• In this exercise I show how to interpolate points using Bezier curves.
• The formula for a 4-point Bezier Curve is the parametric function B(t) = (1-t)^3*p0+t(1-t)^2*p1 +t^2*(1-t)*p2 + t^3*p3 where t goes from 0 to 1.
• The derivative of a 4-point Bezier curve is B'(t) = 3(1-t)^2(p1-p0) + 6(1-t)t(p2-p1)+3t^2(p3-p2) +2t(p2 - p1).
• The second derivative of a 4-point Bezier curve is B''(t) = 6(1-t)(p2-2p1+p0)+6t(p3-2p2+p1)
• If I want to interpolate between the end point of a Bezier Curve B_a and a new point b with a new curve B_b, the first and second derivatives of the curves must be the same where they meet because we want the curve to be smooth. Therefore, B'_a(1) = B'_b(0) and B''_a(1) = B''_b(0).
• Solving the equations above gives P_b1 = 2 * P_a3 - P_a2, P_b2 = P_a1 - 4 * P_a2 + 4 * P_a3, where Pb_1 and Pb_2 are the control points of the new curve B_b

Drag the points to change the shape of the Bezier Curve.