Chapter 5 : the trajectory

Abstraction made of the time, the geometry is preponderant. In this section, we will discuss a lot about vectors and we define the tangent, the normal and the binormal.

The tangent

The trajectory is the path a moving object follows through space. Let’s analyse this curve ζ from the geometric point of view. The trajectory curve can be defined by

If we take two consecutive positions,  and , the average speed is the vector carried by the rope  The instantaneous speed on her side is a vector tangent to the trajectory.

The direction of the speed vector defined thus the tangent to the trajectory. We can get rid of the length of the speed vector and keep its direction by dividing it by its intensity. As a result, we obtain a vector of length 1 tangent to the trajectory.

In the Cartesian system, it corresponds to

In this last expression, the time is not explicitly present. If we defined ds as the instantaneous displacement  we can write

 

The length L of the trajectory is the sum of the lengths of the average displacements and, in the limit, to the integration of the lengths of the instantaneous displacements.

The length of the trajectory s is a parameter intrinsically related to the curve ζ as it doesn’t depend upon the choice of the origin point O, nor of the parameterisation of the curve ζ. It shows, in its infinitesimal version ds, the distance to make on the tangent of the trajectory.

The normal

The vector  being defined, how do we express the geometric aspect related to the acceleration? By construction, the speed vector is tangent to the trajectory of the position vector. The acceleration vector is thus also tangent to the trajectory of the speed vector. How do we represent that in the space?

Let’s derivate the speed vector, giving explicitly its norm V and the direction :

The first term is proportional to  and is thus parallel to the speed vector. The second term is necessarily perpendicular to it. Indeed, the trajectory of  cannot be anywhere else than on a sphere because, by definition, its length does not change. As its derivative is tangent to this trajectory, it is thus tangent to the sphere and thus perpendicular to the radius of the sphere, i.e. to the vector .

We can define the normal vector  as the derivative of the tangent vector.

λ is still to be determined.

• Algebraically

The acceleration  is thus the sum of one tangent part  and one normal part , perpendicular to the first part.

The length of the square of the acceleration is thus given by (the vector product of two perpendicular vectors gives zero)

And then

• Geometrically

Given two consecutive positions of the point P on the trajectory ζ:  and , and the direction of the speed  and  at the corresponding positions. The tangent unit vector , by definition, doesn’t change its length. However, its direction changes: it turns by an angle .

The two tangent vectors delimitate a plane P in which is also their variation . With these three vectors, we can build an isosceles triangle and thus evaluate the length of : 2 sin(/2) and its angle with : π/2-/2.

At the limit, when P₂ tends towards P₁, the length of  becomes  and the angle becomes thus π/2, i.e.  becomes perpendicular to . As a result we can write

λ corresponds thus to the angular speed of rotation of the tangent vector. To be more accurate, we see that if we extend the normal in P₁ and in P₂, they meet at the point C, and that this point C is, at the limit of , the centrum of the tangent circle of the curve ζ in P₁ and P₂. The angle  equals the length of the arc ∆s divided by the radius R of the circle. As a result,

The radius R is called the curvature radius of the trajectory ζ at a given point. 1/R is called the curvature of ζ and the limit plane P formed by the tangent vector and the normal is called the osculating plane of the trajectory ζ.

The binormal

As the tangent  and the normal  are defined, we can build a vector perpendicular to those two vectors: the binormal . To do so, we make the vector product:

This vector is perpendicular to the plane P of the trajectory and is oriented so that the three vectors  can be overimposed to the three Cartesian vectors . Knowing the definitions of the speed and of the acceleration, we also can say that

What allow us to determine the equation of the curvature radius R

The dot over a variable means that it is its derivative over the time. Two dots will mean the second derivative, etc. The readability of the equation can be eased this way. In terms of components of the speed and of the acceleration, we have