中文摘要：

提出采用几何要素的控制点位置变动表示尺寸和几何公差的方法。定义点线面几何要素的尺寸误差、位置误差和方向误差的控制点分别为点、直线的两个端点、平面的矩形包围盒的三个顶点，定义直线和平面的形状误差的控制点分别为直线和平面的内部组成点。根据目标要素和基准要素的几何类型、相互位置和基准优先顺序建立公差坐标系，根据公差坐标系建立平面要素的矩形包围盒，对几何要素的自由度进行分类和定义，以平移自由度方向作为控制点的变动方向，通过控制点变动的各种组合模拟几何要素的各种误差形式。根据几何要素的误差概率分布作为控制点的变动规律，控制点变动参数的定义域就是公差带，目标要素的控制点变动参数之间的相互制约关系可以表示方向公差和位置公差的相互作用关系，而目标要素和基准要素之间的控制点变动参数关系则可以表示各种公差原则。该公差数学模型既符合公差标准和实际惯例，又能够适用于公差分析、设计、检测等应用领域的各种数学分析方法。

英文摘要：

A new tolerance model by representing the position variation of control points of geometric element is proposed. The control points of point, line and plane geometric element are defined as point itself, two end points of line, any three vertices of bounding box of plane and the interior constituting points of line and plane, which are used to present the dimensional error, the position error, the orientation error and the form error. The tolerance coordinate system of target geometric element is established according to the geometric type of target geometry and datum geometry, the relative position between target geometry and datum geometry, and the precedence of datum reference frames. The degree of freedoms （DOFs） of geometry element are classified and defined under the tolerance coordinate system. The bounding box of plane is set up at the tolerance coordinate system and the variation direction of control points is defined along translational DOF of geometry element, and the variation combination of all control points are used to represent various error of geometric element. The variation of control points is controlled to follow the error probability distribution of geometric element and the definitional domain of variation parameter of control point is the tolerance zone, the restrict relationship among the control points of target element are used to represent the interaction relationship between position and orientation tolerance, and the relationship between the variation parameters of control point of target and datum element are used to represent the tolerance principles. The model is compatible with the tolerance standards and practice rules, it also suits the mathematical analysis method of many applications such as tolerance analysis, tolerance allocation and verification.