We will also use a new algorithm, called the Predictor-Corrector algorithm.

Start a new session of Pisces, and open the Model Panel, the Predictor-Corrector Control Panel, and the View Window.

Select the Cubic 2D model from the
**Model** menu on the Main Panel.

We will study the two-parameter family of quadratic polynomials given by

*x^2 + ay + by^2*

as the parameters *a* and *b*
are varied.

Open the Parametric
Animation Panel by selecting **Parametric Animate**
from the **Utilities** menu of the Main Panel. Change the very first entry
marked **Steps** so that we will take 16 animation steps.
About halfway down the panel are two menus that allow the user to
select the parameters to be animated. Select
**Model.coef_y** and **Model.coef_y^2**.
Under the menu labeled **Algorithm**, select
**Predcorr**, press the button marked **Show
Parametric Path** so that the pth in paramter space is shown, and then
press the button marked **Go**.

What you will see is a parabola transform into a series of ellipses, another parabola, hyperbolas, and intersecting lines.

To compute the set of parameter values corresponding to singular curves,

**Create a Derived Model**- Let
*f*be the current model. Then singular curves correspond to the zero set of the function*F=(f, df/dx, df/dy)*. This is a new model which is*derived*from the model*f*.To create this model, select

**Derived Model**from the**Settings**menu on the Main Panel. The Derived Models Panel will appear. On this panel, select the menu item labeled**Planar Singularity**, then press the**Update**button that appears. **Inflate the Model Domain**- At this point, we have a model with a two-dimensional domain,
*(x,y)*, and a three-dimensional range,*(f, df/dx, df/dy)*. In order to use Pisces, however, the dimension of the domain must be greater than the dimensions of the range. The solution of this problem is to*inflate*the domain by two parameters. At the bottom of the Model Panel, press the**Permute**button to bring up the Permutation Panel. Use the Permutation Panel to tell Pisces that the variables**coef_y**and**coef_y^2**should be added to the domain, rather than held fixed as parameters. There are now four variables selected to be in the domain, so press the**OK**button in order to dismiss the Permutation Panel and apply the permutation. **Compute the Zero Set of the Derived Model**- Press the
**Go**button in the Predictor Corrector Control Panel. Pisces will compute a curve in*R^4*; the View Window displays the projection of this curve into*(x,y)*-space. We are more concerned with the projection of this curve into parameter space, so use the**Hor**and**Ver**menus of the View Window in order to change the View Window so that the solution curve is projected onto a vertical line in*(coef_y, coef_y^2)*-space. It is easy to show analytically that the set of singular curves for the family*x^2 + a y + b y^2*occurs for*a=0*, so Pisces has correctly computed the singular set.

Comments to: pisces@geom.umn.edu

Last modified: Wed Nov 22 15:12:44 1995

Copyright © 1995 by The Geometry Center, all rights reserved.