why are Cubic Bezier functions not accurate compared too windows api PolyBezier? - c++

Question

I have been trying to find a way to draw a curved line/cubic bezier line using a custom function. However, all the examples and such found on the internet, differ a little from each other and usually produce different results, why? . None of the ones i have tried produce the same result as windows api PolyBezier which is what i need.
This is my current code for drawing cubic bezier lines:
double Factorial(int number)
{
double factorial = 1;
if (number > 1)
{
for (int count = 1; count <= number; count++) factorial = factorial * count;
}
return factorial;
}
double choose(double a, double b)
{
return Factorial(a) / (Factorial(b) * Factorial(a - b));
}
VOID MyPolyBezier(HDC hdc, PPOINT Pts, int Total)
{
float x, y;
MoveToEx(hdc, Pts[0].x, Pts[0].y, 0);
Total -= 1;
//for (float t = 0; t <= 1; t += (1./128.))
for (float t = 0; t <= 1; t += 0.0078125)
{
x = 0;
y = 0;
for (int I = 0; I <= Total; I++)
{
x += Pts[I].x * choose(Total, I) * pow(1 - t, Total - I) * pow(t, I);
y += Pts[I].y * choose(Total, I) * pow(1 - t, Total - I) * pow(t, I);
}
LineTo(hdc, x, y);
}
}
And here is the code for testing it.
POINT TestPts[4];
BYTE TestType[4] = {PT_MOVETO, PT_BEZIERTO, PT_BEZIERTO, PT_BEZIERTO};
//set x, y points for the curved line.
TestPts[0].x = 50;
TestPts[0].y = 200;
TestPts[1].x = 100;
TestPts[1].y = 100;
TestPts[2].x = 150;
TestPts[2].y = 200;
TestPts[3].x = 200;
TestPts[3].y = 200;
//Draw using custom function.
MyPolyBezier(hdc, TestPts, 4);
//Move the curve down some.
TestPts[0].y += 10;
TestPts[1].y += 10;
TestPts[2].y += 10;
TestPts[3].y += 10;
//Draw using windows api.
//PolyDraw(hdc, TestPts, TestType, 4); //PolyDraw gives the same result as PolyBezier.
PolyBezier(hdc, TestPts, 4);
And an attached image of my bad results:
Note: the bottom bezier line is windows(PolyBezier) version.
Edit:
the final goal, Windows(On the left) VS custom funtion. Hopefully this helps in some way.

Answer 1

So a cubic bezier is a mathematical curve. The cubic bezier is a specific case of a more general curve.
The cubic bezier is defined by 4 control points -- a start and end point, and 2 control points. In general, a bezier has n control points in order.
The line is drawn as a time parameter t goes from 0 to 1.
To find out where a general bezier of degree n is at time t:
For each adjacent pair of control points in your bezier, find the weighted average of them, as controlled by t. So at + b(1-t) for control points a before b.
Use these n-1 points to form a degree n-1 bezier.
Solve the new bezier at time t.
when you hit a degree 1 bezier, stop. That is your point.
Try writing an algorithm based off the true definition of bezier, and see where it differs from the windows curve. This may ne less frustrating than taking some approximation and having two sets of errors to reconcile.