如何在 C++ 中实现贝塞尔曲线？
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How do I implement a Bézier curve in C++?
提问by Nick Bolton
I'd like to implement a Bézier curve. I've done this in C# before, but I'm totally unfamiliar with the C++ libraries. How should I go about creating a quadratic curve?
我想实现一个贝塞尔曲线。我以前在 C# 中做过这个，但我完全不熟悉 C++ 库。我应该如何创建二次曲线？
void printQuadCurve(float delta, Vector2f p0, Vector2f p1, Vector2f p2);
Clearly we'd need to use linear interpolation, but does this exist in the standard math library? If not, where can I find it?
显然我们需要使用线性插值，但是标准数学库中是否存在这种方法？如果没有，我在哪里可以找到它？
Update 1:
更新 1：
Sorry, I forgot to mention I'm using Linux.
抱歉，我忘了提及我正在使用 Linux。
采纳答案by dirkgently
Did you use a C# library earlier?
您之前使用过 C# 库吗？
In C++, no standard library function for Bezier curves is available (yet). You can of course roll your own (CodeProject sample) or look for a math library.
在 C++ 中，没有贝塞尔曲线的标准库函数可用（目前）。您当然可以推出自己的（CodeProject示例）或寻找数学库。
Thisblogpost explains the idea nicely but in Actionscript. Translation should not be much of a problem.
这篇博文很好地解释了这个想法，但在 Actionscript 中。翻译应该问题不大。
回答by Jakob Riedle
Recently I ran across the same question and wanted to implemented it on my own. This image from Wikipedia helped me:
最近我遇到了同样的问题，想自己实现它。来自维基百科的这张图片帮助了我：
The following code is written in C++ and shows how to compute a quadratic bezier.
以下代码是用 C++ 编写的，展示了如何计算二次贝塞尔曲线。
int getPt( int n1 , int n2 , float perc )
{
int diff = n2 - n1;
return n1 + ( diff * perc );
}
for( float i = 0 ; i < 1 ; i += 0.01 )
{
// The Green Line
xa = getPt( x1 , x2 , i );
ya = getPt( y1 , y2 , i );
xb = getPt( x2 , x3 , i );
yb = getPt( y2 , y3 , i );
// The Black Dot
x = getPt( xa , xb , i );
y = getPt( ya , yb , i );
drawPixel( x , y , COLOR_RED );
}
With (x1|y1), (x2|y2) and (x3|y3) being P0, P1 and P2 in the image. Just for showing the basic idea...
其中 (x1|y1)、(x2|y2) 和 (x3|y3) 是图像中的 P0、P1 和 P2。只是为了展示基本的想法......
For the ones who ask for the cubic bezier, it just works analogue (also from Wikipedia):
对于那些要求三次贝塞尔曲线的人，它只是模拟（也来自维基百科）：
Thisanswer provides Code for it.
这个答案为它提供了代码。
回答by iforce2d
Here is a general implementation for a curve with any number of points.
这是具有任意数量点的曲线的一般实现。
vec2 getBezierPoint( vec2* points, int numPoints, float t ) {
vec2* tmp = new vec2[numPoints];
memcpy(tmp, points, numPoints * sizeof(vec2));
int i = numPoints - 1;
while (i > 0) {
for (int k = 0; k < i; k++)
tmp[k] = tmp[k] + t * ( tmp[k+1] - tmp[k] );
i--;
}
vec2 answer = tmp[0];
delete[] tmp;
return answer;
}
Note that it uses heap memory for a temporary array which is not all that efficient. If you only need to deal with a fixed number of points you could hard-code the numPoints value and use stack memory instead.
请注意，它使用堆内存作为临时数组，但效率并不高。如果您只需要处理固定数量的点，您可以对 numPoints 值进行硬编码并改用堆栈内存。
Of course, the above assumes you have a vec2 structure and operators for it like this:
当然，上面假设您有一个 vec2 结构和运算符，如下所示：
struct vec2 {
float x, y;
vec2(float x, float y) : x(x), y(y) {}
};
vec2 operator + (vec2 a, vec2 b) {
return vec2(a.x + b.x, a.y + b.y);
}
vec2 operator - (vec2 a, vec2 b) {
return vec2(a.x - b.x, a.y - b.y);
}
vec2 operator * (float s, vec2 a) {
return vec2(s * a.x, s * a.y);
}
回答by Malcolm McLean
You have a choice between de Casteljau's method, which is to recursively split the control path until you arrive at the point using a linear interpolation, as explained above, or Bezier's method which is to blend the control points.
您可以选择 de Casteljau 的方法，即递归分割控制路径，直到使用线性插值到达该点，如上所述，或者 Bezier 的方法，即混合控制点。
Bezier's method is
贝塞尔方法是
p = (1-t)^3 *P0 + 3*t*(1-t)^2*P1 + 3*t^2*(1-t)*P2 + t^3*P3
for cubics and
对于立方体和
p = (1-t)^2 *P0 + 2*(1-t)*t*P1 + t*t*P2
for quadratics.
为二次方。
t is usually on 0-1 but that's not an essential - in fact the curves extend to infinity. P0, P1, etc are the control points. The curve goes through the two end points but not usually through the other points.
t 通常在 0-1 上，但这不是必需的 - 事实上曲线延伸到无穷大。P0、P1 等是控制点。曲线通过两个端点，但通常不通过其他点。
回答by dirkgently
If you just want to display a Bezier curve, you can use something like PolyBezierfor Windows.
If you want to implement the routine yourself, you can find linear interpolation codeall over the Intarnetz.
I believe the Boost librarieshave support for this. Linear interpolation, not Beziers specifically. Don't quote me on this, however.
如果您只想显示贝塞尔曲线，您可以使用PolyBezierfor Windows 之类的东西。
如果你想自己实现例程，你可以在 Intarnetz 上找到线性插值代码。
我相信Boost 库对此有支持。线性插值，而不是专门的贝塞尔曲线。但是，不要在这方面引用我的话。
