Random obtuse triangles and convex quadrilaterals

We intend to discuss in detail two well known geometrical probability problems. The first one deals with finding the probability that a random triangle is obtuse in nature. We initially discuss the various ways of choosing a random triangle. The problem is at first analyzed based on random angles (adding to 180 degrees) and random sides (obeying the triangle inequality) which is a direct modification of the Broken Stick Problem. We then study the effect of shape on the probability that when three random points are chosen inside a figure of that shape they will form an obtuse triangle. Literature survey reveals the existence of the analytical formulae only in the cases of square, circle and rectangle. We used Monte Carlo simulation to solve this problem in various shapes. We intend to show by means of simulation that the given probabilatity will reach its minimum value when the random points are taken inside a circle. We then introduce the concept of Random Walk in Triangles and show that the probability that a triangle formed during the process is obtuse is itself random. We also propose the idea of Differential Equation in Triangle Space and study the variation of angles during this dynamic process. We then propose to extend this to the problem of calculating the probability of the quadrilateral formed by four random points is convex. The effects of shape are distinctly different than those obtained in the random triangle problem. The effort of true random numbers and normally generated pseudorandom numbers are also compared for both the problems considered.

Department dc:contributor.department

Massachusetts Institute of Technology. Computation for Design and Optimization Program.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2009

Author dc:creator

• Banerjee, Nirjhar

Advisor dc:contributor.advisor

• Gilbert Strang.

• Computation for Design and Optimization Program.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/54212

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/54212