A parametric curve in 2D space is really made of two parameteric functions, x(t) and y(t), which describe how the x and y coordinates change with the parameter, t:
p(t) = ( x(t), y(t) )You can build an intuition about space curves by considering how changes in the parametric functions, x(t) and y(t), cause changes in the curve.
x(t) and y(t) appear below on the right. The 2D space curve, p(t), appears on the left.
You can drag any point with the mouse to see how changes in one curve affect the other curves.
Below, the one-dimensional function, p(t), is defined as a sum of weighted Catmull-Rom blending functions, Bi(t):
p(t) = B0(t) x0 + B1(t) x1 + ... + Bn(t) xn
Each Catmull-Rom blending function in the lower graph is weighted by the corresponding point in the upper graph, so p(t) is the sum of the weighted blending functions in the lower graph.
Drag the points, xi, in the upper graph to see the relative contributions of the blending functions.
(Other applets by James Stewart)