Kinematics, the geometry of motion, studies the positions of geometric objects parameterized by time. This is a 3D space with functions representing the path or trajectory as the locus of places occupied by points. It has a dual mathematics of 3D time with functions representing the course of motion as the locus of times occupied by events. Below is an introduction to both, following the exposition in *Principles of Engineering Mechanics: Kinematics* by Millard Beatty Jr.

1.3 Motion and Particle Path

To locate an object in space, we need a reference system. The only reference we have is other objects. Therefore, the physical nature of what we shall call a *reference frame* is an assigned set of objects whose mutual distances do not change with [dis]time – at least not very much. …

We define a three-dimensional Euclidean reference frame *φ* as a set consisting of a point *O* of space, called the *origin* of the reference frame, and a vector basis {**e*** _{i}*} ≡ {

**e**

_{1},

**e**

_{2},

**e**

_{3}}. That is,

*φ*= {

*O*;

**e**

*}. We shall require for convenience that the basis is an orthonormal basis, i.e., a triple of mutually perpendicular unit vectors.*

_{i}