eJMT Abstract

Title 2D and 3D Loci Inspired by an Entrance Problem and Technologies
Author Wei-Chi YANG and Vladimir SHELOMOVSKII
Volume 11
Number 3

In this paper, we discuss a locus problem that was originated from Chinese college entrance exam practice problems and it has been discussed in. We will see how the 2D locus problem can be explored using the dynamic geometry software (DGS) Geometry in Mathematical Arts with different strategies. Next, we extend the 2D locus problem to more challenging corresponding problems in 3D with the help of a DGS. We also illustrate how a computer algebra system (CAS), Maple, can be used to derive our locus analytically. We shall see that the use of a DGS in constructing the locus is very accessible to students when they can visualize what the locus might look like first. On the other hand, when readers need to use a CAS for verifying if results are consistent with our visualization, the task becomes much more challenging. In particular, the process of finding three points on the ellipsoid systematically and constructing a set of three linearly independent vectors requires the knowledge of a rotation matrix, whose computation is tedious if a CAS is not available. Once the rotation matrix is known, it is then simple to visualize the rotation of a vector about an axis, an important concept in computer graphics. The paper shows that with appropriate aids of technological tools, challenging and applicable mathematics can be made more fun and accessible.