Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm designing a game where objects have to move along a series of waypoints. The object has a speed and a maximum turn rate. When moving between points p1 and p2 it will move in a circular curve like so:

enter image description here

Angle a is the angle the object must rotate through. Therefore we can calculate the time it will take to rotate through this angle as t = a / turn rate.

However, if the object's current speed means it will cover distance d in quicker than this time, it must slow down on this corner or it will miss p2. I need to calculate the maximum speed it can take this corner at, which means calculating the distance of the curve d (so I can calculate the maximum corner speed as d / t).

I think I've figured out b = 2a, but to determine d I need to know the radius of the circle. How can I determine it from this information?

share|improve this question
    
Do we know the distance between $p_1$ and $p_2$? –  awllower Jan 24 '13 at 14:28
    
Yes, they're both known. –  AshleysBrain Jan 24 '13 at 14:28
    
Since we know $b$, we could find the radius by noting that $p_1 p_2 b$ forms an equilateral triangle. Thus, denoting the distance between $p_1$ and $p_2$ by $\Pi$, we find the radius to be $\rho = \Pi *arcsin(a)/2$. –  awllower Jan 24 '13 at 14:31
1  
Note that looking at your diagram carefully, (and the arrows showing the direction of movement at the start and end of the arc) the angle you have marked as $a$ is only one half of the total angle the object has to turn through. –  Shard Jan 24 '13 at 20:52
    
@awllower: I think you mean $1/\sin a$ where you say $\arcsin a$. It doesn't make sense to take the arcsine of an angle. –  Rahul Jan 25 '13 at 8:16

1 Answer 1

[Sorry, figure-fu misplaced]

The angle between the line $P_1 P_2$ and the radius is $\frac{\pi}{2} - \alpha$, where $\alpha$ is the angle you seem to call $a$. By symmetry, the angle at $P_2$ is the same, so the angle $\beta$ at the center is just $2 \alpha$. By the law of cosines, if $d$ is the linear distance between $P_1$ and $P_2$: $$ d^2 = 2 r^2 (1 - \cos (2 \alpha)) $$ I'm sure this can be simplified further...

share|improve this answer
    
I think it would be polite to use the same variable names that the asker used. $d$ is already the length of the arc, so let's call the straight-line length of the chord $c$. Then we have $c^2=2r^2(1-\cos2a)=4r^2\sin^2a$, so $r=c/(2\sin a)$. –  Rahul Jan 25 '13 at 8:14
    
Sorry about that. –  vonbrand Jan 25 '13 at 8:15

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.