﻿ Pencils of Curves

# Pencils of Curves

Note: To view these figures, you need a PC (or PC emulator) running the free DPGraphViewer (or the moderately priced DPGraph) available at DPGraph.

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

α f(x, y, z) + β g(x, y, z) = 0
is a called a pencil of curves.

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

cos(θ) f(x, y, z) + sin(θ) g(x, y, z) = 0

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

θ = (time - sin(time))/8
This has the effect of crossing the interesting angles, θ = n π/4, with zero speed. You see colors because a thin slice of the 3d surface is shown: The part above z = 1 is red and the part below z = 1 is blue. The black curves are lines that appear at special values of θ. These are hyperplanes in 3d, and this arrangement of hyperplanes is what gives each pencil its name.

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 A3, or tetrahedron)
f(x, y, z) = x2 - z2 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 B3)
f(x, y, z) = y2 ( x2 - z2) and g(x, y, z) = f(y, x, z)
cube_2d     cube_3d     cube_3d_no_planes

Terao-Falk arrangement
f(x, y, z) = ( x2 - z2)( 9 y2 - z2) 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.

Jim Swift's DPGraph page     Jim Swift's home page     Department of Mathematics     NAU
e-mail: Jim.Swift@nau.edu