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.