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Here is my interpretation of Anton's idea to capture the ray. I found it almost impossible to illustrate; I could only show three sections of the tube, separated by two rotations, the first counterclockwise the second clockwise. (The actual turn angles I selected are irrelevant.)
Tube Turns

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Here is my interpretation of Anton's idea to capture the ray. I found it almost impossible to illustrate; I could only show three sections of the tube, separated by two rotations, the first counterclockwise the second clockwise. (The actual turn angles I selected are irrelevant.)
Tube Turns