Inspection of reflection line patterns is a standard way to check the quality of free form surfaces. In this paper, we describe an approach which enables the designer to control during the modeling process the reflection line pattern by approximating a specified family of reflection lines. Our method directly operates on the commonly used B-spline format, and produces fair surfaces. The approach is based on the minimization of a suitable error functional. A minimization algorithm based on data dependent quadratic error functionals provides an efficient numerical solution. The potential of the method will be demonstrated by applying it to two standard modeling problems: "surface fairing" and "surface reconstruction".

Progressive addition lenses are a relatively new approach to compensate for defects of the human visual system. While traditional spectacles use rotationally symmetric lenses, progressive lenses require the specification of free‐form surfaces. This poses difficult problems for the optimal design and its visual evaluation. This paper presents two new techniques for the visualization of optical systems and the optimization of progressive lenses. Both are based on the same wavefront tracing approach to accurately evaluate the refraction properties of complex optical systems. We use the results of wavefront tracing for continuously re‐focusing the eye during rendering. Together with distribution ray tracing, this yields high‐quality images that accurately simulate the visual quality of an optical system. The design of progressive lenses is difficult due to the trade‐off between the desired properties of the lens and unavoidable optical errors, such as astigmatism and distortions. We use wavefront tracing to derive an accurate error functional describing the desired properties and the optical error across a lens. Minimizing this error yields optimal free‐form lens surfaces. While the basic approach is much more general, in this paper, we describe its application to the particular problem of designing and evaluating progressive lenses and demonstrate the benefits of the new approach with several example images.

Abstract When modeling spline surfaces of complex shape, one has to deal with an overwhelming number of control points. Modeling by direct manipulation of the control points is a tedious task. In particular, it is very difficult to maintain a generally pleasant looking surface shape. It becomes therefore increasingly important to build tools that allow the designer to specify only a few geometric constraints while automatically determining the explicit representation of the surface. The basic concept of such a tool is simple. In a first step one has to somehow measure the “fairness” (=quality) of a surface. Once this is achieved, an optimization process selects the one surface with optimal fairness from all surfaces satisfying the user specified geometric constraints. To measure the fairness, thin plate energy functionals are a good choice. However, for interactive use these functionals are far too complex. W e will present appropriate approximations to these functionals that allow an optimization nearly in real time. Thefunctionals are obtained by introducing reference surfaces thus leading to data dependent, quadratic approximations to the exact thin plate energy functionals. We apply the method to interactive surface manipulations based on energy constraints.