**Parametric curves and surfaces applet**

This is an applet to demonstrate and give some examples of parametric curves and surfaces.

**Instructions for the 3D Bezier curve**

Click the "Bezier curve" button and the top panel should change
to allow you to enter the control points. The four control points __p___{i}
of the Bezier curve define the curves shape by the equation:

__r__(t) = S(i=0 to 3) __p___{i} B_{i}(t)

where the B_{i}(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".

**Instructions for the Bezier surface**

Clicking the "Bezier surface" button initially draws a flat surface defined by 16 control points in a 4x4 grid. This is actually just a
single Bezier "patch" and is used in computer graphics by joining many together to form surfaces that actually look like something.

Setting the control points is the same as for the Bezier curve.
**Instructions for moving the camera**

Ok it could be better but to move the camera, or change its direction, do the following:

Choose the component you want to change from the drop down list eg "Camera pos z."

Enter the value you want to change it to on the right, and press the button in the middle
**Instructions for the general Fourier curve**

Click the "Fourier curve" button. The general Fourier curve is defined by the equations,

x(t) = S(i=0 to inf.) a_{i} cos it + b_{i} sin it

y(t) = S(i=0 to inf.) c_{i} cos it + d_{i} sin it

z(t) = S(i=0 to inf.) f_{i} cos it + g_{i} 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, sin^{2} t = (1/2)(1 - cos 2t)

we get,

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)

**More information**

To learn more about the parametric curves shown here, click me

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