It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle subtended by the arc is $>2\pi$). Is there a name for higher-order polynomials mapped onto the surface of a sphere?
Could this be related to geodesics? Does there exist a generalization of geodesics parallel to the polynomial generalization of lines? I'd also appreciate some pointers on where to look for more information on the idea of doing flat math on the surface of a sphere: e.g., if you choose a point on the surface of a sphere and define some arbitrary orthonormal basis set to create 2d a coordinate system on the surface, then you can follow some set of rules for how that coordinate system changes as you plot out a function. Sure the coordinate system will collide with itself, but that's fine as long as any point of a function knows what its own local coordinate frame is. Does that description even make sense?
Edit to clarify mapping:
In Euclidean space, you can define a quadratic spline segment with two linear segments that share an endpoint. If we parametrize the spline with $t$ and have points $P(0)$, $K(.5)$, and $P(1)$, then $P(t) = (1-t)^2P(0) + 2t(1-t)K(.5) + t^2P(1)$ describes a quadratic curve that interpolates $P(0)$ and $P(1)$ and has first derivatives defined by $K(.5)$.
I can define an arc on the unit sphere with two orthonormal vectors and an angle (so that I can define arcs subtending $\pi$ radians). If those vectors are $A(0)$ and $B(0)$ with angle $\theta$, then my "first order spherical spline is defined as $Q(t) = \cos(t\theta)A(0) + \sin(t\theta)B(0)$. I can use the same sort of approach as used to make a quadratic spline from two linear splines by setting $A(1) = Q(1)$ with $B(1)$ and $\theta_2$ as free parameters to define the curve (although $B(1)$ must be orthogonal to $A(1)$). There's an extra degree of freedom in the spherical case that must be dealt with (how many times the $A(0),B(0)$ plane flips around as it interpolates onto the $A(1),B(1)$ plane), but the mapping is pretty straightforward. So it would seem that this is a quadratic curve on a spherical surface. It looks cool. Does it have a name?
I can find references to Bernstein-Bezier polynomials on a sphere, but this seems to be something different.
In this example, $A(0)$ is red, $B(0)$ (usually covered) is pink, $A(1)$ is green, $B(1)$ is cyan, the final endpoint is blue. Yellow and orange represent the plane through which $A(0),B(0)$ rotate to interpolate onto $A(1),B(1)$. I change the curve parameters while it rotates.