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

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

*x^3 - y^2 + ax + b*

as the parameters *a* and *b*
are varied. To begin, set all of the parameters in the Model Panel to zero, except
for the parameters **coef_x^3** (set equal to 1) and
**coef_y^2** (set equal to -1).

Change the parameter **Singular_Points** on the
Predictor-Corrector Control Panel to read 1. This sets up a
*1 x 1* array of initial points which Pisces will use to attempt to
detect singularities in the current model. Pressing the
**Go** button on the Predictor-Corrector Control Panel
will cause Pisces to detect the singularity at the origin, to locally
represent the structure of the singularity, and then to trace the
remainder of the level set.

We will once again use two-parameter animation and let
parameters vary along a circle in parameter space.
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_x** and **Model.const_coef**.
Under the menu labeled **Algorithm**, select
**Predcorr**, then press the button marked **Go
**.

What you will see is a single curve transform into a pair of curves, and then back again.

As in the previous problem:

**Create a Derived 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 the bottom of the
Model Panel, press the
**Permute**button to bring up the Permutation Panel. Use the Permutation Panel to add the variables**coef_x**and**const_coef**to the domain. (Don't forget to 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. Change the**Hor**and**Ver**coordinates of the View Window in order to project the solution curve onto*(a,b)*-space. This curve should look like a cusp.The curve separates the parameter space into two regions. For level curves indexed by parameters in the region containing

*(a,b)=(1,0)*, the topology of the level set is that of a line; for level curves indexed by parameters in the region containing*(a,b)=(-1,0)*are topologically equivalent to a circle and a line. For parameter values on the separating curve, the corresponding level set is singular. This information is summarized in the image below.

Comments to: pisces@geom.umn.edu

Last modified: Sun Nov 26 15:35:10 1995

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