This page is joint work with Michael Falk.

Suppose f(x, y, z) and g(x, y, z) are both homogeneous polynomials of the same degree. Then

This page shows 4 different pencils. In each case there are 3 different ways of visualizing the pencil

**arrangement_2d** shows a 2-dimensional animation
were θ increases with time by the formula

**arrangement_3d** shows the 3-dimensional animation.

**arrangement_3d_no_planes** is a 3-dimensional animation without
the planes of the singular curves.

To understand the pencil, ``2d'' is probably the best. The prettiest pictures are obtained with ``3d_no_planes.''

The arrangements are:

**Braid arrangement** (or A_{3}, or tetrahedron)

f(x, y, z) = x^{2} - z^{2} and g(x, y, z) = f(y, x, z)

braid_2d
braid_3d
braid_3d_no_planes

To see the DPGraph figure, click on the link. To get back to this page, exit DPGraph.

**Cube arrangement** (or B_{3})

f(x, y, z) = y^{2} ( x^{2} - z^{2}) and g(x, y, z) = f(y, x, z)

cube_2d
cube_3d
cube_3d_no_planes

**Terao-Falk arrangement**

f(x, y, z) = ( x^{2} - z^{2})( 9 y^{2} - z^{2})
and g(x, y, z) = f(y, x, z)

TF_2d
TF_3d
**TF_3d_no_planes** (This is the graph featured at the top of the page.)

**Pappus arrangement**

f(x, y, z) = (y-z)(2y+x+z)(2y-x+z) and g(x, y, z) = f(x, y, -z)

Pappus_2d
Pappus_3d
Pappus_3d_no_planes

Here's a series of stills from the "featured" graph, TF_3d_no_planes.

These graphs were produced by
DPGraph,
which is a ** fast** program for
viewing 3D objects.

e-mail: Jim.Swift@nau.edu