Define a circle as a geometric circle of positive, finite radius. I am interested in classifying surfaces $S$ embedded in $\mathbb{R}^3$ in two categories:

1. Define $S$ to be a circled surface if every point of $S$ is contained in a circle that lies in $S$. (I choose the name "circled" in analogy with ruled surfaces.) Thus, $S$ is a union of circles; it is covered by circles.
2. Define $S$ to be a hoop surface if it may be partitioned into circles, i.e., it is a disjoint union of circles (hoops). Every point of $S$ is contained in a unique circle that lies in $S$.

A specialization would be to restrict the circles in either class to be congruent, unit-radius circles. Generalizations include: (a) relaxing embedded to immersed; (b) replacing circles by spheres in $\mathbb{R}^d$; (c) replacing geometric circles with topological circles.

Setting aside these variants, let me list the hoop surfaces I know:
Tubular Spline http://cs.smith.edu/%7Eorourke/MathOverflow/TubularSpline.jpg

1. Tubular surfaces, topological cylinders, either infinite in both directions, or with a circular boundary at one or both ends. These are sometimes called tubular splines in computer graphics.
2. Tubular surfaces capped at either or both ends but with one point missing from the cap.
3. Torus surfaces; genus 1.
4. Topological disks (composed of nested circles) with one interior point removed.
5. A surface homemorphic to the plane with one point removed.

Is this list complete?

Every hoop surface is a circled surface, but planes and spheres are circled surfaces. I cannot see how to form a circled surface of genus greater than 1.

Have these concepts been explored before, and if so, under what names? My searches have been unsuccessful. Thanks for any ideas or literature pointers!