Describe the what and how of a design to determine order of cars on a race track among other things using geometry

Question

The figure above is representative of the tracks used in a racing game. The track is 2 dimensional, has uniform width and does not cross over itself (i.e. no “figure 8’s”). A race includes up to 4 drivers completing 3 laps from the Start Line driving clockwise. Cars can drive anywhere on the track and cannot drive off the track.

Using words, diagrams, math formulas etc. provide a concise geometrically based design that addresses the items below. Describe both the what and the how of your design.

1. Describe how to determine the accurate order of the 4 drivers as the race is in progress
2. Describe how to determine if a driver is moving forward or backward
3. Describe how to determine if two cars have collided and what the various consequences would be

Answer 1

We need a coordinate system for reference, so that the position of each car can be uniquely defined. The track can be simplified by using a circle with a circumference equal to the length of the track. The track has a fixed width W and lengths L₁ (inner lap length) and L₀ (outer lap length). If R is the radius of the inner side track then R+W is the length of the outer side of the track. L₁=2πR, so R=L₁/2π and L₀=2π(R+W)=2π(L₁/2π+W). The radial position of a car on the track will be R<r<R+W. If the start position is when θ=0 then, since there are 3 laps the position of a car will be (r,θ) where θ is measured clockwise from the start position and 0≤θ≤6π. If the cars are labelled A, B, C, D then their respective positions can be represented by P_A(r_A,θ_A), P_B(r_B,θ_B), P_C(r_C,θ_C), P_D(r_D,θ_D).

(1) The order is determined by θ. If car A is ahead of car B, θ_A> θ_B, bearing in mind that 0≤θ≤6π. If two cars have the same θ value but different r values they are running neck and neck. If they have the same sinθ and cosθ values but different r values, they are on different laps and are running side by side. But see (3).

(2) The rate of change of θ (ω=dθ/dt) is the angular velocity. When positive the car is moving forwards, when zero, it's stationary, and when negative it's moving backwards, bearing in mind that θ is measured in a clockwise direction. There is no instantaneous change of direction, so to go from moving backwards to moving forwards, or vice versa, there must be a period, however brief, when dθ/dt=0. (Possibly a collision condition.)

(3) A collision occurs if the r value of the cars is the same and sinθ and cosθ the same for each car (at least two cars). Other pairs of trig ratios can be used.

The consequences of a collision are:

(a) The cars involved all come to a full stop (dθ/dt=0 for all cars involved). These cars would be considered to be out of the race.

(b) No cars come to a full stop but only to a temporary stop, then continue forwards (dθ/dt>0) or backwards (dθ/dt<0). The temporary stop may be different for each car involved. See also (2).

(c) Some cars come to a full stop (and are out of the race) while the other(s) continue (dθ/dt≠0).