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Examples of parameterised curves

For details of the basic operation of the program see the Main Help page. Some special options specific to the calculation of parameterised curves is below. Example definitions plus some explanations are also below.

X steps

This parameters control the of number of points calculated along the curve.


A very simple algorithm is used to ensure that the curve does not run off to infinity. If any value is greater than the clipping parameter, then the value is set to that parameter.


A circle
X =  x0 + r cos(Pi t);
Y =  y0 + r sin(Pi t); 
Z = 0; 
r = 1; t =x;
X = a * cos(2 Pi t) + 2 COs(Pi t);
Y = a * sin(2 Pi t) + 2 sin(Pi t);
Z = 0;
a = 1.5; t = x;
Rhamphoid Cusp
X = x^2;
Y = a x^4 + b x^5;
Z = x;
a = 1.0; b = 0.5;
When looking directly down the z-axis the projection of this curve will give a rhamphoid cusp. Along other directions cusps and pairs of crossings can appear.
X = COs(2 Pi t);
Y = sin(2 Pi t);
Z = t;
t = x;

Web page, applet and Algebraic Surface program by Richard Morris
home page Email Copyright Jan 2001.