泰勒級數展開與圓的軌跡方程曲線
阿新 • • 發佈：2019-01-25
<html> <head> <meta charset="UTF-8"> <title>圓的泰勒級數在x=0處展開</title> <style type="text/css"> .box { /*float: left;*/ background-color: black; } .ctrlBox { float: left; /*border: 1px solid #A0F;*/ width: 400; /*height: 700;*/ } </style> </head> <body> <canvas id="canvas" class="box"> canvas not supported, please use html5 browser. </canvas> <div class="ctrlBox"> <p>泰勒級數展開的階數:<p> <button onclick="update(-1)"> << </button> <span id="lvlLb">1</span> <button onclick="update(1)"> >> </button> </div> </body> <script> var radius = 100; var x0 = Math.SQRT2 * radius / 2; var label = document.querySelector("#lvlLb"); var arr=[1,2,4,6,8,10,12], cur = 0; var canvas = document.querySelector(".box"); var ctx = canvas.getContext('2d'); window.onload = function() { canvas.width = 800; canvas.height = 600; draw(); } function update(val) { var preCur = cur; val > 0 ? cur++ : cur--; if (cur < 0) cur = 0; if (cur >= arr.length - 1) cur = arr.length - 1; if (preCur != cur) { ctx.clearRect(0, 0, canvas.width, canvas.height); draw(); } } function draw() { var cx = 400, cy = 300; label.innerText = arr[cur]; ctx.beginPath(); ctx.strokeStyle = "#EE9611"; ctx.arc(cx, cy, radius, Math.PI*2, false); ctx.stroke(); ctx.moveTo(0, cy); ctx.lineTo(canvas.width, cy); ctx.stroke(); ctx.moveTo(cx, 0); ctx.lineTo(cx, canvas.height); ctx.stroke(); ctx.closePath(); var range = 150; for (var i = -range; i < range; i++) { var j = getTyler_Y(i, arr[cur]); if (i === -range) { ctx.beginPath(); ctx.strokeStyle = "#2BD56F"; ctx.moveTo(cx+i, cy-j); } else { ctx.lineTo(cx+i, cy-j); } } ctx.stroke(); } //自變數x， 泰勒級數展開的階數lvl function getTyler_Y(x, lvl) { var exp; if (lvl >= 1) { exp = radius; } if (lvl >= 2) { exp -= x**2 / (2 * radius); } if (lvl >= 4) { exp -= x**4 / (8 * radius ** 3); } if (lvl >= 6) { exp -= x**6 / (16 * radius ** 5); } if (lvl >= 8) { exp -= 5 * x**8 / (128 * radius ** 7); } if (lvl >= 10) { exp -= 7 * x**10 / (256 * radius ** 9); } if (lvl >= 12) { exp -= 21 * x**12 / (1024 * radius ** 11); } return exp; } </script> </html>
圓的軌跡, 也可以用引數方程來表示
x = r * cos( t )
y = r * sin( t )
一階導數簡單， 就是 (dy/dt) / (dx/dt)
二階導數比較麻煩，參見《高等數學第五版》上冊，第107頁, (5)
y對t一階導數 記作dy_t1
x對t一階導數 記作dx_t1
y對t二階導數 記作dy_t2
x對t二階導數 記作dx_t2
y對x二階導數 記作dy_x2
dy_x2 = (dy_t2 * dx_t1 - dy_t1 * dx_t2) / (dx_t1 ** 3)
就圓的軌跡，的引數方程表達來說
一階導數 dy_x1 = -ctg( t ) 二階導數 dy_x2 = -1 / (r * sin(t) ** 3)
當x=0時，t=PI/2; sin ( t ) = 1, dy_x2 = -1 / r;
和根據 圓的隱函式表達 x * x + y * y = r * r 得出的二階導數是一致的