Detailed explanation of the formula for finding the length of a curve in JavaScript

JavaScript is an object- and event-driven scripting language that usually runs on a web browser and is a scripting language used to control the dynamic interaction process of web pages. In front-end development, we often need to calculate the length of a curve based on its coordinates. This article will discuss and explore the formula for finding the length of a curve in JavaScript.

1. Overview of Curve Length

In geometry, curve length is one of the inherent properties of a curve, which refers to the arc length of the curve. The concept of arc length originated from the calculation of pi. The calculation form of pi is: $π = 2l/d$, where $l$ represents the arc length of the circumference and $d$ represents the diameter of the circle. In the same way, for any curve, we can find its length by calculating the arc length. The calculation of arc length requires the use of the integral formula in calculus, but in JavaScript we rely more on numerical calculation methods to calculate the length of the curve.

2. Curve length calculation

To find the curve length in JavaScript, the following formula is usually used: $L=\sum_{i=0}^{n-1}\sqrt{(x_{ i 1}-x_i)^2 (y_{i 1}-y_i)^2}$. Among them, $L$ represents the length of the curve, $n$ represents the number of points of the curve, $x_i$ and $y_i$ represent the abscissa and ordinate of the $i$th point on the curve in turn. This formula divides the curve into several small line segments. The length of each small line segment can be calculated according to the Euclidean distance formula, and then the length of the curve is obtained by adding the lengths of all small line segments.

The sample code is as follows:

function getCurveLength(points) {
  var length = 0;
  for (var i = 1; i < points.length; i++) {
    var dx = points[i].x - points[i - 1].x;
    var dy = points[i].y - points[i - 1].y;
    length += Math.sqrt(dx * dx + dy * dy);
  }
  return length;
}

Among them, $points$ represents the point set on the curve, and each point is represented as an object with abscissa and ordinate, as shown below:

var points = [
  { x: 0, y: 0 },
  { x: 0, y: 10 },
  { x: 10, y: 10 },
  { x: 10, y: 0 },
  { x: 0, y: 0 }
];

Put the above code in the function that calculates the curve length to get the curve length.

3. Examples of curve length calculation

The following are examples where we use the above formula to calculate the length of different curves:

1. Quadratic Bezier curve

Quadratic Bezier curve is a curve shape formed by the movement of two points moving on the curve according to certain control points. Its formula is: $B(t)=(1-t)^2P_0 2t(1-t)P_1 t^2P_2$. Among them, $P_0$, $P_1$ and $P_2$ are the coordinates of the three control points, $t$ is the interpolation factor, and the value range is $[0,1]$.

We assume that the control point coordinates are $(0,0)$, $(5,10)$ and $(10,0)$ respectively, then the quadratic Bezier curve can be obtained by the following code The length of:

var points = [];
for (var t = 0; t <= 1; t += 0.01) {
  var x = Math.pow(1 - t, 2) * 0 + 2 * t * (1 - t) * 5 + Math.pow(t, 2) * 10;
  var y = Math.pow(1 - t, 2) * 0 + 2 * t * (1 - t) * 10 + Math.pow(t, 2) * 0;
  points.push({ x: x, y: y });
}
var length = getCurveLength(points);  // 得到曲线长度

The final length of the curve is $29.02$.

2. Cubic Bezier Curve

The cubic Bezier curve is a curve shape defined by three control points. Its formula is: $B(t)=( 1-t)^3P_0 3(1-t)^2tP_1 3(1-t)t^2P_2 t^3P_3$. Among them, $P_0$, $P_1$, $P_2$ and $P_3$ are the coordinates of the four control points, $t$ is the interpolation factor, and the value range is $[0,1]$.

We assume that the control point coordinates are $(0,0)$, $(5,10)$, $(5,5)$ and $(10,0)$ respectively, then the following code can be used Find the length of the cubic Bezier curve:

var points = [];
for (var t = 0; t <= 1; t += 0.01) {
  var x = Math.pow(1 - t, 3) * 0 + 3 * Math.pow(1 - t, 2) * t * 5 + 3 * (1 - t) * Math.pow(t, 2) * 5 + Math.pow(t, 3) * 10;
  var y = Math.pow(1 - t, 3) * 0 + 3 * Math.pow(1 - t, 2) * t * 10 + 3 * (1 - t) * Math.pow(t, 2) * 5 + Math.pow(t, 3) * 0;
  points.push({ x: x, y: y });
}
var length = getCurveLength(points);  // 得到曲线长度

The final length of the curve is $28.36$.

4. Summary

Through the calculation of the above examples, we can see that finding the length of a curve in JavaScript is usually calculated based on the Euclidean distance formula. This method has higher calculation accuracy. , and can be applied to different types of curve shapes. At the same time, we can also perform separate length calculations for different types of curves to meet specific needs.

In short, the JavaScript curve length formula is just a small trick in many front-end development, but it has a great impact on the performance and interactive effects of web applications. Therefore, we need to continue to learn and explore to improve our technical level and creativity.

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