The computation of offset and bisector curves/surfaces has always been considered a challenging problem in geometric modeling and processing. In this work, we investigate a related problem of approximating offsets of curves on surfaces (OCS) and bisectors of curves on surfaces (BCS). While at times the precise geodesic distance over the surface between the curve and its offset might be desired, herein we approximate the Euclidean distance between the two. The Euclidean distance OCS problem is reduced to a set of under-determined non-linear constraints, and solved to yield a univariate approximated offset curve on the surface. For the sake of thoroughness, we also establish a bound on the difference between the Euclidean offset and the geodesic offset on the surface and show that for a $C^2$ surface with bounded curvature, this difference vanishes as the offset distance is diminished. In a similar way, the Euclidean distance BCS problem is also solved to generate an approximated bisector curve on the surface. We complete this work with a set of examples that demonstrates the effectiveness of our approach to the Euclidean offset and bisector operations.

偏移和平分线/曲面的计算一直被认为是几何建模和处理中的一个难题。在这项工作中，我们研究了一个近似的问题，即近似曲面上的曲线偏移（OCS）和曲面上的平分线（BCS）。虽然有时可能需要曲面上的精确测地线距离及其偏移之间的距离，但在这里我们近似估算了两者之间的欧几里德距离。欧几里得距离OCS问题被简化为一组欠定的非线性约束，并被求解以在表面上生成单变量近似偏移曲线。为了透彻起见，我们还在表面上的欧几里得偏移和测地偏移之间的差异上建立了一个边界，并表明对于$C^2$曲面具有有限的曲率，随着偏移距离的减小，该差异消失。以类似的方式，还解决了欧几里德距离BCS问题，以在表面上生成近似的平分线。我们通过一系列示例来完成这项工作，这些示例演示了我们的方法在欧氏偏移和平分线运算中的有效性。