Re: [math-fun] What is the tennis-ball curve?
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
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The Enneper-sphere tennis-ball curve is smooth, being the intersection of two algebraic surfaces. However it lacks a rational parameterisation, since (according to Maple) the plane curve |[x, y, z]|^2 = r^2 , qua function of parameters u,v , has genus = 8 rather than 0 .
In practice such considerations are irrelevant, since a template needs to be computed only once, and to working tolerance.
To "unroll" a strip of the corresponding spherical region onto a plane template requires a decision to be made about the appropriate projection. There doesn't appear to be a canonical answer to the latter question: it depends upon how much the cover material can be expected to stretch across its central line of symmetry, as opposed to wrinkling up along its boundary.
Fred Lunnon
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Customary typo correction --- should have read
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +0.1849108100, -0.1380711874] , [0, -0.1849108100, -0.1380711874] , [+0.1849108100, 0, +0.1380711874] , [-0.1849108100, 0, +0.1380711874] ; so the cuboid boxing the curve is somewhat flattened.
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Update on Enneper-sphere intersections:
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +/- 0.1849108100, +/- 0.1380711874] , so the cuboid boxing the curve is somewhat flattened.
For the extreme waisted curve with tacnodes at its tips, dividing the sphere into four teardrop regions, radius 1 turns out to be exact.
Note that the usual parameterisation for Enneper's surface scales all coordinates involved up by a factor 3 .
Magma script and Maple graphic are available on request.
Fred Lunnon
On 11/4/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I'd always casually assumed that the tennis-ball / baseball curve was probably some simple space quartic that everybody but me knew about. But since it seems this is not the case, I'll put in my two-penn'orth.
The classic minimal Enneper's surface intersects concentric spheres in a family of such curves. With the parameterisation in the Wikipedia page, a sphere of small radius meets it in an approximate circle; a sphere of radius (approx?) 1/4 meets it in a typical tennis-ball curve, with expected symmetry and arcs parallel at the extremities; a sphere of radius (approx?) 1 meets it in a curve with touching arcs; for larger radius the curve has 4 self-intersections.
See http://www.indiana.edu/~minimal/maze/enneper.html http://en.wikipedia.org/wiki/Enneper_surface
The degree of Enneper's surface equals 9, so presumably these curves have degree 18.
Fred Lunnon
On 11/3/13, Henry Baker <hbaker1@pipeline.com> wrote:
Is a tennis ball seam the same shape as a baseball seam?
http://math.arizona.edu/~rbt/baseball.PDF
"Designing a Baseball Cover"
Richard B. Thompson
College Mathematics Journal, Jan. 1998.
At 09:09 AM 11/3/2013, rkg wrote: >Dear funsters, > A tennis-ball appears to be made from 2 congruent >pieces of material, seamed together in a curve. Are possible >equations to the curve known? I'd like a smooth algebraic >equation, probably of degree 4, and preferably with a maximum >number of rational points on it. A first approximation might >be to take a sphere of radius root(3) and centre at (0,0,0) >and take the 8 great circle arcs (1,1,1) to (-1,1,1) to >(-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), >(1,1,-1) and back to (1,1,1). However, this isn't smooth >at the 8 corners of the cube, and I think that it doesn't >even partition the sphere into two congruent pieces. > > Is this well-known to those who well know it? What >do the tennis-ball manufacturers do? R.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 11/9/2013 4:10 AM, Bill Gosper wrote:
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.
An interesting idea (although novelty isn't what you want in a tennis ball). The symmetry is different so it might have and affect on the aerodynamics depending on the spin axis. Brent Meeker
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 : 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 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
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The Enneper-sphere tennis-ball curve is smooth, being the intersection of two algebraic surfaces. However it lacks a rational parameterisation, since (according to Maple) the plane curve |[x, y, z]|^2 = r^2 , qua function of parameters u,v , has genus = 8 rather than 0 .
In practice such considerations are irrelevant, since a template needs to be computed only once, and to working tolerance.
To "unroll" a strip of the corresponding spherical region onto a plane template requires a decision to be made about the appropriate projection. There doesn't appear to be a canonical answer to the latter question: it depends upon how much the cover material can be expected to stretch across its central line of symmetry, as opposed to wrinkling up along its boundary.
Fred Lunnon
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Customary typo correction --- should have read
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +0.1849108100, -0.1380711874] , [0, -0.1849108100, -0.1380711874] , [+0.1849108100, 0, +0.1380711874] , [-0.1849108100, 0, +0.1380711874] ; so the cuboid boxing the curve is somewhat flattened.
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Update on Enneper-sphere intersections:
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +/- 0.1849108100, +/- 0.1380711874] , so the cuboid boxing the curve is somewhat flattened.
For the extreme waisted curve with tacnodes at its tips, dividing the sphere into four teardrop regions, radius 1 turns out to be exact.
Note that the usual parameterisation for Enneper's surface scales all coordinates involved up by a factor 3 .
Magma script and Maple graphic are available on request.
Fred Lunnon
On 11/4/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I'd always casually assumed that the tennis-ball / baseball curve was probably some simple space quartic that everybody but me knew about. But since it seems this is not the case, I'll put in my two-penn'orth.
The classic minimal Enneper's surface intersects concentric spheres in a family of such curves. With the parameterisation in the Wikipedia page, a sphere of small radius meets it in an approximate circle; a sphere of radius (approx?) 1/4 meets it in a typical tennis-ball curve, with expected symmetry and arcs parallel at the extremities; a sphere of radius (approx?) 1 meets it in a curve with touching arcs; for larger radius the curve has 4 self-intersections.
See http://www.indiana.edu/~minimal/maze/enneper.html http://en.wikipedia.org/wiki/Enneper_surface
The degree of Enneper's surface equals 9, so presumably these curves have degree 18.
Fred Lunnon
On 11/3/13, Henry Baker <hbaker1@pipeline.com> wrote: > Is a tennis ball seam the same shape as a baseball seam? > > http://math.arizona.edu/~rbt/baseball.PDF > > "Designing a Baseball Cover" > > Richard B. Thompson > > College Mathematics Journal, Jan. 1998. > > At 09:09 AM 11/3/2013, rkg wrote: >>Dear funsters, >> A tennis-ball appears to be made from 2 congruent >>pieces of material, seamed together in a curve. Are possible >>equations to the curve known? I'd like a smooth algebraic >>equation, probably of degree 4, and preferably with a maximum >>number of rational points on it. A first approximation might >>be to take a sphere of radius root(3) and centre at (0,0,0) >>and take the 8 great circle arcs (1,1,1) to (-1,1,1) to >>(-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), >>(1,1,-1) and back to (1,1,1). However, this isn't smooth >>at the 8 corners of the cube, and I think that it doesn't >>even partition the sphere into two congruent pieces. >> >> Is this well-known to those who well know it? What >>do the tennis-ball manufacturers do? R. > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Bill Gosper -
Fred Lunnon -
meekerdb