In this thesis I attempt to give a broad overview of non-photorealistic rendering with graphical hatching. First the connection of this computer graphics technique is discussed to the traditional art media. By analyzing the graphical compositions and their details some principles are established about the preferred method of rendering patches of hatch lines, and some insight is given into their esthetics.
I then discuss the possibility of, and propose a method for constructing hatched line drawings based solely on the basis of the extent the faces are lit or shadowed. Result of this investigation is the conclusion, that such an approach mostly suits three dimensional objects constructed of large scale planar faces. Such are models of the ambient world of a 3d scene or pieces of architectural.
It is obvious that such models do not cover the range of all possible applications of non-photorealistic rendering. This brings forth the motivation to explore the possibility of creating a set of hatch lines on a surface based on its geometrical properties. Such fundamental metrics on a surface are the principal curvatures, and the direction they belong to, the principal directions.
At the time of the inception of the thesis I was not aware how broad the subject of differential geometry is. For a working understanding of the basic concepts and tools a large mathematical apparatus is required, some elements of which do not make up the core curriculum of software engineering BSc education. Without having a grasp of the underlying concepts the formulas for curvature are hardly applicable. This matter is made worse by a considerable percentage of available text on the subjects. Some concentrate more on manipulating the formulas, than to reveal the implied meaning. Some others are clearly not aimed at undergraduate level of studies.
To provide a solid basis for working with surfaces, I attempt to give a broad survey of differential geometry's principles with a focus on the geometric meaning. For this I have studied several texts, compared the different approaches to the various problems, and where necessary I have given additional geometric insight. To first familiarize myself and the reader with the concepts of curvature and torsion the subject of planar and space curves is discussed. Much of the results of this chapter is necessary and relevant to surfaces. After a brief overview of differentiability, differentials of maps, and self adjoint transformations, the notion of regular surfaces is defined. This is the basis for developing the first and second fundamental forms of the surface. These are the tools that are essential for describing the local properties defining geometry.
It is shown, that the second fundamental form is a self adjoint linear map. A such under certain restrictions it has minimum and maximum values, that can be obtained by finding its eigenvalues and eigenvectors. It is also shown that these eigenvectors form the principal directions and they provide comfortable means for describing the surface locally.
Methods of differential geometry are suitable for continuous surfaces. For the discrete case, such as the rather ubiquitous triangle meshes, methods have to be devised to use the tools of the fundamental forms. After a brief overview of papers on the subject a method is proposed for approximating curvature at the vertices of the mesh.
These chapters give the bulk of the thesis. In the concluding chapter details of a concrete implementation are detailed with additional suggestions on handling the contour and displaying cast shadows.