Vol. 104, No. 2, 1983

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Double tangent ball embeddings of curves in E3

Lowell Duane Loveland

Vol. 104 (1983), No. 2, 391–399

An arc or curve J in E3 has congruent double tangent balls if there exists a positive number δ such that for each p J, there are two three-dimensional balls B and B, each with radius δ, such that {p} = B B= (B B) J. Such an arc or simple closed curve is shown to be tamely embedded in E3. An example is given to show that the “uniform” radii are required for this conclusion and to show the necessity of having two tangent balls at each point rather than just one. The proof applies as well to show that any subset of E3 having these congruent double tangent balls must locally lie on a tame 2-sphere.

Mathematical Subject Classification 2000
Primary: 57N12
Secondary: 57N45
Received: 27 April 1981
Revised: 26 August 1981
Published: 1 February 1983
Lowell Duane Loveland