epsilon-Guarantee of a covering of 2D domains using random-looking curves

The problem of covering a given 2D convex domain D with a C¹ random-looking curve C is considered. C within D is said to cover D up to ϵ > 0 if all points of D are within ϵ distance of C. This problem has applications, for example, in manufacturing, 3D printing, automated spray-painting, polishing, and also in devising a (pseudo) random patrol-path that will visit (i.e. cover) all of D using a sensor of ϵ distance span. Our distance bound approach enumerates the complete set of local distance extrema, enumeration that is used to provide a tight bound on the covering distance. This involves computing bi/tri-normals, or circles tangent to C at two/three different points, etc. A constructive algorithm is then proposed to iteratively refine and modify C until C covers a given convex domain D and examples are given to illustrate the effectiveness of our algorithm.