This is an applet to demonstrate and give some examples of parametric curves and surfaces.
Click the "Bezier curve" button and the top panel should change
to allow you to enter the control points. The four control points pi
of the Bezier curve define the curves shape by the equation:
r(t) = S(i=0 to 3) pi Bi(t)
where the Bi(t) are special blending functions used by Mr. Bezier to construct his smooth curves.
Initially the control points have been placed at (0,0,0) (1,0,0) (2,0,0) and (3,0,0), but you can move them to anywhere you want by selecting the control point from the drop down list, entering the x, y, and z coordinates, and clicking "Set control point".
Click the "Fourier curve" button. The general Fourier curve is defined by the equations,
x(t) = S(i=0 to inf.) ai cos it + bi sin it
y(t) = S(i=0 to inf.) ci cos it + di sin it
z(t) = S(i=0 to inf.) fi cos it + gi sin it
You can enter the coefficients (up to i = 20) in the top panel.For example, to construct the cardiod:
x = (cos t)(1 - sin t)
y = (sin t)(1 - sin t)
z = 0
using, cos t sin t = (1/2) sin 2t
and, sin2 t = (1/2)(1 - cos 2t)
x = cos t - 0.5 sin 2t
y = -0.5 + 0.5 cos 2t + sin t
so next to "x cos coeffs" select 1 from the drop down list, enter 1.0 in the text field and click the "x cos coeffs" button to enter the value, and you've done the "cos t" term. The other four terms can be done in the same way. (The number, n say, in the drop down list refers to the coefficient of t, cos nt or sin nt)