In this paper, we consider smooth, closed, strictly convex curves contained inside the unit circle $\mathbb{T}$ and their associated curves of geodesic centers (or dual curves). We obtain a formula for the boundary and envelope of the union of the region bounded by the geodesic circles associated with our curves. This description includes a formula for the boundary of the hyperbolic convex hull of points identified by finite Blaschke products. We describe a setting and conditions under which one can produce an infinite chain of ellipses such that each ellipse is inscribed in a convex polygon that is itself inscribed in another polygon (a chain of Poncelet ellipses). We apply these results to operators that are compressions of the shift operator on model spaces and describe necessary and sufficient conditions for the numerical range to contain 0.

在本文中，我们考虑包含在单位圆内的光滑、闭合、严格凸曲线 $\mathbb{吨}$及其相关的测地线中心曲线（或双曲线）。我们获得了由与我们的曲线相关联的测地线圆所包围的区域的边界和包络的公式。此描述包括由有限 Blaschke 产品标识的点的双曲凸包边界的公式。我们描述了一种设置和条件，在该设置和条件下，人们可以产生无限的椭圆链，使得每个椭圆都内接在一个凸多边形中，而凸多边形本身又内接在另一个多边形（Poncelet 椭圆链）中。我们将这些结果应用于模型空间上移位运算符的压缩运算符，并描述数值范围包含 0 的必要和充分条件。