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October 25

What shape is a (gridiron) football ?

Mathematical approximation, that is. I'm thinking a rotation of some type of conic section about the central axis ? SinisterLefty (talk) 02:27, 25 October 2019 (UTC)[reply]

It's commonly described as a prolate spheroid, which is the rotation of an ellipse around its longer axis. CodeTalker (talk) 02:46, 25 October 2019 (UTC)[reply]

Thanks. How about an American football, which is more pointed at the ends ? SinisterLefty (talk) 05:18, 25 October 2019 (UTC)[reply]

It's not any shape with a mathematical name, not precisely anyway. If you want an approximation with a pithy mathematical name, I don't think you're going to do better than "prolate spheroid". Fight for Harvard's glorious name; won't it be peachy if we win the game? --Trovatore (talk) 05:35, 25 October 2019 (UTC)[reply]

How about if you draw an ellipse, offset the major axis on both sides, trim out the portion of the ellipse between the two offset lines, bring the two remaining portions of the ellipse together, then rotate those about the axis ? With suitable ellipse dimensions and offsets, wouldn't that provide a good approximation of an American football ? SinisterLefty (talk) 05:47, 25 October 2019 (UTC)[reply]

Well, that would have a sharp point, whereas a football has a dull point, and in fact turns inward in a sort of "dimple" in the spot where your figure would have the sharp point.

But sure, if you give me enough parameters to play with, I can probably fit a football pretty well; that isn't surprising. It isn't going to have a catchy mathematical-sounding name, though. It'll just be a way to describe a rotationally symmetric shape. --Trovatore (talk) 06:00, 25 October 2019 (UTC)[reply]

Since we're speaking of ideals: do NFL icons have enough resolution to tell whether the balls are pointy or blunt? —Tamfang (talk) 18:55, 25 October 2019 (UTC)[reply]

It can be inconsistent. For example, around the logo of the New York Jets there has almost always been a football-shaped outline (different from the actual football on the logo). Back in the 1960s and 1970s it came to a point. The version used since about 2000 is much more rounded. As far as the actual NFL "shield" logo, you can see versions of it throughout history here, and again it seems to vary from rounded to pointy. --Jayron 32 15:41, 31 October 2019 (UTC)[reply]

Rotation of a Lens (geometry) —Tamfang (talk) 18:52, 25 October 2019 (UTC)[reply]

That would mean the cross section of the football is two circular arcs. Is that correct ? SinisterLefty (talk) 13:02, 27 October 2019 (UTC)[reply]

Approximately? Sure. Take a circular arc that meets the ball at the midpoint (on a seam or a panel? you decide) with the same tangent line and the same curvature. That will fit the profile, with an error proportional to the cube of the distance to the tangent point.

You could get the same error exponent by fitting it to a parabola. With an ellipse or a hyperbola, you have an extra parameter, so maybe you could get the fourth power, if agreement near the tangent point is what you're trying to match.

Which it probably isn't; maybe you really want to minimize the integral of the square of the error. Again, the more parameters you have, the closer you can get.

But the football isn't really any of these shapes; you're just fitting the shape to the ball, to optimize some figure of merit. --Trovatore (talk) 01:09, 28 October 2019 (UTC)[reply]

The actual ball is constructed from several pieces of leather that are cut to a specific shape, and then stitched together. So, it would seem that there might be some way to describe the topology of a set of flat pieces described by a particular two-dimensional shape, and then stuffed, extending (embedding?) these surface pieces into three dimensions, with additional piecewise connectivity constraints at the seams. In the real world, surfaces can flex and bend, while an idealized topological surface should be rigid, or at least should be describable by some deformation metric (stretch factor?). As Trovatore has pointed out, if we permit a large number of parameters, we can fit parametric curves to the leather pieces; and we can characterize the deformation using matrices or tensors or something like that.

I think even with this intuition, there is still no "pithy" mathematical word for it.

From the viewpoint of computer graphic modeling, such a surface could be represented using a NURBS surface - specifically because this allows a mathematically-inclined artist to specify smoothness of the surface, and of its first-, second-, and higher-order derivatives (in other words, they can build smooth, well-defined object models that have "pointy" parts). The NURBS surface is one of many ways that we can formally describe a highly-parameterized model (with as many parameters as we want), using a standard format that is well-understood by many 3D scientific and artistic software tools.

Nimur (talk) 19:06, 28 October 2019 (UTC)[reply]

The pieces of leather are not flat; though. They have to have some native curvature of their own; a three-dimensional curved object cannot be cut into purely flat shapes; this is the map/globe problem and why map projections always introduce a distortion. Since the football has curvature, the individual pieces need to maintain that curvature when cut apart. There is a sort of "law of conservation of curvature" that is a corollary of sorts to the Theorema Egregium. --Jayron 32 12:33, 30 October 2019 (UTC)[reply]

I believe the way "around" this problem is to take specific advantage of the piecewise discontinuity of the surface (and its curvature) at the seams - allowing perfectly-flat pieces to connect together if they are carefully cut to a specific shape. This is the key insight - the piecewise smoothness of the Bezier spline and its related functions - which makes them so useful in engineering. The ball has surface continuity along the line of the seam; but that line also defines a discontinuity in the surface curvature, which means both sides can be "flat" in their own coordinate system. I have to do some thinking to determine whether the perfect idealized geometric version of the surface requires any stretching or deformation during assembly - of course, in the real world, we might not notice a minuscule stretch of the material, or a tiny flexure that allows it to curve - but mathematically, that would mean some type of differential change to the surface-element. This is a great thought-experiment - I'll let you know if I get a chance to figure it out! Nimur (talk) 14:50, 30 October 2019 (UTC)[reply]

a three-dimensional curved object cannot be cut into purely flat shapes Most people would agree that a cylinder is curved (despite having zero Gaussian curvature). --JBL (talk) 21:34, 30 October 2019 (UTC)[reply]

Yes, sorry, thanks for the clarification. Objects with a non-zero Gaussian curvature cannot be reduced to flat shapes without stretching or distorting those shapes. American footballs do, however, have a non-zero Gaussian curvature. --Jayron 32 12:37, 31 October 2019 (UTC)[reply]

John Urschel might know. j/k 173.228.123.207 (talk) 01:38, 27 October 2019 (UTC)[reply]

Thanks all. SinisterLefty (talk) 19:43, 31 October 2019 (UTC)[reply]