- #141
kshitij
- 218
- 27
But only ##d## and ##\theta## are variable, what do you mean by 3 degrees of freedom?That's different again. And it's messier because now there are three degrees of freedom to integrate over instead of two.
But only ##d## and ##\theta## are variable, what do you mean by 3 degrees of freedom?That's different again. And it's messier because now there are three degrees of freedom to integrate over instead of two.
If the ball is coming at angle theta from a uniform distribution of points in the semicircle then there are two degrees of freedom just to specify the point.But only ##d## and ##\theta## are variable, what do you mean by 3 degrees of freedom?
Yes, but isn't it correct that for a given point in that semicircular area, there is only one path possible that passes through that point and is perpendicular to the originIf the ball is coming at angle theta from a uniform distribution of points in the semicircle then there are two degrees of freedom just to specify the point.
Perpendicular to a point? What does that mean?Yes, but isn't it correct that for a given point in that semicircular area, there is only one path possible that passes through that point and is perpendicular to the origin
Sorry I meant, perpendicular to the line joining that point and originPerpendicular to a point? What does that mean?
For every point you pick we can define a unique line which is perpendicular to the line joining that point and originWhy can't I pick any point in the semicircle and any angle from that point that's in the 180 degree range?
If you pick a point from a uniform distribution over the semicircle and an angle uniformly distributed ##(0,\pi)## then look at the distance d that path is from the origin you will not get a uniform distribution for d in (-D,D), so the probability of a goal will be different from what we previously calculated.I was trying to define the equation of the line in parametric form with a given slope and given distance (perpendicular distance) from origin
the distance of the point from origin will also be the distance of the line from origin as the line is perpendicular to the line joining origin and that point so you don't need to look at ##d## again since you already fixed that while picking the pointIf you pick a point from a uniform distribution over the semicircle and an angle uniformly distributed ##(0,\pi)## then look at the distance d that path is from the origin you will not get a uniform distribution for d in (-D,D), so the probability of a goal will be different from what we previously calculated.
So you are saying the point you are picking in the semicircle is the point of closest approach to the origin.the distance of the point from origin will also be the distance of the line from origin as the line is perpendicular to the line joining origin and that point so you don't need to look at ##d## again since you already fixed that while picking the point
What I wanted to do was to include all the paths of the ball that we allowed in our probability calculation, i.e., all possible straight lines whose perpendicular distance from origin is ##\leq D##So you are saying the point you are picking in the semicircle is the point of closest approach to the origin.
That's not what you wrote in post #139. There it was the point the shot was taken from.
You cannot have it both ways. For some legal shots the point of closest approach to the origin would be behind the goal line.
But that does not specify the probability distribution. To use the result obtained earlier you need to specify that the distance from the path to the origin is uniformly distributed over the possible values.all possible paths of the ball (straight lines)
That is, add the assumption that ##d## is uniformly distributed in ##(-D,D)## then it should be fine right?But that does not specify the probability distribution. To use the result obtained earlier you need to specify that the distance from the path to the origin is uniformly distributed over the possible values.
u/gegenpressing91 on reddit does some beautiful illustrations like the one below,Edit: in practice, a striker wide of the goalmouth might aim for the far end of the goal in order to avoid the keeper. In that case square wins again.
That is, add the assumption that ##d## is uniformly distributed in ##(-D,D)## then it should be fine right?
I would like to thank you for your patient and insightful responses throughout this thread and the other oneYes.