Harmonic Bézier Surfaces Passing Given Geodesic

Minimal surfaces and geodesics are two important kinds of elements in the field of architecture geometry, and the harmonic surface is often considered as a linear approximation of minimal surface.In this paper, by the combination of geodesic and harmonic surface, we propose a new method which generates harmonic Bézier surfaces determined by agiven geodesic.We originally prove that the control points on the first and second layers of the control mesh are the only factors that determine the shape of a certain harmonic Bézier surface.With this property of harmonic surfaces, we further prove that the shape of harmonic Bézier surface through a given geodesic is entirely determined by the given geodesic and two control points on opposite end of the second layer.At the end of this paper, the effectiveness of the proposed method is illustrated by several examples with data analysis.The proposed construction method is derived from the geometric properties of geodesic and harmonic surfaces, which is valuable for the design of tension membrane structure.