Spherical helix

SPHERICAL HELIX

case q = 5/2, k » 0,56, slope » 75%
case q = 2/5, k » 0,17, slope » 25%

Notion studied by H.J. Jonas in 1905 and W. Blaschke.
See Loria 3d pp. 84 and 160.

Cartesian parametrization: .
i.e. with , , t(new)=(1-k)t.

The spherical helices are the helices, i.e. the curves with constant slope with respect to a given plane P, traced on a sphere.

It can be proved that they are the curves described by a point on a great circle of a sphere rolling without slipping on a fixed circle of the sphere, parallel to the plane P; therefore, they are special cases of spherical cycloids, as well as satellite curves; they have cuspidal points located on the fixed circle and its symmetrical image with respect to the center of the sphere.

Here, R is the radius of the sphere, O its center, r = k R the radius of the fixed circle; the constant slope is equal to

The second parametrization above shows that the projections on the plane of the fixed circle are the epicycloids with parameter q defined by

; therefore, spherical helices are spherical lifts of epicycloids.

The spherical helices are also the involutes of cones of revolution (loci of a point of a plane rolling without slipping on the cone); the above helix is an involute of the cone of revolution containing the two rolling circles.

Do not mistake these curves for the rhumb lines, the tangents of which form a constant angle, not with a plane, but with the meridians. Do not mistake them either for the clelias.
Spherical helix with 10% slope; it looks like a rhumb line, but as opposed to the latter, the extreme points are not asymptotic points.

See also the curves of constant precession, the indicatrices of curvature of which are spherical helices.

Model of spherical helix obtained as an involute of a cone.
It is a helix that makes one turn between two cuspidal points, therefore it is the case q = 1 (the horizontal projection is a cardioid); hence: circle at the summit with radius R/3, slope . This staircase on a storage sphere has constant-size steps, and therefore follows a spherical helix.
It is a helix that makes a half-turn between two cuspidal points, therefore it is the case q = 2 (the horizontal projection is a nephroid); hence: circle at the summit with radius R/2, slope .

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Model of spherical helix obtained as an involute of a cone. It is a helix that makes one turn between two cuspidal points, therefore it is the case q = 1 (the horizontal projection is a cardioid); hence: circle at the summit with radius R/3, slope .	This staircase on a storage sphere has constant-size steps, and therefore follows a spherical helix. It is a helix that makes a half-turn between two cuspidal points, therefore it is the case q = 2 (the horizontal projection is a nephroid); hence: circle at the summit with radius R/2, slope .