Voronoi Diagramming Without CPR

Duncan Chiu(dchiu@dimacs.rutgers.edu),(dchiu@mail.idt.net)

General Information

The Problem
The class begins in 3 minutes. You are about to go to your class and teach Voronoi diagramming. “Where are my compasses, protractors or rulers!?” To your horror they are nowhere in sight! What do you do?

Suggested Materials
Don’t panic! All you need are pencils and paper.

Save scrap paper whenever you can. Colored paper is helpful, but not necessary.

Prerequisites
Clean finger nails. Nails of one inch or longer are problematic. Not responsible for broken nails.

Activity Description

Guided Exploration
1. Homemade ruler

(a) Fold scrap paper in half, approximately. (fig. 1a)

(b) Fold it again, parallel to the first fold. (fig. 1b)

[Fold it a third time if the paper is too thin.]

(c) Crease the fold with your fingernail and you have a viable ruler.

2. Homemade protractor

(a) Fold a piece of paper in half. (fig. 2a)

Crease the fold with your fingernail. (For the long fingernailer: hold the folded paper with two hands and go over the fold on the edge of the desk a couple of times.)

(b) Fold the paper again by overlapping one end of the crease to the other. Crease the fold. (fig. 2b)

(c) You should get a 90 degree corner --- that is your perpendicular protractor. (fig. 2c)

3. Draw a Voronoi edge between two points A & B.

(a) Place the corner of a piece of paper on point A and line up the edge through point B. (fig. 3a)

(b) Mark off point B on the edge of the paper. (fig. 3b)

(c) Fold the paper so that point A is over B. Crease the fold carefully (this is your perpendicular bisector of AB.) (fig. 3c)

(d) Draw the perpendicular bisector along the fold and extend the line with your paper ruler. (fig. 3d)

[Check to make sure you have lined up the paper over AB properly before you draw the bisector.]

(e) Check the result with your homemade protractor. Put the paper ruler through AB and slide your protractor along to verify the perpendicular line. (fig. 3e)

4. If you have only rulers…

(a) Place your ruler through points A and B and mark the midpoint M of AB on your diagram.

(b) Slide your paper protractor along the ruler to the midpoint M. (fig. 4b)

(c) Draw the perpendicular bisector of AB along the paper protractor.

Concluding the Exploration
Try the technique a few times. You will find it easier for the student to see and understand the concept of bisecting and the notion of perpendicular bisector. The diagram will be neater than when a compass is used. With a compass, try bisecting a distance that is 5 to 6 inches apart with the end points close to the edges of the paper. Compare the result with the above method of bisecting and drawing the perpendicular bisector of AB.

Sample Results from Exploration
fig. 5a

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