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- Derivatives
- Tangents and normals
- Working with 3D normals -
- Component functions +
- 分量函数
- Finding extremities: root finding
- Bounding boxes
- Aligning curves @@ -3501,31 +3501,18 @@ generateRMFrames(steps) -> frames: - Component functions + 分量函数
- One of the first things people run into when they start using Bézier curves in their own programs is "I know how to draw the curve, but - how do I determine the bounding box?". It's actually reasonably straightforward to do so, but it requires having some knowledge on - exploiting math to get the values we need. For bounding boxes, we aren't actually interested in the curve itself, but only in its - "extremities": the minimum and maximum values the curve has for its x- and y-axis values. If you remember your calculus (provided you ever - took calculus, otherwise it's going to be hard to remember) we can determine function extremities using the first derivative of that - function, but this poses a problem, since our function is parametric: every axis has its own function. + 当人们开始在自己的程序中使用贝塞尔曲线时，首先遇到的问题之一是：“我虽然知道怎么画曲线，但是怎么确定包围盒？”其实做法颇为直接，但需要知道如何利用一些数学知识得到所需的值。对于包围盒而言，所需的其实并不是曲线本身，而只是曲线的“极值”——曲线的x轴和y轴分量的最小值和最大值。如果还记得微积分的话（前提是学过微积分，否则更难记），那么函数的极值可以用函数的一阶导数所确定，但由于“曲线函数”有不只一个分量，这就产生了一个问题——每个分量都有自己的函数。
+解决办法：对每个分量分别计算导数，再按照原来的分量顺序重新拼在一起。
- The solution: compute the derivative for each axis separately, and then fit them back together in the same way we do for the original. -
-- Let's look at how a parametric Bézier curve "splits up" into two normal functions, one for the x-axis and one for the y-axis. Note the - leftmost figure is again an interactive curve, without labeled axes (you get coordinates in the graph instead). The center and rightmost - figures are the component functions for computing the x-axis value, given a value for t (between 0 and 1 inclusive), and the y-axis - value, respectively. -
-- If you move points in a curve sideways, you should only see the middle graph change; likewise, moving points vertically should only show a - change in the right graph. + 以下演示参数化的贝塞尔曲线如何“分解”为两个正常的函数，一个对应于x轴，一个对应于y轴。注意左侧的图像依然是可交互的曲线，但没有标出坐标轴（坐标显示在图中）；中间和右侧的图像是分量函数，分别对应于指定t值（介于0和1之间，含端点）后求出的x轴和y轴分量。
+如果水平移动曲线上的点，那么应当只有中间的图像在变化；同样，如果竖直移动曲线上的点，那么应当只有右侧的图像在变化。
