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 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.
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