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Can we define differential equation and parametrization of a smooth continuous line on a unit ball with symmetry about two orthogonal axes ? And also have same geodesic curvature $k_g$ at four equidistant points on the curve on the two planes of symmetry? With a relation between curvature and torsion of the space curve seam?

Can this space curve be unique?

Imagine the seams of a common baseball or tennis ball that divides surface area into two equal parts with two identical/congruent regions. However it has discontinuities at four locations... with curvature jumps at join of four small circles... these should translate to inflection points in the smooth curve sought.

Looking for a relation between their curvature and torsion..

From an earlier question here we have a relation

$$ \theta = (\pi/3)\sin 2\phi $$

that had been suggested (by @ minopret). It appears by visual inspection to me that not all $k_g$ are equal in magnitude and moreover there are six points of maximum geodesic curvature, not four.

Thanks for all thoughts and hope would it be some fun too.

BaseballSeam

Narasimham
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  • Can add Mathematica code if interested. – Narasimham May 11 '19 at 16:14
  • could you add the code and the reference to generate the 3d seam, please? – Feras Oct 27 '22 at 16:03
  • Very sorry, lost it 3 yrs ago seeing no responses. Coded minopret's relation once again now, but it does not look alright. – Narasimham Nov 01 '22 at 16:56
  • @Feras: So I took up the problem just now afresh, got an improved symmetrical path with azimuth correction (second answer below) but still needs better formulation. Shall post it if it improves after some more path adjustments. – Narasimham Nov 02 '22 at 13:24

1 Answers1

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Ode of baseball seams taken as ( ball radius a, latitude $~\phi$, geodesic curvature $~\kappa_g ~ $ )

$$ a ~\kappa_g= \sin (2\phi) $$

Almost there with trial and error on azimuth, but however the orbit needs some minor tweaking.

enter image description here

Narasimham
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