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}

The spatial location of a particle *P* in the frame *φ* at time *t* is given by the *position vector* **x*** _{φ}* of

*P*from

*O*. Of course, as time goes on the place occupied by

*P*generally will vary. The time sequence of positions of

*P*in

*φ*is called a

*motion*of

*P*

*relative to φ*, and it is defined by the equation

**x** = **x*** _{φ}*(

*P*,

*t*), (1.1)

in which **x** is the place in frame *φ* occupied at time *t* by the particle *P* in its motion relative to *φ*. The locus *L* of places occupied by *P* in the motion is called the path or trajectory of *P*. The place **x**_{0} that *P* occupies along its path at some instant *t* = *t*_{0}, say, which we may sometimes consider as the initial instant *t*_{0} = 0, is determined by the motion (1.1):

**x**_{0} = **x*** _{φ}*(

*P*,

*t*

_{0}). (1.2)

When the particular choice of *φ* is clear, as it is when only one frame is being considered, for the sake of simplicity we shall discard the subscript *φ* and write (1.1) and (1.2) as

**x** = **x*** _{φ}*(

*P*,

*t*), (1.3a)

**x**_{0} = **x*** _{φ}*(

*P*,

*t*

_{0}). (1.3b)

Frequently we shall identify {**e*** _{m}*} as the familiar Cartesian basis {

**i**

*} = {*

_{m}**i**

_{1},

**i**

_{2},

**i**

_{3}} with

**i**

_{1}=

**i**,

**i**

_{2}, =

**j**and

**i**

_{3}=

**k**, as usual.

*Dual of above*

Motion and Eventicle Path

To locate an subject in time, we need a reference system. The only reference we have is other subjects. Therefore, the physical nature of what we shall call a *reference timeframe* is an assigned set of subjects whose mutual distimes do not change with distance – at least not very much. …

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

**e**

_{1},

**e**

_{2},

**e**

_{3}}. That is,

*ψ*= {

*O*;

**e**

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

_{i}The temporal location of an eventicle *Q* in the timeframe *ψ* at distance *s* is given by the *time position vector* **x**_{ψ} of *Q* from *O*. Of course, as distance goes on the place in time occupied by *Q* generally will vary. The distance sequence of time positions of *Q* in *ψ* is called a *motion* of *Q* *relative to ψ*, and it is defined by the equation

**w** = **w*** _{ψ}*(

*Q*,

*s*), (1.1)

in which **w** is the place in timeframe *ψ* occupied at distance *s* by the eventicle *Q* in its motion relative to *ψ*. The locus *L* of places in time occupied by *Q* in the motion is called the time path or trajectory of *Q*. The place in time **w**_{0} that *Q* occupies along its time path at some point *s* = *s*_{0}, say, which we may sometimes consider as the initial point *s*_{0} = 0, is determined by the motion (1.1):

**w**_{0} = **w*** _{ψ}*(

*Q*,

*s*

_{0}). (1.2)

When the particular choice of *ψ* is clear, as it is when only one timeframe is being considered, for the sake of simplicity we shall discard the subscript *ψ* and write (1.1) and (1.2) as

**w** = **w*** _{ψ}*(

*Q*,

*s*), (1.3a)

**w**_{0} = **w*** _{ψ}*(

*Q*,

*s*

_{0}). (1.3b)

Frequently we shall identify {**e*** _{m}*} as the familiar Cartesian basis {

**i**

*} = {*

_{m}**i**

_{1},

**i**

_{2},

**i**

_{3}} with

**i**

_{1}=

**i**,

**i**

_{2}, =

**j**and

**i**

_{3}=

**k**, as usual.