A method for interpolating scattered data based upon a minimum norm network

A method for interpolating scattered data is described. Given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript i Baseline comma y Subscript i Baseline comma z Subscript i Baseline right-parenthesis comma i equals 1 comma ellipsis comma upper N"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">({x_i},{y_i},{z_i}),i = 1, \ldots , N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a bivariate function <italic>S</italic> with continuous first order partial derivatives is defined which has the property that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis x Subscript i Baseline comma y Subscript i Baseline right-parenthesis equals z Subscript i Baseline comma i equals 1 comma ellipsis comma upper N"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S({x_i},{y_i}) = {z_i},i = 1, \ldots ,N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The method is based upon a triangulation of the domain and a curve network which has certain minimum pseudonorm properties. Algorithms and examples are included.