# Rational 2D Trigonometric Spline

### Click and drag control points to change curve. Modify weights in boxes below the curve.

For more information, check out the post on my blog: Bezier Curves.
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W1: W2:
W3: W4:

This rational trigonometric spline has 4 control points, a weight per control point (4 total), and total up the values of the 4 functions below to get the final point at time t. t ranges from 0 to pi/2 instead of the usual 0 to 1.
1. A * W1 * (0.5*cos(x)*(cos(x)+1.0))
2. B * W2 * (0.5*sin(x)*(sin(x)-1.0))
3. C * W3 * (0.5*cos(x)*(cos(x)-1.0))
4. D * W4 * (0.5*sin(x)*(sin(x)+1.0))
It then divides that by the total of these 4 functions.
1. W1 * (0.5*cos(x)*(cos(x)+1.0))
2. W2 * (0.5*sin(x)*(sin(x)-1.0))
3. W3 * (0.5*cos(x)*(cos(x)-1.0))
4. W4 * (0.5*sin(x)*(sin(x)+1.0))
Parameters:
t - "Time", this value goes from 0 to pi/2 to generate each point on the curve
A - The first control point, also the starting point of the curve.
B - The second control point.
C - The third control point.
D - The fourth control point, also the ending point of the curve.
W1 - The weighting of control point A.
W2 - The weighting of control point B.
W3 - The weighting of control point C.
W4 - The weighting of control point D.

In other words, if you have 4 control points A,B,C and D, 4 weights W1,W2,W3,W4 and a time t:
CurvePoint = (A*W1*(0.5*cos(t)*(cos(t)+1.0)) + B*W2*(0.5*sin(t)*(sin(t)-1.0)) + C*W2*(0.5*cos(t)*(cos(t)-1.0)) + D*W3*(0.5*sin(t)*(sin(t)+1.0)))/(W1*(0.5*cos(t)*(cos(t)+1.0)) + W2*(0.5*sin(t)*(sin(t)-1.0)) + W2*(0.5*cos(t)*(cos(t)-1.0)) + W3*(0.5*sin(t)*(sin(t)+1.0)))

Note that this spline is 2 dimensional because A,B,C,D are 2 dimensional, but you could use these same equations in any dimenion!