Steiner minimal trees on zig-zag lines

A Steiner minimal tree for a given set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of points in the Euclidean plane is a shortest network interconnecting <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose vertex set may include some additional points. The construction of Steiner minimal trees has been proved to be an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N upper P"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">NP</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-complete problem for general <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. However, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N upper P"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">NP</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-completeness does not exclude the possibility that Steiner trees for sets of points with special structures can be efficiently determined. In this paper we determine the Steiner mimmal trees for zig-zag lines with certain regularity properties. We also give an explicit formula for the length of such a tree.