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
Start a new session of Pisces, and open the
Model Panel, the
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
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
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
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
Next: The Unfolding of a Cusp
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Last modified: Wed Nov 22 15:12:44 1995
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The Geometry Center,
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