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.