I would like to calculate the angle of a corner in an old snes video game. I am convinced the tracks do not loop back into the start of the track and I would like to prove this by creating some software to display what the tracks really look like compared to what the game says they look like.
I can write the software, but I dont know the math to determine the angle of a corner.
I do have a theory for how to figure it out. Let me know your thoughts. The theory goes.. drive the car at 100mph in the inside lane of a corner. When you hit the corner, time how long it takes the car to move into the other lane. Since lanes on highways in the usa are 12 feet (3.7m) and we know the speed and direction of the car, it should be possible to calculate the curve of the corner.. right?
Yes, it's possible. Say the radius of the curvature is $R$, and that you are 12 feet away from the far side of the road when you start the cornering. If your velocity is $v$, and the time is $t$, you move $vt$ in a straight line. If you draw this on a circle, and the angle difference between when you start and where you exit the road is $\alpha$, you have $$vt=R\sin\alpha\\12=R-R\cos\alpha$$ In the second equation you move the terms to have only $\cos\alpha$ term on one side, then you square the equations $$v^2t^2=R^2\sin^2\alpha\\(R-12)^2=R^2\cos^2\alpha$$ When you add these equations, you can use $\sin^2\alpha+\cos^2\alpha=1$ to get $$R=\frac{v^2t^2+144}{24}$$