This post follows the introduction to intrinsic coordinates given *here*, and changes it for time-space (1D space + 3D time). So rather than the 3D space position vector **r**, we’ll use the 3D time position vector, **w**, and we’ll switch the arc time, *t*, with the arc length, *s*.

We follow the motion of a point using a time position vector **w**(*s*) whose position along a known trajectory in time is given by the *scalar function* *t*(*s*) where *t*(*s*) is the arc time along the curve. We obtain the celerity, *u*, from the space rate of change of the vector **w**(*s*) following the particle:

We identify the scalar as the magnitude of the celerity *u* and *d***w**/*dt* as the unit vector tangent to the curve at the point *t*(*s*). Therefore we have

**u** = *u* **e**_{t},

where **w**(*s*) is the time position vector, is the pace, **e**_{t} is the unit tangent vector to the trajectory, and *t* is the arc time coordinate along the trajectory.

The unit tangent vector can be written as **e**_{t} = *d***w**/*dt*. The prestination vector, **b**, is the derivative of the celerity vector with respect to arc length. Since d**w**/d*t* depends only on *t*, using the chain rule we can write

The second derivative *d*²**w**/*dt*² is another property of the arc time. We shall see that it is related to the radius of curvature.

Taking the space derivative of **u**, an alternate expression can be written in terms of the unit vector **e**_{t} as

The vector **e**_{t} is the local unit tangent vector to the curve which changes from point to point in time. Consequently, the space derivative of **e**_{t} will, in general, be nonzero. The space derivative of **e**_{t} can be written as

In order to calculate the derivative of **e**_{t}, we note that, since the magnitude of **e**_{t} is constant and equal to one, the only changes that **e**_{t} can have are due to rotation, or *swinging*. When we move from *t* to *t* + *dt*, the tangent vector changes from **e**_{t} to **e**_{t} + d**e**_{t}. The change in direction can be related to the angle *dβ* (the differential angle).

The direction of *d***e**_{t}, which is perpendicular to **e**_{t}, is called the *normal* direction. On the other hand, the magnitude of *d***e**_{t} will be equal to the length of **e**_{t} (which is one), times *dβ*. Thus, if **e**_{n} is a *unit normal vector* in the direction of *d***e**_{t}, we can write

*d***e**_{t} = *dβ ***e**_{n}.

Dividing by *dt* yields

Here *κ* = *dβ*/*dt* is a local property of the curve, called the *time curvature*, and *ρ* = 1/*κ* is called the *radius of time curvature*.

From the above we have that

Finally, we have that the prestination can be written as

Here is the *tangential* component of the prestination, and *b*_{n} = *u*²/*ρ* is the *normal* component of the prestination. Since *b*_{n} is the component of the prestination pointing towards the center of curvature, it is sometimes referred to as the *centripetal* prestination. When *b*_{t} is nonzero, the celerity vector changes magnitude, or *stretches*. When *b*_{n} is nonzero, the celerity vector changes direction, or *swings*. The modulus of the total prestination can be calculated as

Relationship between *t*, *u*, and *b*

The quantities *t*, *u*, and *b*_{t} are related in the same manner as the quantities *t*, *u*, and *b* for rectilinear motion. In particular we have that This means that if we have a way of knowing *b*_{t}, we may be able to integrate the tangential component of the motion independently.

The vectors **e**_{t} and **e**_{n}, and their respective coordinates *t* (tangent) and *n* (normal), define two orthogonal directions. The plane defined by these two directions is called the *osculating* plane. This plane changes from point to point and can be thought of as the plane that locally contains the trajectory (Note that the tangent is the current direction of the celerity, and the normal is the direction into which the celerity is changing).

In order to define a right-handed set of axes we need to introduce an additional unit vector which is orthogonal to **e**_{t} and **e**_{n}. This vector is called the *binormal*, and is defined as **e**_{b} = **e**_{t} × **e**_{n}.

At any point in the trajectory, the time position vector, the celerity, and prestination can be referred to these axes. In particular, the celerity and prestination take very simple forms:

**u** = *u ***e**_{t}

The difficulty of working with this reference frame stems from the fact that the orientation of the axis depends on the trajectory itself. The time position vector, **w**, needs to be found by integrating the relation *d***w**/*ds* = **u** as follows:

where **w**_{0} = **w**(0) is given by the initial condition.

We note that, by construction, the component of the prestination along the binormal is always zero. When the trajectory is planar, the binormal stays constant (orthogonal to the plane of motion). However, when the trajectory is a time curve, the binormal changes with *t*. It will be shown below that the derivative of the binormal is always along the direction of the normal. The rate of change of the binormal with *t* is called the *time torsion*, *σ*. Thus

We see that whenever the torsion is zero, the trajectory is planar, and whenever the curvature is zero, the trajectory is linear.

Radius of curvature and torsion for a trajectory

In some situations the trajectory will be known as a curve of the form *y* = *f*(*x*). The radius of curvature in this case can be computed according to the expression,

Since *y* = *f*(*x*) defines a planar curve, the torsion *σ* is zero. On the other hand, if the trajectory is known in parametric form as a curve of the form **w**(*s*), where *s* can be arc length or any other parameter, then the radius of curvature *ρ* and the torsion *σ* can be computed as

Equations of motion in intrinsic coordinates

Newton’s second law is a vector equation, **Γ** = *n***b** (with surge **Γ** and vass *n*; cf. **F** = *m***a**), which can now be written in intrinsic coordinates. In tangent, normal, and binormal components, *tnb*, we write **Γ** = *Γ*_{t} **e**_{t} +* Γ*_{n} **e**_{n} and **b** = *b*_{t} **e**_{t} + *b*_{n} **e**_{n}. We observe that the positive direction of the normal coordinate is that pointing toward the center of curvature. Thus, in component form, we have

Note that, by definition, the component of the prestination along the binormal direction, **e**_{b}, is always zero, and consequently the binormal component of the surge must also be zero.