This paper describes a novel procedure for visualization (contouring) of implicit functions. It is based on a combination of the Newton algorithm and the ‘arc-length’ method. The new procedure has many advantages e.g. alleviates the singularity problem of the Newton algorithm. The resulting points that belong to the function are automatically ordered along the line (it can be difficult to properly connect the points resulting from some other algorithms). In addition, the new algorithm is self- starting and has no problems with multiple domains. The developed contouring procedure can be extended to visualize space surfaces by intersecting them with a number of arbitrary planes. Such a procedure, when parallelised belongs to the ‘embarrassingly parallel algorithms’, i.e. the n-planes intersection problems are solved separately and asynchronously from each other.