Many 3D mesh processing tasks revolve around generating and manipulating curves on surface meshes. While it is intuitive to explicitly model these curves using mesh edges or parametric curves in the ambient space, these methods often suffer from numerical instability or inaccuracy due to the projection operation. Another natural strategy is to adapt spline based tools, these methods are quite fast but are hard to be extended to more versatile constraints and need heavy manual interactions. In this paper, we present an efficient and versatile approach to curve design based on an implicit representation known as the level set. While previous works have explored the use of the level set to generate curves with minimal length, they typically have limitations in accommodating additional conditions for rich and robust control. To address these challenges, we formulate curve editing with constraints like smoothness, interpolation, tangent control, etc., via a level set based variational problem by constraining the values or derivatives of the level set function. However, the widely used gradient flow strategy converges very slowly for this complicated variational problem compared to the classical geodesic one. Thus, we propose to solve it via Newton's method enhanced by local Hessian correction and a trust-region strategy. As a result, our method not only enables versatile control, but also excels in terms of performance due to nearly quadratic convergence and almost linear complexity in each iteration via narrow band acceleration. In practice, these advantages effectively benefit various applications, such as interactive curve manipulation, boundary smoothing for surface segmentation and path planning with obstacles as demonstrated.