Conic Sections

This section of the tutorial assumes that you have completed the section on computing level sets of the default model. In this section we describe how to animate the level sets of a two-parameter family of functions in order to see the transition between various conic sections (ellipses, hyperbolas, parabolas, etc.)

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.

Two Parameter Animation

Let's compute the level set of x^2 + a y + b y^2 as the parameters a and b both vary. One simple way to let the parameters vary is to vary them along a circle in parameter space. A parametrization of a circle in parameter space is a(t) = cos(t), b(t) = sin(t), so let's animate the level sets as a and b vary along this path.

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.

Computing Parameters Corresponding to Singular Curves

In the previous animation, there were several level sets that were singular in the sense that there was a point on the curve for which the gradient of the model vanished. The parameter values for which a level curve is singular will typically form a hypersurface in parameter space (in this case, a curve in (a,b)-space). This hypersurface is often (but not always) indicative of a change in the topology of the level curve.

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.

Ending the Session

When you are finished, end this Pisces session by selecting Quit from the File menu on the Main Panel.

Next: The Unfolding of a Cusp

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