We introduce a user defined probabilistic polygonal representation for plane curves. Given a curve, we select vertices on the curve and connect consecutive vertices by line segments to obtain a polygonal approximation. Each segment is equipped with a user defined uncertainty parameter in the normal direction. This yields a collection of thin probabilistic geometric primitives that retain the geometrz of the underlying curve while extending it beyond the idealized deterministic one dimensional formulation. For each segment, we define a Random Variable that is uniform distributed in the tangent direction of the segment and Gaussian distributed in the normal direction of the segment. By matching the first and the second central moments, this construction induces a Gaussian component whose mean lies at the segment midpoint and whose covariance encodes both tangential and normal uncertainty. Combining the segment wise components with appropriate weights yields a Gaussian Mixture Model (GMM) representation of the user defined probabilistic polygonal representation of the plane curve. The proposed framework provides an analytically tractable probabilistic model that preserves local geometry, and uncertainty in the normal direction. It applies to smooth, closed, open, non regular, and self intersecting plane curves, allows adaptive discretization and varying uncertainty in the normal direction, and as a result supports uncertainty aware geometric modeling. Experiments on a collection of canonical plane curves show that the resulting GMM capture local tangent, local normal, and local arc length; resulting in the global shape of the underlying curves to be truthfully captured as well. The representation is particularly relevant for applications in uncertainty aware CAD and digital twins, probabilistic obstacle modeling in robotics, and probabilistic trajectory planning.

翻译：本文提出了一种用户自定义的平面曲线概率多边形表示方法。给定一条曲线，我们在曲线上选取若干顶点，并通过线段依次连接相邻顶点，从而获得其多边形近似。每条线段在法线方向配有一个用户定义的不确定度参数。由此得到一组细长概率几何基元，这些基元在保留原始曲线几何形状的同时，将其推广至理想化确定性一维表达之外的范畴。针对每条线段，我们定义了一个在其切向服从均匀分布、法向服从高斯分布的随机变量。通过匹配一阶和二阶中心矩，该构造诱导出一个高斯分量，其均值位于线段中点，协方差矩阵同时编码了切向与法向的不确定性。将各线段的分量以适当权重组合，即得到平面曲线的用户定义概率多边形表示的高斯混合模型。该框架提供了一个解析可处理的概率模型，既可保持局部几何结构，又可描述法向不确定度。它适用于光滑、封闭、开放、非正则及自相交的平面曲线，支持自适应离散化与法向不确定度的变化，从而支持不确定性感知的几何建模。对一组典型平面曲线的实验表明，所构建的高斯混合模型能够有效捕捉局部切向、局部法向及局部弧长信息，从而如实再现原始曲线的整体形状。该表示方法尤其适用于不确定性感知的计算机辅助设计与数字孪生、机器人学中的概率障碍建模以及概率轨迹规划等应用场景。