_ec / rp * alpha; const ec = make_cycloid(rp, r_ec, -th_max, -angle_slope); ec.points.forEach(p => l.push(p)); const l_no_uc = l; if (has_undercut) { // undercut curve const th_undercut_start = hc_loa_end_th * r_hc / rp; const undercut_corner = r_th_to_p(r_hc, hc_loa_end_th); undercut_corner.x += rp - r_hc; const slop = dedendum - addendum; const undercut = make_undercut_circle( rp, rp_other, th_undercut_start, undercut_corner, slop, round_bottom); if (undercut[0]) { l = min_curve(l, undercut, 0).points; } var i = 0; while (l[i].y > angle_slope * l[i].x) ++i; if (i > 0) { l.splice(0, i, intersect_segment(l[i - 1], l[i], angle_slope)); } } return {points: l, points_no_uc: l_no_uc, r_ec: r_ec, r_hc: r_hc, loa_end_th: hc_loa_end_th}; }3d:T968,// Make sort of, kind of, a cycloid. The curve is traced by // the edge of a circle instead of a point. // the curve isn't guaranteed to intersect the pitch circle at Z=0, instead // it is based on a fixed start position of the lantern bar. function make_lantern_cycloid(r_pitch, r_pitch_other, r_lantern, tooth_theta, limit_slope, N = 20) { var l = []; var loa = [] var has_point = false; // initial position of lantern bar const th1_start = tooth_theta / 4; // we stop when the lantern centre reaches the addendum circle, plus // one lantern radius on the other pitch circle. // that is a bit approximate but good enough // First calculate the angles in our end state with the law of cosines. const ad = r_lantern * 2; var alpha = triangle_angle(r_pitch + ad, r_pitch_other, r_pitch + r_pitch_other) || Math.PI; var th1_end = r_pitch_other / r_pitch * (alpha + r_lantern / r_pitch_other); const d_th = (th1_end - th1_start) / N; // We rely here on a property of gears: the normal of the contact point // goes through the pitch point. For the lantern bar this means the contact // point is on the line between the lantern bar centre and the pitch point. var p_dist = r_pitch; var done = false; for (var step = 0; step <= N && !done; ++step) { var d_th1 = step * d_th; var th1 = th1_start + d_th1; var th2 = Math.PI - r_pitch / r_pitch_other * th1; // position of lantern bar var l_p = r_th_to_p(r_pitch_other, th2); l_p.x += r_pitch + r_pitch_other; // the touch point is on the line to the pitch point (r_pitch, 0) var loa_touch = {x: r_pitch - l_p.x, y: -l_p.y}; var loa_touch_scale = r_lantern / Math.hypot(loa_touch.x, loa_touch.y); var p = {x: l_p.x + loa_touch.x * loa_touch_scale, y: l_p.y + loa_touch.y * loa_touch_scale}; loa.push(p); p = tr(make_rot(tooth_theta / 4 - th1), p); const p_dist_prev = p_dist; p_dist = Math.hypot(p.x, p.y); if (p_dist > r_pitch + ad) { var t = (r_pitch + ad - p_dist_prev) / (p_dist - p_dist_prev); var p1 = l[l.length - 1]; p = {x: p1.x * (1-t) + p.x * t, y: p1.y * (1-t) + p.y * t}; done = true; } if (p.y < limit_slope * p.x) { has_point = true; p = intersect_segment(l[l.length - 1], p, limit_slope); done = true; } l.push(p); } return {points: l, has_point: has_point, loa: loa}; }3e:T568,undercut_scheme = { var pa = undercut_graph_settings.pressure_angle; var teeth = undercut_graph_settings.tooth_count; const settings = {tooth_shape: "cycloid", zoom_to: "teeth_2", tooth_count_1: teeth, tooth_count_2: 10, pressure_angle: pa, cycloid_round_bottom: true, show: ["pitch", "loa", "base", "uc"]} const c = make_gear_canvas(400, settings); update_gear_canvas(c, settings); draw_gears_on_canvas(c, settings, rotate_undercut); const g1 = c.gear_1; var rot_rad = rotate_undercut * 2 * Math.PI / teeth; var rotM = make_rot(rot_rad); const ctx = c.node.getContext("2d"); // undercut curve const th_undercut_start = g1.loa_end_th * g1.r_hc / g1.radius; const undercut_corner = r_th_to_p(g1.r_hc, g1.loa_end_th); undercut_corner.x += g1.radius - g1.r_hc; const undercut = make_undercut_circle(g1.radius, c.gear_2.radius, th_undercut_start, undercut_corner, g1.dd - g1.ad, true); ctx.beginPath(); undercut.forEach(p => { p = tr(rotM, p); ctx.lineTo(p.x + g1.x, p.y + g1.y); }); ctx.lineWidth = 2; ctx.strokeStyle = "#f30"; ctx.stroke(); ctx.beginPath(); if (g1.tooth_no_uc) { g1.tooth_no_uc.forEach(p => { p = tr(rotM, p); ctx.lineTo(p.x + g1.x, p.y + g1.y); }); ctx.lineWidth = 2; ctx.strokeStyle = "#03f"; ctx.stroke(); } return c.node; }3f:T46d,function draw_axle_hole(ctx, max_radius) { // cross shaped axle, the size corresponds to that of axles in // certain construction brick sets, assuming gears with a module // of 1mm (aka 1/8th stud). const rHole = Math.min(2.39, max_radius); const x1 = 0.89; // half width of one arm const r2 = 0.5; // radius for rounding between arms const x2 = x1 + r2; // centre of the circle for that rounding ctx.moveTo(rHole, 0); if (rHole < x2 + r2) { ctx.arc(0, 0, rHole, 2 * Math.PI, 0, true); } else { const a1 = Math.asin(x1 / rHole); // end angle for outer circle arc segments const a90 = Math.PI / 2; ctx.arc(0, 0, rHole, 0, -a1, true); ctx.arc(x2, -x2, r2, a90, 2 * a90, false); ctx.arc(0, 0, rHole, -a90 + a1, -a90 - a1, true); ctx.arc(-x2, -x2, r2, 0, a90, false); ctx.arc(0, 0, rHole, -2 * a90 + a1, -2 * a90 - a1, true); ctx.arc(-x2, x2, r2, -a90, 0, false); ctx.arc(0, 0, rHole, -3 * a90 + a1, -3 * a90 - a1, true); ctx.arc(x2, x2, r2, -2 * a90, -a90, false); ctx.arc(0, 0, rHole, -4 * a90 + a1, -4 * a90, true); ctx.closePath(); } }40:T6f1,/// combines 2 curves: for each X coordinate, have the lowest Y coordinate of the two. /// (approximately, we don't make up new points here). /// both curves must be sorted by increasing X. /// Returns: /// - the combined curve /// - the first point of points2, if any previous one was under points1 by more than the given tolerance. function min_curve(points1, points2, tolerance) { var i1 = 0; var i2 = 0; var p2_first = undefined; var p2_dropped = false; var result = []; // non-overlapping region for (; i1 < points1.length && points1[i1].x < points2[0].x; ++i1){ result.push(points1[i1]); } for (; i2 < points2.length && points2[i2].x < points1[0].x; ++i2){ result.push(points2[i2]); } function intersect_y(p1, p2, x) { var t = (x - p1.x) / (p2.x - p1.x); return p1.y + t * (p2.y - p1.y); } // overlapping, take lowest Y while (i1 < points1.length && i2 < points2.length) { if (points1[i1].x < points2[i2].x) { var i2_1 = Math.max(0, i2 - 1); var y2 = intersect_y(points2[i2_1], points2[i2_1 + 1], points1[i1].x); if (points1[i1].y < y2) { result.push(points1[i1]); } ++i1; } else { var i1_1 = Math.max(0, i1 - 1); var y1 = intersect_y(points1[i1_1], points1[i1_1 + 1], points2[i2].x); if (points2[i2].y < y1) { result.push(points2[i2]); if (p2_dropped && p2_first == undefined) p2_first = points2[i2]; } else if (points2[i2].y >= y1 + tolerance) { p2_dropped = true } ++i2; } } // non-overlapping region for (; i1 < points1.length; ++i1){ result.push(points1[i1]); } for (; i2 < points2.length; ++i2){ result.push(points2[i2]); } return {points: result, first_2: p2_first}; }41:[] 32:{"id":"13ce2f4cd38a7ddc","slug":"gears","title":"Gears","thumbnail":"0c5a2e1b5c918873c7f257a1290a3d4f1232dc78acbbb59ab9d2704a99bf00a5","access_level":"public","create_time":"$D2024-07-04T05:18:13.921Z","update_time":"$D2026-03-24T04:46:55.588Z","view_time":null,"first_public_time":"$D2024-07-04T05:19:18.828Z","has_unpublished_edits":null,"fork":false,"stars":7,"starred":false,"owner":{"type":"team","id":"3ebafa4f59a35ad3","login":"roelandschoukens","name":"Roeland Schoukens","avatar_url":"https://avatars-next.observableusercontent.com/5f891a64a5a2649f6b6b94d7d2910cc1399672aa613af247dc080d9b0ecdd63e","tier":"starter_2024"},"version":2451,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"bc0dceea753ec329","login":"roelandschoukens","name":"Roeland Schoukens","avatar_url":"https://avatars-next.observableusercontent.com/5f891a64a5a2649f6b6b94d7d2910cc1399672aa613af247dc080d9b0ecdd63e","tier":null}],"members":[{"type":"team","id":"3ebafa4f59a35ad3","login":"roelandschoukens","name":"Roeland Schoukens","avatar_url":"https://avatars-next.observableusercontent.com/5f891a64a5a2649f6b6b94d7d2910cc1399672aa613af247dc080d9b0ecdd63e","tier":"starter_2024","role":"editor"},{"type":"individual","id":"bc0dceea753ec329","login":"roelandschoukens","name":"Roeland Schoukens","avatar_url":"https://avatars-next.observableusercontent.com/5f891a64a5a2649f6b6b94d7d2910cc1399672aa613af247dc080d9b0ecdd63e","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Gears","cells":[{"id":0,"value":"# Gears","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6,"value":"display_1 = {\n var canvas = gear_canvas_1.node;\n var div = html`

`;\n div.appendChild(canvas)\n div.style.resize = 'horizontal';\n div.style.paddingRight = '1em';\n div.style.width = canvas_default_width+\"px\";\n div.style.boxSizing = 'content-box';\n div.style.maxWidth = '100%';\n div.style.minWidth = '100px';\n div.style.overflow = 'hidden';\n canvas.style.display = 'block';\n canvas.style.maxWidth = '100%';\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":794,"value":"viewof rotate_1 = infiniteRange({label: \"Rotate\", step: 0.001, dragStep: 0.01, value: 0, width: 600});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":200,"value":"viewof settings_base = Inputs.form({\n tooth_count_1: range([2, 500], {label: \"Teeth 1\", step: 1, value: 20, transform: Math.log, width: 600}),\n tooth_count_2: range([2, 500], {label: \"Teeth 2\", step: 1, value: 10, transform: Math.log, width: 600}),\n zoom_to: Inputs.radio(Object.keys(zoom_choice), {label: \"Zoom to:\", value: \"gear_both\", format: x => zoom_choice[x]}),\n show: Inputs.checkbox(Object.keys(show_parts), {label: \"Show:\", format: x => show_parts[x]}),\n pressure_angle: range([5, 35], {label: html`Pressure angle ^[a]`, step: 0.1, value: 20, width: 600}),\n tooth_shape: Inputs.radio(Object.keys(tooth_shape), {label: \"Tooth shape:\", value: \"cycloid\", format: x => tooth_shape[x]}),\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":778,"value":"animate_slider = Inputs.bind(animationSpeedSlider({width: 400, label: \"Animation speed\", max: 5, transform: Math.sqrt, smallestDelta: 0.001}), viewof rotate_1);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1389,"value":"viewof settings_extra = Inputs.form({\n cycloid_round_bottom: Inputs.toggle({label: \"Round cycloid undercut\"}),\n move_involute: range([0, 1], {label: html`Move involute ^[b]`, step: .01, value: 0, width: 600}), \n modify_involute: range([0, 1], {label: html`Modify involute ^[c]`, step: .01, value: 0, width: 600}), \n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2300,"value":"viewof settings_flourish = Inputs.form({\n hole_shape: Inputs.select(Object.keys(hole_choice), {label: \"Hole:\", value: \"round\", format: x => hole_choice[x]}),\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1437,"value":"

^[a] For cycloid gears this slider changes the way the generating circle diameters are chosen: at the default (20) the circles are half the pitch radius of the gears, so the hypocycloids become straight lines. At the maximum, it is the size where the entire cycloid between two cusps is exactly as wide as a gear tooth. This shape is used in Roots blowers. At the minimum setting, the hypocycloid generating circles equals the pitch circles, creating point tooth forms.

^[b] Involute gears still mesh properly if you move them slightly apart.

^[c] This modification refers to gears with (usually) a smaller dedendum and larger addendum. The resulting gear is effectively a gear with a slightly larger module, a slightly steeper pitch angle, and more narrow teeth. The key property that allows it to mesh properly is its base circle radius, which is the same as that of the unmodified gear.

","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1487,"value":"get_svg_button = {\n var button = html``;\n button.addEventListener(\"click\", function(e){\n\n var settings = {...settings_1, zoom_to: \"gear_both\"};\n settings.show = [...settings.show, \"axes\"];\n var g_canvas = make_gear_svg_canvas(1000);\n update_gear_canvas(g_canvas, settings);\n // use `viewof rotate_1` here, so the cell doesn't re-evaluate every time the gears rotate\n draw_gears_on_canvas(g_canvas, settings, (viewof rotate_1).value);\n \n const blob = new Blob([g_canvas.ctx.getSerializedSvg()], {type: \"image/svg+xml\"});\n const element = document.createElement(\"a\");\n element.download = `gears-${settings.tooth_shape}-${settings.tooth_count_1}-${settings.tooth_count_2}.svg`;\n element.href = window.URL.createObjectURL(blob);\n element.click();\n element.remove();\n \n e.preventDefault();\n })\n return button;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"gear_canvas_1 = make_gear_canvas(canvas_default_width)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2192,"value":"settings_1 = ({...settings_base, ...settings_extra, ...settings_flourish});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":358,"value":"// this separate updater is important for the UI, it makes sure that the display\n// doesn't get reinitialized (thus resetting its size)\ngear_canvas_1_updater = {\n update_gear_canvas(gear_canvas_1, settings_1);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2221,"value":"animation_period_updater = {\n function gcd(a,b) {\n if (b > a) [a, b] = [b, a];\n while (true) {\n if (b == 0) return a;\n a %= b;\n if (a == 0) return b;\n b %= a;\n }\n }\n \n function lcm(a, b) { return a * b / gcd(a, b); }\n var p = lcm(settings_base.tooth_count_1, settings_base.tooth_count_2);\n animate_slider.period = p;\n return p;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1092,"value":"// this separate updater is important for the UI, it makes sure that the display\n// doesn't get reinitialized (thus resetting its size)\ngear_canvas_1_rotator = {\n gear_canvas_1_updater; // must update first\n draw_gears_on_canvas(gear_canvas_1, settings_1, rotate_1);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2152,"value":"function get_involute_extra_dist(settings) {\n // check for different tooth shape, undefined or NaN.\n return settings.tooth_shape == \"involute\" ?\n (settings.modify_involute || 0) + (settings.move_involute || 0) :\n 0;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"function make_gear_canvas(chart_size) {\n chart_size = Math.min(width, chart_size);\n const dpr = window.devicePixelRatio;\n const ctx = DOM.context2d(chart_size, chart_size, dpr);\n \n return {ctx: ctx, node: ctx.canvas, chart_size: chart_size, dpr: dpr};\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2413,"value":"function make_gear_svg_canvas(chart_size) {\n var ctx = new C2S(chart_size, chart_size);\n return {ctx:ctx, dpr:1, chart_size: chart_size};\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1095,"value":"$34","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1820,"value":"pixel_ratio = window.devicePixelRatio","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":344,"value":"$35","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2172,"value":"$36","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":272,"value":"$37","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1944,"value":"function opposite_gear_shape(s) {\n if (s == \"lantern\") return \"lantern-cycloid\";\n if (s == \"lantern-cycloid\") return \"lantern\";\n return s\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1464,"value":"$38","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":222,"value":"show_parts = ({\n pitch : \"Pitch c.\",\n base : \"Base c.\",\n loa : \"Line of action\",\n ad_dd : \"add/ded c.\",\n uc : \"Undercut\",\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":385,"value":"zoom_choice = ({\n gear_1: \"Gear 1\",\n gear_2: \"Gear 2\",\n gear_both: \"2 gears\",\n teeth_1: \"Teeth\",\n teeth_2: \"Teeth detail\"\n })","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2293,"value":"hole_choice = ({\n \"\": \"None\",\n mark: \"Mark\",\n round: \"Round\",\n square: \"Square\",\n keyed: \"Keyed\",\n cross: \"Cross axle\"\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":850,"value":"tooth_shape = ({ involute: \"Involute\", cycloid: \"Cycloid\", lantern: \"Lantern\"})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"canvas_default_width = 800","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1797,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1799,"value":"viewof cut_graph_settings = Inputs.form({\n draw: Inputs.button(\"Draw scheme\"),\n teeth1: range([6, 100], {label: \"Tooth count 1\", value: 10, step: 1, width: 500, transform: Math.log}),\n teeth2: range([6, 1000], {label: \"Tooth count 2\", value: 10, step: 1, width: 500, transform: Math.log, value: 1000}),\n modify_involute: range([-0.4, 1.0], {label: \"Modify involute\", value: 0, step: .01, width: 500}),\n pressure_angle: range([5, 35], {label: \"Pressure angle\", value: 20, step: 0.1, width: 500}),\n frames: range([1, 101], {label: \"Frames\", value: 11, step: 1, width: 500}),\n time: range([-1, 1], {label: \"Offset\", value: 0, step: .001, width: 500}),\n tooth_shape: Inputs.radio(Object.keys(tooth_shape), {label: \"Tooth shape:\", value: \"involute\", format: x => tooth_shape[x]}),\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2244,"value":"## Tooth shapes","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":2251,"value":"### Involute","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":37,"value":"$3a","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":399,"value":"$3b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2253,"value":"### Cycloid","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1445,"value":"// Make a cycloid. If r2 is negative, the result is a hypocycloid.\n// Stop if either we roll for a certain angle (th_max) or until we\n// intersect with the line with a given slope (limit_slope)\n// the cusp is at (r1, 0). If th_max is negative the generating\n// circle will roll downwards.\nfunction make_cycloid(r1, r2, th_max, limit_slope, N = 20) {\n var l = [];\n var has_point = false;\n const d_th = th_max / N;\n const sign = th_max < 0 ? -1 : 1;\n for (var step = 0; step <= N; ++step)\n {\n var th = step * d_th;\n var th_hc = th * r1 / r2;\n var p1 = r_th_to_p(r1 + r2, th);\n var p2 = r_th_to_p(r2, th + th_hc);\n var p = {x: p1.x - p2.x, y: p1.y - p2.y};\n if (sign * p.y > sign * limit_slope * p.x) {\n if (l.length)\n {\n has_point = true;\n p = intersect_segment(l[l.length - 1], p, limit_slope);\n l.push(p);\n }\n break;\n }\n else {\n l.push(p);\n }\n }\n return {points: l, has_point: has_point};\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":854,"value":"$3c","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2255,"value":"### Lantern","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1608,"value":"$3d","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":401,"value":"$3e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":807,"value":"viewof rotate_undercut = range([-2, 2], {label: \"Rotate\", step: 0.001, value: 0, width: 500});\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":597,"value":"viewof undercut_graph_settings = Inputs.form({\n tooth_count: range([6, 500], {label: \"Tooth count\", value: 8, step: 1, width: 500, transform: Math.log}),\n pressure_angle: range([5, 35], {label: \"Pressure angle\", value: 20, step: 0.1, width: 500}),\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1140,"value":"function make_undercut_circle(r1, r2, th_1_start, corner_p, slop, everything) {\n // make an undercut given a corner point where the undercut starts.\n // corner_p must be given relative to gear 1, and must be on the\n // line of action.\n // corner relative to gear 2\n const x2 = r1 + r2;\n const p0_c_2 = {x:corner_p.x - x2, y:-corner_p.y};\n var undercut = [];\n const t_max = everything ? 1.001 : .501;\n for (var t = 0; t < t_max; t += 0.02) {\n var th_1 = (1 - t) * th_1_start; // gear 1: max to 0\n var th_2 = t * th_1_start * r1 / r2; // gear 2: 0 to max\n // rotation of gear 2\n var p = tr(make_rot(-th_2), p0_c_2);\n if (everything && p.y > 0) break;\n // make relative to gear 1\n p.x += x2;\n // undo rotation of gear 1\n p = tr(make_rot(th_1), p);\n // add slop\n p.x -= slop * t;\n p.y -= slop * t;\n if (!everything && p.y > 0) break;\n undercut.push(p);\n }\n undercut.reverse();\n return undercut;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2238,"value":"## Holes","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":2305,"value":"function draw_round_hole(ctx, max_radius) {\n const r = Math.min(1, max_radius);\n ctx.moveTo(r, 0);\n ctx.arc(0, 0, r, 2 * Math.PI, 0, true);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2447,"value":"function draw_mark_hole(ctx, max_radius) {\n const r = Math.min(.5, max_radius);\n ctx.moveTo(r, 0);\n ctx.lineTo(-r, 0);\n ctx.moveTo(0, r);\n ctx.lineTo(0, -r);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2240,"value":"$3f","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2259,"value":"function draw_keyed_hole(ctx, max_radius) {\n // circular hole with square keys\n if (max_radius < 1.1)\n {\n const r = Math.min(1, max_radius);\n ctx.moveTo(r, 0);\n ctx.arc(0, 0, r, 2 * Math.PI, 0, true);\n }\n else {\n // x and y coordinates for the intersection fo the arc and the square\n const y1 = .2;\n const x1 = Math.sqrt(1 - y1*y1);\n // x coorinate for the depth of the key\n const x2 = Math.min(1.2, Math.sqrt(max_radius * max_radius - y1 * y1));\n // start and end angle for the arc\n const a1 = Math.asin(y1);\n ctx.moveTo(x2, -y1);\n ctx.lineTo(x1, -y1);\n ctx.arc(0, 0, 1, 2 * Math.PI - a1, Math.PI + a1, true);\n ctx.lineTo(-x1, -y1);\n ctx.lineTo(-x2, -y1);\n ctx.lineTo(-x2, y1);\n ctx.lineTo(-x1, y1);\n ctx.arc(0, 0, 1, Math.PI - a1, a1, true);\n ctx.lineTo(x1, y1);\n ctx.lineTo(x2, y1);\n ctx.closePath();\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2272,"value":"function draw_square_hole(ctx, max_radius) {\n // for a hole optically the same size as the circle hole, we need a bit less than 1 times that radius as the side.\n const r = Math.min(1.25, max_radius);\n ctx.moveTo(0, r);\n ctx.lineTo(r, 0);\n ctx.lineTo(0, -r);\n ctx.lineTo(-r, 0);\n ctx.closePath();\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2236,"value":"## Misc geometry","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":448,"value":"function r_th_to_p(r, th) {\n var ct = Math.cos(th);\n var st = Math.sin(th);\n return {x: ct * r, y: st * r};\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":61,"value":"function make_rot(th) {\n var ct = Math.cos(th);\n var st = Math.sin(th);\n return {a: ct, c: -st, e: 0,\n b: st, d: ct, f: 0};\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":32,"value":"// ⎡a c⎤ ⎡p.x⎤ ⎡e⎤\n// ⎣b d⎦ ⎣p.y⎦ + ⎣f⎦\nfunction tr(matrix, point) {\n return {\n x: matrix.a * point.x + matrix.c * point.y + matrix.e,\n y: matrix.b * point.x + matrix.d * point.y + matrix.f,\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":610,"value":"$40","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1322,"value":"function intersect_segment(p1, p2, slope) {\n var dy1 = slope * p1.x - p1.y;\n var dy2 = slope * p2.x - p2.y;\n var t = -dy1 / (dy2 - dy1);\n var x = p1.x + (p2.x - p1.x) * t;\n return {x: x, y: slope * x};\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1130,"value":"// returns the distance to the centre of the r1 circle.\nfunction intersect_circles(d, r1, r2) {\n var x_intersect = (d*d + r1*r1 - r2*r2) / (2 * d);\n return x_intersect;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1441,"value":"/// Angle of a triangle with three given sides. The angle of the corner opposite of a is returned.\nfunction triangle_angle(a, b, c) {\n return Math.acos((b*b + c*c - a*a) / (2 * b * c));\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":366,"value":"import { range, infiniteRange, animationSpeedSlider } from \"@roelandschoukens/inputs\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2403,"value":"// The original implementation was made by Gliffy Inc. but is no longer maintained\n// The module we're loading here is a TypeScript fork\nC2S = (await import(\"https://cdn.jsdelivr.net/npm/canvas-to-svg@1/+esm\")).default","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W41","stdlib":"1","titling":"auto"}} 33:"$32:body"