Motion along a curve

👋 About the lesson

Welcome to one of the final lectures of this semester. This time we will be talking about curves. Do not panic, curves are very similar concept to what we were talking about the last time - they map each value (read some real number) in given domain $D$ to a corresponding \(n\)-dimensional vector.

Within the scope of this lecture, we will be focusing on 2 and 3 dimensional curves which simulate some movement in time. Therefore, the values in the domain represent time and for each particular time we want to define for instance where the given object was located, i.e., we want to map each \(t\) to a \(3\) dimensional vector.


📓 Notes

Basic definition of the curve

As already mentioned in the intro, curve is a mapping of all numbers in the given domain \(D\) to a corresponding \(n\) dimensional vector, therefore more formally we define this mapping as:

\[R^1 \rightarrow R^n\]

An example of such curve would be:

\[r(t) = <t^2, t>\]


First and second derivative of the curve (and their interpretation)

Finding first and second derivative of a given curve should be fairly simple since we just differentiate each component separately. For instance, given \(r(t) = <t^2, t>\), its first derivative would be:

\[r'(t) = <2t, 1>\]

and similarly, its second derivative would be:

\[r''(t) = <2, 0>\]

Now, if you imagine that the curve models movement of some object in time, then:

  • first derivative of the given curve can be interpretted as velocity
  • second derivative of the given curve can be interpretted as acceleration

Recall that:

  • velocity defines rate and direction of objects’ movement while speed only defines rate of the movement. One can obtain speed of a given object by simply computing the size of the velocity vector: \(\text{speed} = \left\|r^{\prime}(t)\right\|\).

  • acceleration defines rate of change a velocity of a given object


Arc length

Now, in practice, we might also be interested in computing how long the given curve is on a given interval \([a, b]\). This can be done by integrating over the speed of the given object within the given interval:

\[\int_a^b\left\|r^{\prime}(t)\right\| d t\]

⛳️ Learning goals checklist

In this session, we have built on the foundations laid in the previous one. More specifically, we introduced the concept of a curve and how it can help us model different physical phenomena such as velocity, speed and acceleration of given object in time. Thus, after the session you should be able to:

  • explain what a curve is and how it is mathematically defined
  • compute velocity, speed and acceleration given some function \(r(t)\) which defines a movement of a given object
  • compute arc lengthk of given curve within the specified interval

If you feel shaky in any of these concepts, I suggest you check exercises for this week. With this being said, I conclude the session, see you next week when we will talk about partial derivatives - essential topic in machine learning!