CSC274b
Spring 2000
Ileana Streinu

1. Cinderella
2. Theory
3. Machine shop

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).