# Search results for query: *

1. ### What is the best shape for a soccer goal post?

I would like to thank you for your patient and insightful responses throughout this thread and the other one :smile: 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...
2. ### What is the best shape for a soccer goal post?

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
3. ### What is the best shape for a soccer goal post?

That is, add the assumption that ##d## is uniformly distributed in ##(-D,D)## then it should be fine right?
4. ### What is the best shape for a soccer goal post?

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...
5. ### What is the best shape for a soccer goal post?

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
6. ### What is the best shape for a soccer goal post?

I was trying to define the equation of the line in parametric form with a given slope and given distance (perpendicular distance) from origin
7. ### What is the best shape for a soccer goal post?

For every point you pick we can define a unique line which is perpendicular to the line joining that point and origin
8. ### What is the best shape for a soccer goal post?

Sorry I meant, perpendicular to the line joining that point and origin
9. ### What is the best shape for a soccer goal post?

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
10. ### What is the best shape for a soccer goal post?

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...
11. ### What is the best shape for a soccer goal post?

But only ##d## and ##\theta## are variable, what do you mean by 3 degrees of freedom?
12. ### What is the best shape for a soccer goal post?

I think that you assumed that a shot is hit from a point on the arc of a semicircle of radius ##D## but I meant that a shot hit from any point in the area of the semicircle of radius ##D##, because now we see that ##\theta## is uniformly distributed in ##(0,\pi)## and ##d## (displacement form...
13. ### What is the best shape for a soccer goal post?

I didn't get how you did this? Also if that is not how we can think of this probability then what is the physical meaning of the probability that we calculated?
14. ### What is the best shape for a soccer goal post?

Okay, I think now I understand (but still it doesn't feel right) Anyway, let me ask one final thing, for a given post round or square, ##\dfrac{D\pi-2S}{2D\pi}=\dfrac{D\pi-2R}{2D\pi}## is the probability of a shot that is hit within the semicircular area of radius ##D## going in goal assuming...
15. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

Oh yes, I see it now, If we did the first fold horizontally (and the horizontal fold should be in opposite direction to the unfolding otherwise we will end up with what we started with) instead of vertically then the circles won't be linked and thus that is not what is happening in inverting a...
16. ### What is the best shape for a soccer goal post?

Why is the model I'm talking about a subset of your model? for ##R \gt d \gt -R## aren't both the models equivalent?
17. ### What is the best shape for a soccer goal post?

I am thinking with a different model, consider all possible shots that hit the post, now the probability that a shot hits the sector AB is ##\frac14##, right? the whole post is divided into 4 quarters and hitting anyone of these quarter is equally likely? correct? Now back to your model, we...
18. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

Well first I learned that making a cylinder is relatively easier but making a toroid is way too difficult, the paper doesn't want to change its surface's curvature and IIRC there is a theorem like that by Gauss (??). Anyway, to the conclusion, the blue circle are red circle are not linked like...
19. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

I think that the linkage should be different in both cases but I should check it myself
20. ### What is the best shape for a soccer goal post?

Also, I have another doubt, Say if I set the value of ##D=R## in the round post, then it should mean that now I'm considering only the shots that hit the round post, so now I get the probability of a shot hitting the round post going in as ##P(round)=\dfrac{D\pi-2R}{2D\pi}=0.18## But that...
21. ### What is the best shape for a soccer goal post?

I need a numerical value of ##D## because that will give us a number, i.e., if ##D=R=6cm## then we get 18% chance as calculated in post #124 instead of getting just that the probability in square and round post are equal, we get that probability in square and round post are equal which is equal...
22. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

After stitching it back I cannot believe how the red circle will still be linked with the blue one?
23. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

As I said before that now it makes sense why the circles won't be linked after we stitch it back the other way because we are literally cutting both the circle such that they changed from a circle to a line, but the cutting isn't allowed
24. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

But why is cutting the strip allowed? wouldn't that change the shape we are dealing with?
25. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

I was thinking about this a lot and even though I don't know how it will happen but I think that it should happen, i.e., the red and blue circles shouldn't be linked after inverting the tube inside out. This is because, the red circle which was initially outside the tube, after flipping has to...
26. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

Wait, can we cut it such that we get a rectangle? doesn't that break any rules? is a toroid with single hole isomorphic to rectangle? I agree that with can cut it in both ways that you mentioned, i.e., cutting it such that we get a short fat cylinder and cutting it such that we get a long...
27. ### What is the best shape for a soccer goal post?

Just tell me whether my this conclusion is correct or not?
28. ### What is the best shape for a soccer goal post?

But our selection depends on what value we choose for ##D## (see post #124) and if we chose ##D## to be large enough then it would also include the cases where the ball didn't hit the post. Isn't setting the same constant value of ##D## for the probability calculation of both posts a like to...
29. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

So that means that we cannot increase the length of the hole such that the tube is converted to a strip? Because by this method it does completely makes sense why the red circle passed through the blue one (because we literally cut and stitched the red circle in the process) But yes if that is...
30. ### What is the best shape for a soccer goal post?

Again I can't understand what you mean here, how am I changing the distribution of angles? And why can't this be used to compare the two post shapes? Let me state the whole method as I understand it. Now, I need diagrams and figures to understand pretty much everything so I will now talk w.r.t...
31. ### What is the best shape for a soccer goal post?

*correction, ##P(square)=\dfrac{D\pi-2S}{2D\pi}## ##P(round)=\dfrac{D\pi-2R}{2D\pi}## Now for ##D \to \infty##, we get ##P(square)=P(round)=\dfrac{1}{2}=0.5##
32. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

So it should look somewhat like this,
33. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

Okay so here is my second attempt I use the same process as earlier that is I increase the size of the hole such that the tube is now a strip, and the hole is increased such that blue circle remains intact but the red circle is cut into a line, now if we flip this strip and stitch it back...
34. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

Oh I get it now! (hopefully)
35. ### What is the best shape for a soccer goal post?

Oh yes I forgot the ##\pi## in the numerator but now the probability is 50% right?
36. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

That means that you painted a circle along the outer circumference (because its the long way round) of the tube but on the inside surface of the tube And this means that you painted the circle on the inside circumference of the tube but this time on the outside surface
37. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

What?? How are they linked like a chain? I thought that those two circles are like rings one surrounded by the other and there is a layer of rubber in between them separating each other
38. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

I don't have a proof but I have heard that changing the size of a hole in any shape doesn't change it homeomorphically (idk if that's the word) that is why a ring is homomorphic to a cylinder or a disc is homomorphic to a cone or a cup

40. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

By those two circles being linked do you mean that they are concentric?? If so, then to answer your problem I think maybe that first we extend that small hole we made so that we can slit the tube (torus) along its length (along its circumference) such that now instead of a tube we are left with...
41. ### What is the best shape for a soccer goal post?

So ##D## is a large constant and ##d## belongs to ##(S\sqrt 2\cos(\frac{\pi}4-\theta),D)## for square post and for round post it lies between ##(R\cos(\frac \theta 2),D)## So still, when ##D \to \infty## it should surely give us the probability when the path of the ball can lie anywhere in the...
42. ### What is the best shape for a soccer goal post?

Well I'm too confused so let me state what I think I understood then you correct me where I'm wrong. ##d## is the distance of a given path of the ball from origin and origin is the center of one of the posts (right post from our point of view and left from the ball's) and the maximum value of...
43. ### What is the best shape for a soccer goal post?

I realized that 33% and 25% where the probabilities of a shot going in after hitting the post whereas in your method the ball doesn't need to hit the post, but now this confuses me even more! that means 15% is the probability for any shot (no matter where it is hit in the 2D plane) to go in! So...
44. ### What is the best shape for a soccer goal post?

And if we take ##D=7.32m## which is equal to the distance between the two posts and ##R=S=6cm## according to the laws of the game, we get, ##P(square)=P(round)=0.1516%## Which still doesn't make any sense to me, I mean Probability for the square ones should be close to 33% and for the round...
45. ### What is the best shape for a soccer goal post?

I evaluated your integral for ##\theta \in (0,\pi)## and got ##P(square)=\dfrac{D-2S}{2D\pi}## ##P(round)=\dfrac{D-2R}{2D\pi}## Now for ##D \to \infty##, we get ##P(square)=P(round)=\dfrac{1}{2\pi}=0.1591%## regardless of what the dimensions of the square and round posts are as long as they...
46. ### How to find centroid of a hemisphere using Pappus's centroid theorem?

I don't understand what you said there about understanding Pappus theorem, but I just wanted to find the area/volume enclosed by a hollow hemisphere, solid hemisphere, hollow cone and a solid cone on rotating these about their base, i.e., how did you get Volume of 4D ball = ##\frac 12\pi^2r^4##...
47. ### What is the best shape for a soccer goal post?

I updated the graphic for the circular post case, now you don't have to move two points separately to get all possible paths you can get it by moving only point B in the link below, https://www.geogebra.org/calculator/khj7d6hk
48. ### What is the best shape for a soccer goal post?

That is so clever! First we calculated favourable cases for line whose slope is fixed and then we fixed the distance of that line from origin then calculated favourable cases for all possible values of slope! But if we consider the maximum value of ##D## to be ##\infty## (i.e. the incident...
49. ### What is the best shape for a soccer goal post?

According to what I understood reading post #91, I similarly created a graphic for the circular post as well!
50. ### What is the best shape for a soccer goal post?

Now finally I understand what you said here! But now how did you get this? And also, please assume ##D=S\sqrt 2=R\sqrt 2## in further calculations because the diameter of the round post is equal to side length of square post according to the laws of the game