Length of quadratic Bezier curve

Quadratic Bezier curves are usually constructed in the following manner, given a fixed point p0,p1,p2 on a two-dimensional plane, and B (t) is used to represent the curve

With an animation to demonstrate, you can more clearly indicate the construction process of this curve






If the T-variable itself is linear, the generation of this Bezier curve itself is not uniform, usually at both ends of the fast middle slow.




If the T-variable itself is linear, the generation of this Bezier curve itself is not uniform, usually at both ends of the fast middle slow.

How do you want to get a constant Bezier motion? For example, we design a path to a Bezier curve in a game, how to achieve the player at a constant speed in this path movement.
Thinking of this algorithm is quite a bit of a brain, in the meantime also get math cow Charlesgao help, thank him very much (more embarrassing is, I ask the question when I put one of the formula mistaken, laughed at the-_-!).

First, we need to obtain the velocity formula S (t) of B (t) relative to T.

s (t) = Math.sqrt (Bx (t) ^2 + by (t) ^2)

To simplify the formula, we define the following variables:

A = p0-2*p1 + P2 B = 2*p1-2p0

The calculated s (T) can be expressed as:

s (t) =math.sqrt (a*t^2 + b*t + C)

Where a,b,c is a constant calculated according to the P0,P1,P2:

A = 4 * (ax^2 + by^2)

B = 4 * (ax* by + ay* by)

C = bx^2 + by^2

According to this formula, the length formula of the Bezier curve (t) is obtained:


Set T ' is an independent variable that enables L to achieve uniform motion, then obviously l (t ') =l (1.0) *t, namely:


Since the L (t) function is very complex, it is almost impossible to find the expression of the inverse function directly, but fortunately we can know that its derivative is s (t), in practice, we can use Newton tangent method to find the approximate solution. Its iterative algorithm can be expressed as:


I wrote a test program to verify the algorithm, the results are as follows, you can see that this curve has been generated in a uniform manner ^_^:

[CPP] View plain copy #include  <stdio.h>      #include  <math.h>      #include  <windows.h>           //three control points       point p0={50,50},p1={500,600},p2={800,200};            int ax = p0.x-2*p1.x+p2.x;      int ay = p0.y-2*p1.y+p2.y;       int bx = 2*p1.x-2*p0.x;      int by =  2*p1.y-2*p0.y;            double a = 4* (Ax*ax+ay*ay);       double b = 4* (ax*bx+ay*by);      double c =  bx*bx+by*by;           //Curve total length       Double total_ length = 0.0;           //Curve split number of copies &NBSP;&NBSp    const int step = 70;           // Array to hold plot point data       point pixels[step];           // -------------------------------------------------------------------------------------     / /Speed function      /*    S (t_)  = Sqrt[A*t*t+B*t+C]    */       Double s (double t)       {           Return sqrt (a*t*t+b*t+c);     }           //----- --------------------------------------------------------------------------------     /Length function      /*        L (t)  = Integrate[s[t], t]      & nbsp L (T_)  =  (2*sqrt[a]* (2*a*t*sqrt[c + t* (b + a*t)]&Nbsp;+ b* (-sqrt[c] + sqrt[c + t* (b + a*t)))  +                   (b^2 - 4*a*c)   (log[b +  2*sqrt[a]*sqrt[c]] - log[b + 2*a*t + 2 sqrt[a]*sqrt[c + t* ( B&NBSP;+&NBSP;A*T)]])                      /(8* a^ (3/2));    */      double l (double t)       {          double temp1 = sqrt (c+ t* (b+a*t));          double temp2 =  (2*a*t*temp1+b* ( TEMP1-SQRT (C));          double temp3 = log (B+2*sqrt (A) * sqrt (C));          double temp4 = log (b+2*a*t+2*sqrt (A) * Temp1);          double temp5 = 2*sqrt (A) *temp2;          double temp6 =  (b*b-4*a*c) * (temp3-temp4);                     return  (TEMP5+TEMP6)/(8*pow (a,1.5));      }           //----------------------------------- --------------------------------------------------     /length function inverse function, using Newton tangent method to solve      /*        x (n+1)  = xn - f (Xn)/F ' (Xn)     */       Double invertl (double t, double l)       {           double t1=t, t2;                     do          {              t2 = t1  -  (L (T1)-L)/s (T1);              if (ABS ( T1-T2) <0.000001)  break;              t1=t2;           }while (True);           return t2;     }           //-------------------- -----------------------------------------------------------------      Lresult callback  _wndproc (Hwnd hwnd, uint message, wparam wparam, lparam lparam)        {          switch  (message)            {          case wm_timer:              {                   static nIndex = 0;                  if (nindex>=0 & & nindex<=step)                    {                       double t =  (double) nindex/step;                       //If you grow linearly, the corresponding curve length                        double l = t*total_length;                       //according to the inverse function of l function, l corresponding T value is obtained                        t = invertl (T,  l);                             //according to the Bessel curve function, obtain the x,y coordinates at this time                        double x =  (1-t) * (1-t) *p0.x +2* (1-t) *t*p1.x + t*t*p2.x;                       double y =  (1-T) * (1-T) * p0.y +2* (1-t) *t*p1.y + t*t*p2.y;                             //Rounding        &NBSP;&NBsp;              pixels[nindex].x =   (int) (x+0.5);                       pixels[nIndex].y =  (int) (y+0.5);                             nindex++;                       invalidaterect (hwnd, 0, 0);                   }                   else                   {                        killtimer (hwnd, 101);                   }               }              break;           case WM_PAINT:               {                   paintstruct ps;                   hdc hdc = beginpaint (hwnd, &ps);                   ::movetoex (hdc, p0.x, p0.y, 0);                  lineto (hdc, P1.x,  p1.Y);                  lineto (HDC, &NBSP;P2.X,&NBSP;P2.Y);                         for (int i=0; i<step; i++)                    {                       const point &pt  = pixels[i];                       if (pt.x==0 && pt.y==0)  break;                              ::movetoex (hdc, pt.x-2, pt.y, 0);                        ::lineto (HDC,&NBSP;PT.X+2,&NBSP;PT.Y);                        ::movetoex (hdc, pt.x, pt.y-2, 0);                       ::lineto (hdc, pt.x, pt.y+2);                   }                  endpaint (hWnd, & PS);              }               break;          case wm_ destroy:              postquitmessage (0);     &nbsp         break;          default:               return defwindowproc (hwnd,  Message, wparam, lparam);          }          return 0;     }            //-------------------------------------------------------------------------------------      Int apientry winmain (hinstance hinstance,                             hinstance hprevinstance,                            LPTSTR     lpCmdLine,