In this article, we will introduce methods of non-standard analysis into projective geometry and within this introduction we discretize continuous theory without the usual discretization error. Especially, we will analyze the properties of a projective space over a non-Archimedeanfield. Non-Archimedean fields contain numbers that are smaller than every real number: the so called “infinitesimal” numbers. The theory is well known from non-standard analysis. This enables us to define projective objects that deviate only infinitesimally (in an appropriate metric) from each other. And we show that in most cases operations involving such objects that deviate infinitesimally also experience only infinitesimal change. Another focus will be where this property does not hold true and show that this usually involves discontinuity or degeneration. Further-more, we will explore common projective concepts like projective transformations, cross-ratios and conics in a non-standard setting.

The CindyJS Project brings interactive mathematical visualization to a broad variety of devices. Using projective geometry, homotopy methods and well tuned algorithms the CindyJS project is one of the first real time capable software projects in this eld that at the same time approaches high-level mathematical descriptions and performance.

We propose a framework for temporally consistent videocompletion. To this end we generalize the exemplar-based inpaintingmethod of Criminisi et al. to video inpainting. Specifically we addresstwo important issues: Firstly, we propose a color and optical flow inpaint-ing to ensure temporal consistency of inpainting even for complex motionof foreground and background. Secondly, rather than requiring the userto hand-label the inpainting region in every single image, we proposea flow-based propagation of user scribbles from the first to subsequentvideo frames which drastically reduces the user input. Experimental com-parisons to state-of-the-art video completion methods demonstrate thebenefits of the proposed approach.