One of the main research topics in scientific visualization is to "visualize the appropriate features" of a certain structure or data set. Geodesics are very important in geometry and physics, but there is one major problem which prevents scientists from using them as a visualization tool: The differential equations for geodesics are very complicated and in most cases numerical algorithms must be used. There is always a certain approximation error involved. How can you be sure to visualize the features and not only the approximation quality. We present here an algorithm to overcome this problem. This paper consists of two parts. In the first, a geometric method for the construction of geodesics of arbitrary surfaces is introduced. This method is based on the fundamental property that geodesics are a generalization of straight lines on plains. In the second part these geodesics are used to generate local nets on the surfaces.