在贝塞尔曲线中加一点，js求这个点的控制点和前后控制点

下面是在JavaScript中计算在三次贝塞尔曲线中加入一个点后的新控制点和起点、终点的代码示例：

function addPointToBezierCurve(point, p0, p1, p2, p3) {
  // Find the closest point Q on the curve to the new point P
  var q = findClosestPointOnBezierCurve(point, p0, p1, p2, p3);

  // Divide the curve into two parts at the point Q
  var curve1 = [p0, p1, q, point];
  var curve2 = [point, q, p2, p3];

  // Calculate new control points for the two curves
  var c1 = calculateControlPoints(curve1);
  var c2 = calculateControlPoints(curve2);

  // Return the new control points and start/end points
  return {
    p0: p0,
    p1: c1[1],
    p2: c2[0],
    p3: p3,
    newPointControlPoints: [c1[2], c2[1]]
  };
}

function findClosestPointOnBezierCurve(point, p0, p1, p2, p3) {
  // Use de Casteljau algorithm to find the closest point on the curve
  var t = 0.5;
  for (var i = 0; i < 10; i++) {
    var Q = deCasteljau(p0, p1, p2, p3, t);
    var tangent = subtractPoints(deCasteljau(p0, p1, p2, p3, t + 0.001), Q);
    var distance = distanceBetweenPoints(point, Q);
    if (distance < 0.1) {
      return Q;
    }
    t -= dotProduct(subtractPoints(Q, point), tangent) / Math.pow(magnitude(tangent), 2);
    t = Math.max(0, Math.min(1, t));
  }
  return deCasteljau(p0, p1, p2, p3, t);
}

function calculateControlPoints(curve) {
  // Calculate new control points for a curve that passes through the given points
  var p0 = curve[0];
  var p1 = curve[1];
  var p2 = curve[2];
  var p3 = curve[3];

  var d1 = subtractPoints(p1, p0);
  var d2 = subtractPoints(p2, p1);
  var d3 = subtractPoints(p3, p2);

  var a1 = magnitude(d1) / (magnitude(d1) + magnitude(d2));
  var a2 = magnitude(d2) / (magnitude(d2) + magnitude(d3));

  var b1 = addPoints(p1, scalePoint(d1, a2 / a1));
  var b2 = subtractPoints(p2, scalePoint(d3, (1 - a2) / (1 - a1)));

  var control1 = addPoints(b1, b2);
  var control2 = subtractPoints(b2, b1);

  return [p1, control1, control2];
}

function deCasteljau(p0, p1, p2, p3, t) {
  // Use de Casteljau algorithm to find the point on the curve at the given parameter t
  var q0 = interpolatePoints(p0, p1, t);
  var q1 = interpolatePoints(p1, p2, t);
  var q2 = interpolatePoints(p2, p3, t);
  var r0 = interpolatePoints(q0, q1, t);
  var r1 = interpolatePoints(q1, q2, t);
  var point = interpolatePoints(r0, r1, t);
  return point;
}

function subtractPoints(point1, point2) {
  return [point1[0] - point2[0], point1[1] - point2[1]];
}

function addPoints(point1, point2) {
  return [point1[0] + point2[0], point1[1] + point2[1]];
}

function scalePoint(point, scalar) {
  return [point[0] * scalar, point[1] * scalar];
}

function dotProduct(point1, point2) {
  return point1[0] * point2[0] + point1[1] * point2[1];
}

function magnitude(point) {
  return Math.sqrt(Math.pow(point[0], 2) + Math.pow(point[1], 2));
}

function distanceBetweenPoints(point1, point2) {
  var dx = point1[0] - point2[0];
  var dy = point1[1] - point2[1];
  return Math.sqrt(dx * dx + dy * dy);
}

function interpolatePoints(point1, point2, t) {
  return [point1[0] + (point2[0] - point1[0]) * t, point1[1] + (point2[1] - point1[1]) * t];
}

这段代码中的addPointToBezierCurve()函数接受五个参数：新点的坐标point和原始三次贝塞尔曲线的四个点p0、p1、p2、p3。它使用findClosestPointOnBezierCurve()函数找到离新点最近的点Q，然后将曲线分成两个部分，分别计算新的控制点和起点、终点。最后，函数返回新的控制点和起点、终点的坐标。

需要注意的是，这段代码中的计算方法可能不是最优的，因为在实际应用中，可能需要考虑更多的情况。但是，这段代码可以帮助你理解在三次贝塞尔曲线中加入一个点时，如何计算新的控制点和起点、终点。