Construct Control Meshes of Helicoids over Trapezium Domain

In this paper, we present a geometric construction of control meshes of helicoids over trapezium domain. We first introduce the quasi-Bézier basis in the space spanned by {1, t, cos t, sin t, t sin t, t cos t}, with t ∈ [0, α], α ∈ [0, 2π). We denote the curves expressed by the quasi-Bézier basis as algebraic-trigonometric Bézier curves, for short ATBézier curves. Then we find out the transform matrices between the quasi-Bézier basis and {1, t, cos t, sin t, t sin t, t cos t}. Finally, we present the control mesh representation of the helicoids and the geometric construction of the control mesh. In detail, we construct the control polygon of the planar Archimedean solenoid, which is also expressed with the quasi-Bézier basis, and then generate the mesh vertices by translating points of the control polygon.