Week Three Practical Problem

In this assignment, you examine a practical procedure used in computer-aided design and computational fluid dynamics. You will make some assessments regarding this procedure.

The word triangulation has two definitions. The first, and most common, is the use of trigonometry to establish the position of an object relative to two or more fixed, known locations. This is common in navigation. The second definition is the decomposition of a polygon into triangles. This provides a convenient representation of a polygon that can be used in a variety of computational contexts, such as those mentioned above. For this assignment you will not be concerned about computer science; rather, you will study the variety of ways in which polygons may be triangulated.

For the first three questions, consider the polygons to be convex. If you select any pair of points inside or on the boundary of the polygon, and join them with a line segment, that line segment will remain inside or on the boundary of the polygon; it will never cross the boundary and be outside the polygon. The final question asks you to consider what sort of effect the loosening of this restriction might have on your efforts.

Below is a series of diagrams showing the ways in which the first few polygons may be triangulated. At the start of this task, consider the vertices of the polygon as distinct; that is, they are distinguished from one another, perhaps by a label, letter, or number. The possible triangulations T(n) of an n-gon, for n = 3, 4, and 5, are illustrated here:

T(3) = 1 (A triangle is its own triangulation.)

T(4) = 2 (A convex quadrilateral can be triangulated diagonally on each of two diagonals.)

T(5) = 5 (A pentagon can be triangulated with two segments joining each vertex to its two opposite vertices.)

Questions

A. Determine T(n) for n = 6, 7, & 8.

B. Do you detect a pattern to these numbers? This pattern may arise out of the numbers or the manner in which you generated triangulations. (A closed-form function for T(n) is relatively straightforward, but is fairly nontrivial to construct; you will not have to explore that here.)

C. How would T(n) change if you ignored the vertices distinctness? That is, if you remove the labels, and say two triangulations are identical if one can be transformed into the other via a rotation or a reflection, how does this change T(n) for n = 4, 5, 6, 7, & 8?

T(4) = T(5) =1

For n > 5, T(n) = n2

Supporting work follows:

Examination of an octagon: 9 possible triangulations topologically distinct from one another.

Examination of a nonagon: 16 possible triangulations topologically distinct from one another.

D. What effect does relaxing the convexity restriction have on T(n)? See how T(n) changes for n = 4, 5, & 6. Do you see a pattern?

D. Take a concave n-sided polygon and

cave it in until the only possible chords are

those toward a single vertex:

This shows that without restrictions on convexity,

any n-gon could be limited to a single

possible triangulation.

In such a case, T(n) = 1 with or without

vertex differentiation!

Therefore, if convexity restrictions are removed T(n) can really range between 1 and its upper limits.