Consider a standard sine curve.
It is periodic with a period of 2pi, and each period has two, for lack
of a better term, lobes, one concave down, one concave up. Revolving a
lobe about the x-axis results in a somewhat lemon-shaped solid, and
revolving the whole sine curve about the x-axis forms a beaded strand of
these solids. Alternatively, revolving a lobe about a vertical line at a
odd multiple of pi/2 results in a bowl shape.
let's consider the lobe from x = 0 to x = pi.
The derivative of sin(x) is cos(x), and evaluating cos(x) at x = 0 and x
= pi, we find the tangents of sin(x) at those values to have slopes of
+1 and -1 respectively. This implies that the tangents intersect in a
right angle, and that four lobes could be joined endpoint-to-endpoint to
form a smooth, convex closed curve.
This curve would have the symmetry of a square, and like a square, could
form two different solids of revolution, one analogous to a bicone, the
other analogous to a cylinder.
But perhaps the more interesting solids are ones where three of these
curves, placed in mutually orthogonal planes form a skeleton of sorts.
this can be done in two ways that result in octahedral symmetry: the
curves intersect where the lobes are joined, or they intersect at the
crests of the lobes... comparing to squares again, this is analogous to
either three orthogonal squares intersecting along the lines through
opposite edge midpoints or intersecting along their diagonals.
One could simply take the convex hull of these skeletal forms, but I
think the more interesting construction would be to create a stack of
similar curves atop one of the curves where the joints or the crests
trace the other two curves... similar to building up a sphere from three
orthogonal circles and stacking smaller circles atop one circle while
following the circumferences of the other two... though, would the
resulting solid have octahedral symmetry still?
And can the amplitude and wavelength of a sine curve be manipulated such
that lobes can be smoothly joined to form curves analogous to other
regular polygons or even star polygons?
I've titled this musing "sinoids", and while sinoid doesn't appear to be
defined in the mathematical literature, I'm not sure if it's the plane
curves or the solids that the term would better fit.