THE BOOK--Playing The Percentages In Baseball

If I remember my math from college correctly (hey I’m only 2 years removed, so I should):

If you could express it as a function (or two) starting at the origin with the foul line as your x-axis, you could just take the area underneath the curve (via integral) and that would be your area. If I wasn’t at work I’d probably be able to do the work.

That would be in the ballpark, so to speak. The park is a quarter-circle. The average radius would be about 371.25. Pi * 371.25^2 / 4 = approximately 108,248 square feet.

I trialed-and-errored my way last night to the “
close to” 370 figure (actually 371.25, which I guess you got the same way I did as 330+405+375*2, divided by 4).

I’m wondering if you skew it alot, like drop it to 300 down the line, and bump it to 435 in center, if that will “cancel out”, as my quick equation noted above would suggest.

It’s not a circle since the radius varies.

The radius r(t) needs to satisfy:
r(0) = 330
r(pi/8) = 375
r(pi/4) = 405
r(3pi/8) = 375
r(pi/2) = 330

And you want r to be differentiable, so the fence is smooth. Of course there are many functions that will work. One is the polynomial:

r(t) = (30/pi^4) (11 pi^4 + 4 pi^3 z + 120 pi^2 z^2 - 512 pi z^3 + 512 z^4)

This one seems like a pretty good fit. It gives an average radius of 372.

To get the area you just integrate r(t)^2/2 from 0 to Pi/2, which gives an area of about 109,191 square feet.

Changing it to 300 down the line and 435 in center doesn’t change the area too much, as you infer. I get an area of about 110,064 square feet with those changes.

As a quick rule of thumb, using the average radius makes sense. But I think a ballpark with a curved outfield wall would have more of a shape like an ellipe than a circle. If you take the distance to center field as the long axis, figure the angle it is between CF and either gap, and use that to help you determine area, you’d get some more accuracy in your calculation. This way would account for the skew scenario as well.

Another thought: there are a number of ball parks that have straight walls (Coors and Citizens Bank comes to mind), and your could parcel those up into triangular segments and calculate that way too.

The really accurate answer is difficult, mainly because ‘smooth’ might well fall apart, probably right at the 405 sign.

That said, here’s what i’d do:

Find the center of the circle containing the foul pole, ‘the gap’ and CF on half the field.  There will be such a circle.  Based on that, determine the angle between FP and CF, and then the area of the segment of the circle between them.  Double that and then add and subtract a bunch of triangles to account for overlap between the two circle segments and the parts of the field that don’t fall into them.

If this is still of interest at lunchtime EST, i’ll take care of it.  I’m assuming that by ‘smooth’ we mean ‘circular’, but if you don’t make that assumption (or one like it), you’ve really got no way to determine anything.

Also, like i said, there’s probably a non-smooth point right at the 405, where the left and right circles meet, but we can just put a mat up against the wall there so no one gets hurt.

By smooth I mean something elliptical.

An area of 109,191 for a baseball field is the same as a quarter circle with a radius of 372.9 feet.

Dropping down the line 30 feet, while moving CF back 30 feet, to match the 110,064 figure would give me a quarter circle with r=374.3 feet.

So, roughly similar.

Click on my name to see a plot of the function I picked, i.e. what the baseball field would look like (from top down). I think it’s a very good choice, and satisfies all the desired conditions (smooth, and the distances are correct).

While on the topic of stadium size, does any know if someone has determined the amount of area in play in foul territory for MLB stadiums.  I had done some work on stadium information (size, park factors, elevation and temp effects) and would like to include this information (http://66.37.156.70/Ballpark_info.xls).

If not could there be a way to see the number foul pop outs per stadium?

If we get a good/general formula for area, I could include it also.

Jeff, here’s a general formula for area:
pi/11340 * (936 CF^2 + 146 L^2 + 896 LC^2 + 56 LC R + 146 R^2 + 256 LC RC + 296 R RC + 896 RC^2 + L (296 LC - 29 R + 56 RC) - 6 CF (29 L + 64 LC + 29 R + 64 RC))

L, LC, CF, RC, R = distance to left, left-center, center, right-center, right (resp.)

- sigh - You crazy kids and your curve-fitting.  I’m taking my straight-edge and compass and going home…

Andy—on most Minor League Parks I am missing some data (i.e. I only have LF, RF and CF).  Can I just remove the LC and RC parts in the equation?

Without those it becomes much simpler:

pi/60 * (8 CF^2 + 2 L^2 + 2 R^2 + 2 CF(L + R) - LR)

Andy—Thanks, I will plug it into the sheet and see if there are any other “unique” cases.

Andy, great stuff!  Much appreciated.

I presume for the non-alley numbers, you are presupposing what the alley numbers would be?  That is, if the standard is 330/405 for down-the-line and straight away, and 375 is the alley, then if you have a minor league park of say 320/415, then your equation is inferring some reasonable number for the alley so you get a reasonable shape?

Damn you Tango, I was thinking about this when I should have been paying attention in my actual math class this morning!  I tried to work it out assuming that the outfield wall was actually a segment of a circle (which is NOT the same as assuming that the entire field is simply a quadrant of a circle).  This comes down to the fun problem of computing the area of the part of a disk cut out by a chord when all you know is the length of the cord and the maximal length from the cord to the boundary of the disk.

I convinced myself that my method works, but I didn’t work through all the details, since they are going to be messy, involving a bunch of trig functions.  Andy’s method, which assumes that the outfield wall is a quartic polynomial instead of a segment of a circle, is very likely to give almost the same result.  Still, if people are interested I can work out the general formula under the circular hypothesis.

Yes, it infers some reasonable numbers, but it doesn’t really assume that the shape will be the same (although it’s not going to differ by a huge amount).

If you wanted the same shape you could just interpolate the alley numbers, e.g.
LC = L + (CF - L) * (3/5)

and then put them into the full equation. This might be a better idea if minor league parks have about the same shape as major league parks.

I suspect for cases like Fenway and Minute Maid Park (or whatever it’s called these days) we’d need to construct separate models for more accuracy, due to things like the Green Monster not being curved and the triangle in center or that hill in Houston.

Andy—One more case (Lansing Field in Casper Wyoming) has L, LC, RC, and R but no C.  Do you have an equation to figure C in this case.  After this, I promise no more requests.

Here is the Google Maps image of Lansing Field, in case that helps with an idea of how the park is shaped:

http://maps.google.com/maps?f=q&hl=en&sll=42.681104,-84.525088&sspn=0.374547,0.747757&ll=42.860528,-106.329428&spn=0.003012,0.00788&t=h&z=18

Wow, we have lots of really smart readers!

BTW, if you have a pretty good to-scale image of the park, or you can draw one, you can use a screen tracing program like Iconico’s Screen Tracing Paper (I use it) to compute the area.  It should be pretty accurate.

Jeff, I use that (Screen Tracing) program and the diagrams from Clem’s Baseball site to compute the foul area of all the major league parks. If you want that info, email me or post a link or email address and I’ll send it.

MGL - Can you send me the information to wydiyd at hotmail dot com?  Thanks.

I believe the most accurate way would be to do it with a computer program that uses the outlay of the stadium or field or whatever it is you are measuring and place it inside a square, rectangle, circle, some object of known area.  Have the computer program generate millions of random points and determine how many points lie within the stadium/field and how many lie outside of it.  All points must be inside the shape of choice of course.  If the area of your shape (s,r,c) has an area of 100 units and 73% of your points fall within your baseball field, then you know the area of your baseball field is 73 units.  The more points you choose, the more accurate the measurement.  And yes there are programs/software that can do this, and no I don’t own any of them.
vr, Xei

What Xei says is good for irregularly shaped objects, like, say Canada, or Italy.  For something as smoothed as a baseball field, I don’t see that’s necessary.

In any case, my interest goes beyond current ballparks and so, I won’t necessarily have ready images of the ballparks.

Tango #25

Are you thinking Far East/Latin American parks, or future MLB parks?  There are scale diagrams available for most parks that are under construction or even still in concept phase, you just have to know where to look.  Although the latter may change…

Greg, I was thinking of older MLB parks.

A stadium like PETCO might also cause some problems with these formulas (I think, I don’t really know for sure), since it has virtually no curved walls. But then again, you cold just break it up into about 6 triangles and get the exact answer by hand.

http://www.petcoparkinsider.com/images/petco-park-seating-chart.gif

I’ve always wanted to add total playing area to my ballparks database, but it was missing for so many, I haven’t.  These formulas should give me enough to approximate the areas.

Then there’s foul territory, which I want do add also, but is almost impossible to calculate for older parks.  I may just go with “small, medium, large” in the database for foul territory.

I’m no math wizard, but might it be as simple to think of the field as a cone and finding the total curved surface area of the cone and then halving that?

In other words, Pi x Radius x S / 2 (where S = 330)?

Or am I way off base?

I assumed the outfield walls were straight instead of curved and that the foul line and left/right field wall made 90 degree angles.  Then you can solve the problem exactly.  It gives an effective radius of 364 ft.

I had some moron ask me this question in my first interview out of college.  He believed it to be 1/4 of a circle (pi X r2/4) which I nervously disagreed with as I have never seen a professional park that measures that way. Needless to say he angrily cut my interview short.  I would love to run into this idiot again 15 years of experience under my belt.

The fields for the LLWS are a uniform 225 feet, an exact quarter circle.  Any other field, no… Maybe you’re better off not having to try to make a boss like that feel smart…

Out of curiosity, I used my fence position data (which I have at 1 degree increments for each park), and modeled each field as a group of 90 1-degree wide triangles with apex at home plate, and altitude equal to the average distance from home plate.  The area of such a triangle works out to A = d^2*tan(0.5 degrees), where d is the average distance to the fence in that 1 degree “wedge”.  here are the areas I get (for fair territory mind you):

Park Area (sq. ft)
Rangers Ballpark in Arlington 110276
Angels Stadium 113246
AT&T Park 110681
Busch Stadium 111741
Citizens Bank Park 104644
Comerica Park 114049
Coors Field 117858
Chase Field 112843
Dolphins Stadium 108976
Fenway Park 104187
Great American Ball Park 104724
Jacobs Field 104343
Kauffman Stadium 116473
McAfee Coliseum 107905
Metrodome 107271
Safeco Field 107704
PNC Park 110579
PETCO Park 110895
Miller Park 107191
New Yankee Stadium 107233
Minute Maid Park 106430
U.S. Cellular Field 105495
Turner Field 112988
Tropicana Field 105600
Rogers Centre 109105
Wrigley Field 108160
Dodger Stadium 106988
Citi Field 114655
Oriole Park at Camden Yards 106843
Nationals Park 107718

If I get around to adding more historical parks to HT, I’ll figure those out too…

I took Greg’s data, and converted it to the “equivalent radius” of a circle.  In effect, if the ballpark was (one-fourth) circular, what would be the radius that would match his square footage.  Or, similarly, and in real lay terms, what is the average distance to the fence.

Fence Park Area (sq. ft)
387 117858 Coors Field
385 116473 Kauffman Stadium
382 114655 Citi Field
381 114049 Comerica Park
380 113246 Angels Stadium
379 112988 Turner Field
379 112843 Chase Field
377 111741 Busch Stadium
376 110895 PETCO Park
375 110681 AT&T Park
375 110579 PNC Park
375 110276 Rangers Ballpark in Arlington
373 109105 Rogers Centre
372 108976 Dolphins Stadium
371 108160 Wrigley Field
371 107905 McAfee Coliseum
370 107718 Nationals Park
370 107704 Safeco Field
370 107271 Metrodome
370 107233 New Yankee Stadium
369 107191 Miller Park
369 106988 Dodger Stadium
369 106843 Oriole Park at Camden Yards
368 106430 Minute Maid Park
367 105600 Tropicana Field
366 105495 U.S. Cellular Field
365 104724 Great American Ball Park
365 104644 Citizens Bank Park
364 104343 Jacobs Field
364 104187 Fenway Park

In Excel terms:
=SQRT(4*[mycell]/PI())