What is the best shape for a soccer goal post?

[kshitij]

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?

 

[kshitij]

I just want to visualise the probability, saying that "this is the probability when the path of the ball is such that $d$ and $\theta$ are uniformly distributed over $(-D,D)$ and $(0,\pi)$ respectively" doesn't give me any hint about what is happening physically. Saying that "this is the probability when the path of the ball is always intersected in a semicircular area of radius $D$" gives a much better understanding of how the ball is hit.

 

[haruspex]

But only $d$ and $\theta$ are variable, what do you mean by 3 degrees of freedom?

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.

 

[kshitij]

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.

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

 

[haruspex]

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

Perpendicular to a point? What does that mean?
Why can't I pick any point in the semicircle and any angle from that point that's in the 180 degree range?

 

[kshitij]

Perpendicular to a point? What does that mean?

Sorry I meant, perpendicular to the line joining that point and origin

 

[kshitij]

Why can't I pick any point in the semicircle and any angle from that point that's in the 180 degree range?

For every point you pick we can define a unique line which is perpendicular to the line joining that point and origin

 

[kshitij]

I was trying to define the equation of the line in parametric form with a given slope and given distance (perpendicular distance) from origin

 

[haruspex]

I was trying to define the equation of the line in parametric form with a given slope and given distance (perpendicular distance) from origin

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.

 

[kshitij]

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.

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

 

[haruspex]

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

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.

 

[haruspex]

Given that S=R, the only effective difference between square and round is corner A in thread post #105, the one where the side facing the pitch meets the side facing the goalmouth.
For a shot taken from wide of the goal, it creates an obstruction, reducing the chance of a goal; for a shot taken from in front of the goal it acts as a sweeper, helping to deflect the ball into the goal.
Since the posts are relatively narrow, it is very reasonable to take d as uniformly distributed, so the major question is whether the striker is more likely to be wide of the goal or in front of it. If wide, round makes the task easier, if in front, square wins.

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.

 

[kshitij]

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.

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 $\leq D$

But now I see that all possible paths that intersect the semi-circle of radius $D$ would not include some paths of the ball so now I say that instead of the semicircle, let us assume a complete circle.

So, I believe that now we can say that the probability that we calculated is,

"all possible paths of the ball (straight lines) that intersect a circle of radius $D$ has a probability $P(square)=\dfrac{D\pi-2S}{2D\pi}=P(round)=\dfrac{D\pi-2R}{2D\pi}$ of going in the back of the net if the other post is at a very large distance to the left of the post whose dimensions we considered"

Edit: the paths of the ball are bidirectional, i.e., for any path that intersects the circle, the direction of the shot is assumed such that it doesn't come from behind the goal line.

Edit II: the distance of paths of the ball from origin is uniformly distributed in $(-D,D)$

 

[haruspex]

all possible paths of the ball (straight lines)

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.

 

[kshitij]

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.

That is, add the assumption that $d$ is uniformly distributed in $(-D,D)$ then it should be fine right?

 

[kshitij]

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.

u/gegenpressing91 on reddit does some beautiful illustrations like the one below,

Here we clearly see that for wider shots, he did aim to the opposite post so your assumption is right

 

[haruspex]

That is, add the assumption that $d$ is uniformly distributed in $(-D,D)$ then it should be fine right?

Yes.

 

[kshitij]

Yes.

I would like to thank you for your patient and insightful responses throughout this thread and the other one

As mentioned earlier this was not any homework question of significance but rather something I casually made up on my own and yet you were committed to help me throughout this which I think is incredible!