# Problem Set 3

#### Thursday, February 25, 2000 Due: Thursday, March 2, 2000 - before class

The homework has 3 parts:
1. Cinderella
2. Theory
3. 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.
1. 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.
2. 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:
1. To decompose P into monotone polygons, r/2 diagonals are sometimes needed.
2. 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.
3. 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).