16 Spline
16 Spline
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Would some like to explain cubic and quadratic splines?
You must forgive me for being a bit thick but the standard
a cubed ... answer goes right over my head. I'm not sure what 'a' is and how did we get a d. What i'd like is answer that relates coefficients with 3 points on a graph, lets call em: Fud(2, 2), Bill(8,16) and Gates(12,4) or F, B & G for short.
The idea of splines is to patch together a series of polynomials that go through your points. For a cubic spline, you determine a cubic equation for each interval between two adjacent points. A cubic equation has the form:
y = a*x^3 + b*x^2 + c*x + d
The procedure of doing the spline involves determining the values for a, b, c, and d for each interval that gives a smooth curve. There will be a different set of constants for each interval between each pair of adjacent points.
For each interval, you solve equations so the curve goes through the two point and the slope of one section matches the slope of the next section.
Take your points F, B, and G. There will be 2 sets of constants a1, b1, c1, and d1 for the curve between F and B; a2, b2, c2, and d2 for the curve between B and G. Use the notation:
y2(x) = a2*x^3 + b2*x^2 + c2*x + d2
dy2/dx = y2' (x) = 3*a2*x^2 + 2*b2*x + c2
The equations to be solved are:
y1(2) = 2
y1(8) = 16
y2(8) = 16
y2(12) = 4
y1' (8) = y2'(8)
y1''(8) = y2''(8)
You'll notice that there are 6 equations but 8 unknowns. There are always two less equations than unknowns. These two are determined by the end conditions. Typically, some slope condition is applied to each end to give the final two equations. The set of equations is coupled and linear so it may be solved via matrix inversion.







































