Re: [math-fun] What is the tennis-ball curve?
On 2013-11-09 12:38, Fred Lunnon wrote:
Good point about the stretch issue. Increasing the radius r makes the Enneper-sphere curve progressively "waisted" in a more traditional fashion. Via symmetry, the angular distance across the boundary at the inflections must always equal pi/2 ; so perhaps the optimal curve should have distance pi/2 across the waist as well, whence r^2 - 5 sqrt2 r + 2 = 0 , r = 0.2951635567... See https://www.dropbox.com/s/7pi27klhpdt6yxk/tennis_ball_2952.jpg
I'm afraid I do not understand how Bill's onepiececover.png maps to tennis.gif :
Neither do I. Try gosper.org/tennis2.gif ?
surely the semicircular ends of the strip should be replaced by copies of the central section, bisected by a line across the cusps?
Fred Lunnon
Brent M> An interesting idea (although novelty isn't what you want in a tennis
ball). The symmetry is different so it might have an[ e]ffect on the aerodynamics depending on the spin axis.
Brent Meeker If anybody complains, just add more fuzz. --rwg
On 11/9/13, Bill Gosper <billgosper@gmail.com> wrote:
On 2013-11-08 18:38, Fred Lunnon wrote:
For arbitrary sphere radius r , the curve has four nodal singularities where x = 0 or y = 0 : these are real and obvious where the curve intersects itself when r > 1 , but safely complex and out of harm's way when r < 1 . Rather unexpectedly, there are four more real singularities at the inflections where z = 0 and x = y : these correspond to a pair of lines on the Enneper surface which superficially appear innocuous.
There is a theorem that any TB curve must have (at least) four inflections, see http://www.qedcat.com/archive/165.html --- it intrigues me that these points turn out to be so special here.
Despite getting down and dirty with Magma's algebraic geometry feature, I haven't yet managed to decide whether there are further singularities. [Don't even think about trying to understand what schemes are --- just scope the examples, then hack them!]
Having gone this far, I couldn't resist putting the r ~ 0.2307718797455473 curve onto a sphere. OK, I know it looks boring --- that's the whole point! https://www.dropbox.com/s/ypsqc07tisw2gf2/tennis_ball.jpg
Fred Lunnon
That's a pretty curve, but it accommodates a large inscribed circle, which requires large stretching to "map" onto the sphere. Minimizing stretching might motivate those racetrack shaped pieces we've seen. Recall years ago when the subject was the baseball stitch, we came up with *one piece* covers of arbitrarily low stretch, based on sphericons and fruit peels. gosper.org/onepiececover.png , gosper.org/tennis.gif It seems to me someone should manufacture tennis balls like this, just for the novelty. --rwg
There is admittedly a limit to the amount of novelty additional fuzz can compensate: gosper.org/tennis3.gif --rwg On Sat, Nov 9, 2013 at 8:21 PM, Bill Gosper <billgosper@gmail.com> wrote:
On 2013-11-09 12:38, Fred Lunnon wrote:
Good point about the stretch issue. Increasing the radius r makes the Enneper-sphere curve progressively "waisted" in a more traditional fashion. Via symmetry, the angular distance across the boundary at the inflections must always equal pi/2 ; so perhaps the optimal curve should have distance pi/2 across the waist as well, whence r^2 - 5 sqrt2 r + 2 = 0 , r = 0.2951635567... See https://www.dropbox.com/s/7pi27klhpdt6yxk/tennis_ball_2952.jpg
I'm afraid I do not understand how Bill's onepiececover.png maps to tennis.gif :
Neither do I. Try gosper.org/tennis2.gif ?
surely the semicircular ends of the strip should be replaced by copies of the central section, bisected by a line across the cusps?
Fred Lunnon
Brent M> An interesting idea (although novelty isn't what you want in a tennis
ball). The symmetry is different so it might have an[ e]ffect on the aerodynamics depending on the spin axis.
Brent Meeker If anybody complains, just add more fuzz. --rwg [...]
Having gone this far, I couldn't resist putting the r ~ 0.2307718797455473 curve onto a sphere. OK, I know it looks boring --- that's the whole point! https://www.dropbox.com/s/ypsqc07tisw2gf2/tennis_ball.jpg
Fred Lunnon
That's a pretty curve, but it accommodates a large inscribed circle, which requires large stretching to "map" onto the sphere. Minimizing stretching might motivate those racetrack shaped pieces we've seen. Recall years ago when the subject was the baseball stitch, we came up with *one piece* covers of arbitrarily low stretch, based on sphericons and fruit peels. gosper.org/onepiececover.png , gosper.org/tennis.gif It seems to me someone should manufacture tennis balls like this, just for the novelty. --rwg
OK. And tennis3.gif is an "orange-peel" curve, which did seem to me a more natural choice anyway. WFL On 11/10/13, Bill Gosper <billgosper@gmail.com> wrote:
There is admittedly a limit to the amount of novelty additional fuzz can compensate: gosper.org/tennis3.gif --rwg
On Sat, Nov 9, 2013 at 8:21 PM, Bill Gosper <billgosper@gmail.com> wrote:
On 2013-11-09 12:38, Fred Lunnon wrote:
Good point about the stretch issue. Increasing the radius r makes the Enneper-sphere curve progressively "waisted" in a more traditional fashion. Via symmetry, the angular distance across the boundary at the inflections must always equal pi/2 ; so perhaps the optimal curve should have distance pi/2 across the waist as well, whence r^2 - 5 sqrt2 r + 2 = 0 , r = 0.2951635567... See https://www.dropbox.com/s/7pi27klhpdt6yxk/tennis_ball_2952.jpg
I'm afraid I do not understand how Bill's onepiececover.png maps to tennis.gif :
Neither do I. Try gosper.org/tennis2.gif ?
surely the semicircular ends of the strip should be replaced by copies of the central section, bisected by a line across the cusps?
Fred Lunnon
Brent M> An interesting idea (although novelty isn't what you want in a tennis
ball). The symmetry is different so it might have an[ e]ffect on the aerodynamics depending on the spin axis.
Brent Meeker If anybody complains, just add more fuzz. --rwg [...]
Having gone this far, I couldn't resist putting the r ~ 0.2307718797455473 curve onto a sphere. OK, I know it looks boring --- that's the whole point! https://www.dropbox.com/s/ypsqc07tisw2gf2/tennis_ball.jpg
Fred Lunnon
That's a pretty curve, but it accommodates a large inscribed circle, which requires large stretching to "map" onto the sphere. Minimizing stretching might motivate those racetrack shaped pieces we've seen. Recall years ago when the subject was the baseball stitch, we came up with *one piece* covers of arbitrarily low stretch, based on sphericons and fruit peels. gosper.org/onepiececover.png , gosper.org/tennis.gif It seems to me someone should manufacture tennis balls like this, just for the novelty. --rwg
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'Ang abaht though --- Brent Meeker's comment is relevant here. All three of Bill's suggestions are enantiomorphic: the choice of spiral sense will surely influence their spin behaviour asymmetrically. WFL On 11/10/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
OK. And tennis3.gif is an "orange-peel" curve, which did seem to me a more natural choice anyway. WFL
On 11/10/13, Bill Gosper <billgosper@gmail.com> wrote:
There is admittedly a limit to the amount of novelty additional fuzz can compensate: gosper.org/tennis3.gif --rwg
On Sat, Nov 9, 2013 at 8:21 PM, Bill Gosper <billgosper@gmail.com> wrote:
On 2013-11-09 12:38, Fred Lunnon wrote:
Good point about the stretch issue. Increasing the radius r makes the Enneper-sphere curve progressively "waisted" in a more traditional fashion. Via symmetry, the angular distance across the boundary at the inflections must always equal pi/2 ; so perhaps the optimal curve should have distance pi/2 across the waist as well, whence r^2 - 5 sqrt2 r + 2 = 0 , r = 0.2951635567... See https://www.dropbox.com/s/7pi27klhpdt6yxk/tennis_ball_2952.jpg
I'm afraid I do not understand how Bill's onepiececover.png maps to tennis.gif :
Neither do I. Try gosper.org/tennis2.gif ?
surely the semicircular ends of the strip should be replaced by copies of the central section, bisected by a line across the cusps?
Fred Lunnon
Brent M> An interesting idea (although novelty isn't what you want in a tennis
ball). The symmetry is different so it might have an[ e]ffect on the aerodynamics depending on the spin axis.
Brent Meeker If anybody complains, just add more fuzz. --rwg [...]
Having gone this far, I couldn't resist putting the r ~ 0.2307718797455473 curve onto a sphere. OK, I know it looks boring --- that's the whole point! https://www.dropbox.com/s/ypsqc07tisw2gf2/tennis_ball.jpg
Fred Lunnon
That's a pretty curve, but it accommodates a large inscribed circle, which requires large stretching to "map" onto the sphere. Minimizing stretching might motivate those racetrack shaped pieces we've seen. Recall years ago when the subject was the baseball stitch, we came up with *one piece* covers of arbitrarily low stretch, based on sphericons and fruit peels. gosper.org/onepiececover.png , gosper.org/tennis.gif It seems to me someone should manufacture tennis balls like this, just for the novelty. --rwg
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participants (2)
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Bill Gosper -
Fred Lunnon