This academic paper outlines various approaches to identifying the ideal racing line of an arbitrary race track.
On page 10, the problem is formulated, and the idea is to minimize total lap time. Expressed as minimizing $\int \frac{1}{v} ds$ where $s$ is the curvilinear distance travelled along the track, and $v$ is the velocity of the car. It has the following constraints ($k$ is the curvature at a specific point).
My question is about the second constraint: $\int kds = \Delta \theta$ as it makes no sense to me. The paper describes the constraint as such:
Ultimately I have 3 main questions:
- Why would $\theta$ be used to parameterize a curve and not something like a $t$ value with a point on the curve being defined as $(x(t),y(t))$?
- How is $\theta$ defined? I.e. what is the angle based off of?
- And finally, what is the point of this constraint?
I'm not sure if the constraints are paper-specific, or if there is some mathematical reasoning behind it.