# $K = 0$ and $K = \text{const}$ surfaces produce $k_g = \text{constant}$ intersections?

Generally speaking, when do constant K (Gauss curvature) and zero K surfaces intersect

to produce lines of constant geodesic curvature $k_g$ ? Small circles on a sphere

are examples.

Or more specifically the question is:

How should a plane be bent and relatively positioned in order to intersect a constant K surface along a constant $k_g$ line ?

EDIT1: A more complicated example with a cone type pseudosphere and a geodesic circle:

EDIT2: A hyperbolic sphere (with cuspidal edge not shown) cut by an almost flat bent cylinder:

In both cases constant $k_g$ geodesic circles are seen rotated around axis of symmetry.

EDIT3:

MY BACKGROUND CONJECTURE:

"A suitably bent right circular cylinder intersects constant Gauss curvature surfaces in $R^3$ along lines of constant geodesic curvature.

It remains to disprove my conjecture and, if validated define what that bending is.

EDIT4: In the particular images above generators of cylinder are perpendicular to symmetry axis of revolution.

EDIT5:

I suppose this edit would make it easier to define case of moving red ring: The ring is made to slide on surface of revolution of K as a special case.

Let a thin parallel circle of ring be mounted on the surface. Let us imagine it is made of an inextensible but flexible material.It can be cut and dislocated to any arbitrary latitude,cut ends repaired to be again glued back elsewhere on the surface.

Perimeter length and $k_g$ do not change after such a transportation. Find cordinates of this oval on an axial plane when one axis of the oval is projected on the axial plane as a function of the new arbitrary latitude.

• I think there are some interesting things here, but you have to be more precise with your question. For example, I'm not sure what you mean by the phrase "how can a plane be bent and relatively positioned." Maybe you could phrase your question in the form: "Given [certain things], does there exist [a specific thing you want] satisfying [exact conditions you want]?" – Jesse Madnick Oct 9 '14 at 8:57
• @ Alright, let me define then a simpler scenario in EDIT5. – Narasimham Oct 9 '14 at 20:03
• Are "ring" and "thin parallel circle of ring" and "oval" all referring to the same thing, namely a simple closed curve with constant $k_g$? Is it obvious that on every surface of revolution with $K$ constant that there actually exists such a curve at all? By "thin" do you mean "$1$-dimensional," because to me a "circle" is something which has only one dimension, and hence exactly zero thickness. And by "axial plane" do you mean "tangent plane" or "normal plane" or "rectifying plane," or something else entirely? And to me, every "plane" is completely flat, without curving or bending. – Jesse Madnick Oct 10 '14 at 4:36
• I don't mean to be difficult, but as the question is phrased, I wouldn't have the slightest idea of how to begin. I imagine that many other mathematicians might have a similar reaction, but I hope I'm wrong; hopefully others are better able to interpret and answer your question. Best of luck! – Jesse Madnick Oct 10 '14 at 4:37