 Examples of algebraic curve

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

Region of interest

These parameters control the range over which the surface is calculated. Its best if unequal values are chosen for the ranges. Otherwise certain degeneracies may occur which can lead to poorer results.

Resolution control

These parameters control how accurate the resulting curve will be. Each of these must be a power of two. The rectangle specified by the range is split into a number of smaller rectangles and points of the edges of these rectangles will be found together with any singularities lying in the middle of them. The Coarse parameter specifies the number of smaller rectangles, by default 8 along the x, y, i.e. 512 boxes. The Fine parameter specifies the smallest rectangle size used to find a singularity.

Generally you will just want to change the Coarse parameter. If the singularities are not calculated very well then you could also increase the Fine parameter.

Examples

It generally a good idea if Top (X-Y) item is selected in the Camera panel in the control window.
Unfolding of a Cusp
x^3-y^2+a x^2=0;
a = 0;
When a = 0 this gives a cusp. An unfolding of the cusp is achieved by changing the value of a, for a > 0 the curves will have a simple crossing (crunode) at the origin, for a < 0 the curve has an isolated point (acnode) at the origin.
Many other examples can be found by selecting one of the predefined curves in the Project panel. See A Visual Dictionary of Special Plane Curves and Famous Curve Index for other algebraic curves.
Web page, applet and Algebraic Surface program by Richard Morris