贝塞尔曲线的OpenGL坐标 [英] OpenGL Coordinates from Bezier Curves

问题描述

基本上，我需要从OpenGL中的Bezier曲线实现中获取所有坐标.具体来说，我需要坐标以沿弯曲的轨迹路径移动场景中的球体对象(棒球).这就是我用来绘制曲线的方式:

Basically, I need to get all the coordinates drawn from a Bezier curve implementation in OpenGL. Specifically, I need the coordinates to move a sphere object (baseball) in my scene along a curved trajectory path. This is what I use to draw my curve:

GL2 gl = drawable.getGL().getGL2();    
float ctrlpoints[][] = new float[][]{
            {0.0f, 0.0f, 60f},
            {0.0f, 3.0f, 45.0f},
            {0.0f, 2.0f, 15.0f},
            {0.0f, 1.0f, 0f}};
    FloatBuffer ctrlpointBuf = FloatBuffer.allocate(ctrlpoints[0].length * ctrlpoints.length);
        for (int i = 0; i < ctrlpoints.length; i++) {
            for (int j = 0; j < 3; j++) {
                ctrlpointBuf.put(ctrlpoints[i][j]);
            }
        }
        ctrlpointBuf.rewind();

        gl.glMap1f(GL2.GL_MAP1_VERTEX_3, 0.0f, 1.0f, 3, numControlPoints, ctrlpointBuf);
        gl.glEnable(GL2.GL_MAP1_VERTEX_3);

        gl.glColor3f(1.0f, 1.0f, 1.0f);
        gl.glBegin(GL2.GL_LINE_STRIP);
        for (int i = 0; i <= 30; i++) {
            gl.glEvalCoord1f((float) i / (float) 30.0);
        }
        gl.glEnd();

有人知道如何从该实现中获得要点吗?

Does any one have any idea how to get the points out of this implementation?

推荐答案

贝塞尔曲线很容易计算.首先是可分离的，这意味着您可以一次计算一个坐标(首先是x，然后是y，然后是z ...).对于给定的坐标，以下是使用该定义的函数:

A Bezier curve is quite easy to compute. First of all is separable, that means that you can compute it one coordinate at a time (first x, then y, then z...). For a given coordinate the following is a function that uses the definition:

double bezier(double A,  // Start value
              double B,  // First control value
              double C,  // Second control value
              double D,  // Ending value
              double t)  // Parameter 0 <= t <= 1
{
    double s = 1 - t;
    double AB = A*s + B*t;
    double BC = B*s + C*t;
    double CD = C*s + D*t;
    double ABC = AB*s + BC*t;
    double BCD = BC*s + CD*t;
    return ABC*s + BCD*t;
}

请注意，在上面的函数中，参数t不是曲线的弧长参数，而是从t=0(点位于曲线起点)到t=1的通用参数. (该点位于曲线的末端).

Note that in the above function the parameter t is not the arc-length parameter for the curve but a generic parameter that goes from t=0 (where the point is at the beginning of the curve) to t=1 (where the point is at the end of the curve).

上面图片的交互式版本，您可以在其中拖动A，B，C，D和AB点.在此处可用.它使用html/js/canvas实现，并且仅在Chrome，Firefox，Safari上进行了测试.

An interactive version of the above picture where you can drag A, B, C, D and AB points is available here. It's implemented with html/js/canvas and tested only on Chrome, Firefox, Safari.

如果需要在XYZ中以受控的特定速度移动对象，一种简单的方法是计算近似的折线(例如，通过对100个t值的曲线进行采样)，然后以恒定的速度在生成的结果上移动折线.

If you need to move your objects at a controlled specific speed in XYZ an easy way is to compute an approximated polyline (for example by sampling the curve for 100 values of t) and then walk at constant speed on the resulting polyline.

贝塞尔三次方的真实弧长参数化(即使用沿曲线测得的长度作为参数)相当烦人(IIRC没有积分的封闭形式解).

The true arc-length parametrization for a Bezier cubic (i.e. using a parameter that is the length measured along the curve) is rather annoying to compute (IIRC there is no closed form solution for the integral).

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