#### CSC274b

Spring 2000

Ileana Streinu

#
Problem Set 3

#### Thursday, February 25, 2000

Due: Thursday, March 2, 2000 - before class

The homework has 3 parts:
- Cinderella
- Theory
- Machine shop

Everybody has to do Part 1 and at least 3 problems on part2, but
if you have difficulty with other theory questions, you have the
option of doing part 3 as a make-up for them.

**Cinderella**:
**Trapezoidalization**
- Draw (in Cinderella) a polygon with 20 vertices or so, with a
spiraled shape (and many reflex vertices which are cusps).
Use the "pin down" feature to fix the vertices, so that they
won't move.
- Draw the horisontal lines induced by the line sweep on this
polygon. Export to HTML to create an applet.
- Copy the same polygon in another picture, and now clip the
lines you create din the previous step, so that you get a
trapezoidal decomposition. Export to HTML.
- Copy the trapezoidal decomposition from the previous step in
another picture, and add the diagonals produced by the algorithm
for decomposition into monotone polygons. Export to HTML.
- When done, put all the three applets/images in one HTML page
and link it from your class web page, under the name of
**Trapezoidalization
in Cinderella**. Edit the HTML page with a few explanatory
sentences describing what each image is about.

**Monotone triangulation**.
- Draw in Cinderella a monotone polygon. Pin the vertices, so
they won't move.
- Draw the horisontal lines corresponding to the line sweep.
- Draw the diagonals of the triangulation produced by the
algorithm for monotone triangulation.
- Make sure you have chosen a polygon which will illustrate all
the cases encountered by the algorithm, and use different colors
for the diagonals produced by the algorithm in each case.
- Export to HTML, add some explanatory text and link from your
calss web page under the name
**Monotone triangulation**

- Email me when you are done with this part.

**Theory**
- Problems from the textbook.
- p. 55, pbs. 5,6. Do the drawings in Cinderella and print
them.
- p. 68, pbs 6,7,8.

- Let
**P** be a polygon with **r** interior cusps. Prove
or disprove:
- To decompose
**P** into monotone polygons,
**r/2** diagonals are sometimes needed.
- To decompose
**P** into a minimum number of monotone
polygons, **r ** diagonals always suffice.

Everybody should do at least problems 7 and 8 on page 68, and
at least one more. If
you have difficulty with the others, then consider doing the last
part of this homework, explained below.

**Machine shop** (optional)
- Construct a polygon from rigid bars (of various sizes),
joined together at vertices in such a way that the edges rotate
freely around the joints. At least two polygons, with 6 or more
edges. Please synchronize with your other colleagues doing this
part of the assignement, so that we get in the end polygons of different
sizes (number of vertices).

Last modified February 25, 2000.