3 point parametric interpolation curve

It is quite simple: the equation (the page you linked) provided is vectorial, let's write it (for simplicity)

P(0)={x0,y0}={a0,b0};
P(s)={xs,ys}={a2*s*s+a1*s+a0, b2*s*s+b1*s+b0}
P(1)={x1,y1}={a2+a1+a0, b2+b1+b0}

Where the P(0), P(s), P(1) are known (0<s<1).
We have to find the six parameters a2,a1,a0, b2,b1,b0. Limiting computation to x (y is the same):

x0 = a0
x1 = a2+a1+x0 => a1 = x1-x0-a2
xs = a2*s*s+x1*s-x0*s-a2*s+x0 => a2 = (xs-x0+(x0-x1)*s)/(s-s*s)

With a2 value you compute a1 using the second equation.
Now a2,a1,a0 (and in the same way b2,b1,b0) are known and you may compute P(t) = {x(t), y(t)} = {a2*t*t+a1*t+a0, b2*t*t+b1*t+b0} for each 0<t<1.
(I hope it is correct...)

Some formulas in the referenced web site are not correct. Especially the result formula for a₁ is wrong.

You must understand that you have the freedom to chose the t₁ for your P₁: by varying t₁ for your chosen P₁, the shape of the curve will vary too.

Assuming you have three defined points:
- start (t = 0) at P₀
- end (t = 1) at P₂
- a point P₁ between start and end where you also decide for a t₁ (e.g. 0.5).

Given these four parameters (P₀, P₂, P₁, t₁), the needed curve coefficients are:
a₀ = A
a₁ = B - t₁ C
a₂ = C - B
Where
A = P₀
B = (P₁ - P₀) / ((1 - t₁) t₁)
C = (P₂ - P₀) / (1 - t₁)

If working in 2D, each coefficient consists of two values, one for each axis - working in 3D, each coefficient consists of 3 values, on per axis.

E.g. for each axis coeficient value, you set the respective axis value of the points P_u plus the parameter t₁ value into the formulas above.

Then you sweep the parameter t from 0 to 1 and calculate the curve points P(t) to draw.

pseudo code:

class Formula
{
    private Point a0;
    private Point a1;
    private Point a2;

    private Point A(Point p0, Point p1, Point p2, double t1) { return p0; }
    private Point B(Point p0, Point p1, Point p2, double t1) { double d = 1/((1-t1)*t1); return (p1-p0)*d; }
    private Point C(Point p0, Point p1, Point p2, double t1) { return d = 1/(1-t1); return (p2-p0)*d; }

    public Formula(Point p0, Point p1, point p2, double t1)
    {
       Point a = A(p0, p1, p2, t1);
       Point b = B(p0, p1, p2, t1);
       Point c = C(p0, p1, p2, t1);

       a0 = a;
       a1 = b-c*t1;
       a2 = c-b;
    }
    public Point P(double t) { return ((a2*t)+a1)*t+a0; }
}
...
var start = new Point(0,2); // p0
var end   = new Point(7,6); // p2
var p1    = new Point(2,3); // p1
double t1 = 0.5;
var Formula curve = new Formula(start, p1, end, t1);
...
int steps = 100;
double step = 1.0/(double)steps;
Point curr = p0;
foreach(int i = 1; i <= steps; ++i)
{
   Point next = curve.P(step*(double)i);
   Draw(curr, next);
   curr = next;
}

Cheers
Andi

private void DrawCurve( Graphics g, PointF a, PointF b, PointF c )
        {
            float t, n = 30;

            float x1 = 4 * b.X - 2 * ( a.X + c.X );
            float x2 = x1 - a.X + c.X;
            float y1 = 4 * b.Y - 2 * ( a.Y + c.Y );
            float y2 = y1 - a.Y + c.Y;

            PointF p1 = a, p2;

            for( float i = 1; i <= n; i++ )
            {
                t = i / n;
                p2 = new PointF( a.X - x1 * t * t + x2 * t, a.Y - y1 * t * t + y2 * t );
                g.DrawLine( pen, p1, p2 );
                p1 = p2;
            }
        }