I'm trying to find an arc between two points that maps a path between them given some set distance. I'm not a math major so pardon if I make some inaccurate statements below. I'm trying to figure out the movement of a robot.
I have point A(0,0) and point B(10, 0). The shortest path would simply be a straight line of distance 10. However, I know the given distance travelled is 15. Therefore, I assume there exists a way to determine the shortest arc between the two points (in the positive xy-axis). Perhaps the word arc may be misleading. It can be any path from point A to point B as long as the path is of the given length 15. Is there a way to find an equation for the domain of possible paths?
Any ideas would be most appreciated. Thank you.
So let me phrase it differently. You want to find the circle arc between $(0,0)$ and $(10,0)$ such that the length of the arc between those two points is $15$.
This is something you could do: The equation of a circle with center in $(a,b)$ (not to be confused with your points $A,B$) and radius $R$ $$ (x-a)^2+(y-b)^2=R^2 $$ In particular, your points have to satisfy this equation, i.e. if I replace $x=0,y=0$ and $x=10,y=0$ the equation holds. That gives you that $a=5$, and $a^2+b^2=R^2$
Now to impose the length condition note that the length of your path is $R\theta$ where $\theta$ is the angle traveled, so you want to impose: $$ R\theta =15 $$
To recapitulate we have the two equations $$ R\theta=15\\ 5^2+b^2=R^2 $$ To solve for the three variables $b,R,\theta$. The equation we are missing simply comes from trigonometry. It gives: $$ \sin (\theta/2)=5/R $$ (I could make a drawing if you don't see it, but I encourage to try to get it by yourself).
You can now solve for $b,R$ to get the equation of the circle.