A Project on Circles in Space

This project asks students to decide if a collection of points in space do or do not lie on a circle. The project is accessible to linear algebra students who have studied Gram–Schmidt orthogonalization and the application of linear algebra to least squares regression. Several pedgogical aspects of the project are discussed including two successful approaches for the solution of the problem. Matlab code for generating data points is included. The project of deciding if a collection of points in space do or do not lie on a circle has been very successful in my recent linear algebra classes. Students understand the problem and, given the background of the problem, the motivation for solving it. The project forces them to integrate their knowledge of several parts of the course and some earlier work in other courses. Moreover, since it sounds easier than it usually turns out to be, students get hooked on the problem. I believe they come to understand to some extent how the mathematics they know could be useful in a “real world” setting. The project was inspired by a presentation given by Michael A. Lachance, University of Michigan at Dearborn, at the Tri-section Meeting of the Michigan, Illinois, and Indiana Sections of the Mathematical Association of America held at Saint Mary’s College (Notre Dame, Indiana) in the spring of 1993. Professor Lachance described this and other problems he had learned of in his consulting work with Ford Motor Company. I would like to thank Professor Lachance and Dr. Edward Moylan, Ford Motor Company, for their assistance in preparing this note. ∗ c ©Carl C. Cowen. Permission is granted for use of this material, with attribution, for instruction, but publication of this material requires prior written consent of the author.