I've been thinking about the nature of motion and have a couple of related questions.
1) If we can think of an angle in terms of rotations, how can we analogously think of solid angles? Is there some $3$ dimensional analog of rotation?
2) We can seemingly decompose motion into two different types: linear motion and rotational motion. Linear motion is described very well by a line (a vector really). Rotational motion is described very well by a magnitude and the plane in which the rotation occurs (if we happen to be talking about a plane in $\Bbb R^3$ then we can also relate this to a normal vector, but I'm really more concerned with motion in $\Bbb R^n$). Is there some type of motion that is described well by a 3-space and a magnitude? Or is every type of motion characterized by linear and rotational parts?
1) In differential geometry integral curvature $ \int \int K d A $ is defined for surfaces. It is the non-dimensional property of 2D surface areas embedded in 3-space.For a spherical segment it equals $ 2 \pi h/R $ where h is the height of spherical segment and R radius of sphere.
2a) A curved line in 3-space has curvature and torsion. Torsion vanishes for curves confined to a plane. Vectors $ T,N,B $ describe the motion by Frenet-Serret differential relations.These geometrical quantities are related to their velocity and acceleration dynamical counter parts.
2b) If you wish to further study space curves on 2D surfaces, you need extra scalar curvatures geodesic curvature,geodesic torsion and normal curvature, $ k_g, \tau_g,k_n $ respectively.They are described by generalized Frenet-Serret relations.