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Curves (curve)
If moving everything along a straight line between two points is all we wanted to do, then this would be enough. However, in the real world, objects move in smooth curves and accelerate and decelerate smoothly. A curve can be represented by three or more control points. For most curves, there are more than three control points, two of which form the endpoints; the others define the shape of the curve. Consider the simple curve shown in Figure 4.14
If you will be able to meet all our needs along a straight line motion, the linear interpolation is sufficient, however, in fact, more objects along the curvilinear motion, slowly accelerating and decelerating. Three or more than three points may be a curve expressed, the majority of three or more curve control points. Wherein the two points used to define the boundaries, the other points to define the shape of the curve, Figure 4.14 shows a simple curve 
The curve shown in Figure 4.14 has three control points, A, B, and C. A and C are theendpoints of the curve and B defines the shape of the curve. If we join points A and B with one line and points B and C together with another line, then we can interpolate along the two lines using a simple linear interpolation to find a new pair of points, D and E. Now, given these two points, we can join them with yet another line and interpolate along it to find a new point, P. As we vary our interpolation parameter, t, point P will move in a smooth curved path from A to D. Expressed mathematically , this is
4.14 curves in FIG. 3 control points, wherein A and C are endpoints, B control the shape of the curve. If we AB and BC all together, we can carry out these two segments along a linear interpolation to find similar D, one point for such E. Connection D, E and two can be a new segment, this line interpolation will obtain the new point p. When we allow t to change, can be obtained from the point A to D p smooth motion, is calculated as follows: 
Substituting for E and D and A Little crunching doing, with The following :( we come up between the same for D, E interpolation, we have the following formula :)
You should recognize this as a quadratic equation in t. The curve that it describes is known as a quadratic Bézier curve (you will find that this is actually the second-order interpolation Bezier curves). We can actually implement this very easily in GLSL using the mix function, as all we're doing is linearly interpolating (mixing) the results of two previous interpolations (it is easy to achieve this in GLSL interpolation inside)
vec4 quadratic_bezier(vec4 A, vec4 B, vec4 C, float t)
{
vec4 D = mix(A, B, t); // D = A + t(B - A)
vec4 E = mix(B, C, t); // E = B + t(C - B)
vec4 P = mix(D, E, t); // P = D + t(E - D)
return P;
}
By adding a fourth control point as shown in Figure 4.15, we can increase the order by 1 and produce a cubic Bézier curve.
当我们添加第四个控制点后，这个插值又变成了三次方的贝塞尔曲线了
We now have four control points, A, B, C, and D. The process for constructing the curve is similar to that for the quadratic Bézier curve. We form a first line from A to B, a second line from B to C, and a third line from C to D. Interpolating along each of the three lines gives rise to three new points, E, F, and G. Using these three points, we form two more lines, one from E to F and another from F to G, interpolating along which gives rise to points H and I, between which we can interpolate to find our final point, P. Therefore, we have
现在我们有四个控制点了，构建曲线的方法跟三个控制点构建曲线的方法类似。我们首先连接AB、BC、CD，然后分别对他们插值，得到新的点E、F、G。然后用这仨点做出新的两条线EF、FG， 然后继续对EF、FG插值得到H、I点，然后我们对HI进行插值，得到了p点，这样一来我们有下面的公式
If you think these equations look familiar, you’re right: Our points E, F, and G form a quadratic Bézier curve that we use to interpolate to our final point P. If we were to substitute the equations for E, F, and G into the equations for H and I, then substitute those into the equation for P, and crunch through the expansions, we would be left with a cubic equation with terms in t—hence the name cubic Bézier curve. Again, we can implement this simply and efficiently in terms of linear interpolations in GLSL using the mix function:
如果你觉得这个等式非常相似，那么你的直觉是正确的，我们E、F、G组成了的二次方的贝塞尔曲线去插值p点的时候，如果把E、F、G求得H、I，那么就可以得到P，如此这般，最后我们就会得到 由因子t影响的插值公式，下面给出了GLSL里面的插值实现算法
vec4 cubic_bezier(vec4 A, vec4 B, vec4 C, vec4 D, float t)
{
vec4 E = mix(A, B, t); // E = A + t(B - A)
vec4 F = mix(B, C, t); // F = B + t(C - B)
vec4 G = mix(C, D, t); // G = C + t(D – C)
vec4 H = mix(E, F, t); // H = E + t(F - E)
vec4 I = mix(F, G, t); // I = F + t(G - F)
vec4 P = mix(H, I, t); // P = H + t(I - H)
return P;
}
Just as the structure of the equations for a cubic Bézier curve “includes” the equations for a quadratic curve, so, too, does the code to implement them. In fact, we can layer these curves on top of each other, using the code for one to build the next
正如三次方与二次方的贝塞尔曲线的关系那样，是包含关系，实际上，咱们的代码也是这样的包含关系。如下所示：
vec4 cubic_bezier(vec4 A, vec4 B, vec4 C, vec4 D, float t)
{
vec4 E = mix(A, B, t); // E = A + t(B - A)
vec4 F = mix(B, C, t); // F = B + t(C - B)
vec4 G = mix(C, D, t); // G = C + t(D - C)
return quadratic_bezier(E, F, G, t);
}
Now that we see this pattern, we can take it further and produce even higher-order curves. For example, a quintic Bézier curve (one with five control points) can be implemented as
类似的，我们可以得到更高阶的曲线的推导公式，长相如下这般
vec4 quintic_bezier(vec4 A, vec4 B, vec4 C, vec4 D, vec4 E, float t)
{
vec4 F = mix(A, B, t); // F = A + t(B - A)
vec4 G = mix(B, C, t); // G = B + t(C - B)
vec4 H = mix(C, D, t); // H = C + t(D - C)
vec4 I = mix(D, E, t); // I = D + t(E - D)
return cubic_bezier(F, G, H, I, t);
}
This layering could theoretically be applied over and over for any number of control points. However, in practice, curves with more than four control points are not commonly used. Rather, we use splines.
This situation can actually curve with layers of nested down, but in reality, we rarely use more than 4 Bezier curve control points, the control points more if we use use spline
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