Konstantin Poelke and Konrad Polthier
Complex-valued functions are fundamental objects in complex analysis, algebra, differential geometry and in many other areas such as numerical mathematics and physics. Visualizing complex functions is a non-trivial task since maps between two-dimensional spaces are involved whose graph would be an unhandy submanifold in four-dimensional space. The present paper improves the technique of domain coloring in several aspects: First, we lift domain coloring from the complex plane to branched Riemann surfaces, which are essentially the correct domain for most complex functions. Second, we extend domain coloring to the visualization of general 2-valued maps on surfaces. As an application of such general maps we visualize the Gauss map of surfaces as domain colored plots and establish a link to current surface parametrization techniques and texture maps. Third, we adjust the color pattern in domain and in image space to produce higher quality domain colorings. The new color schemes specifically enhance the display of singularities, symmetries and path integrals, and give better qualitative measures of the complex map.
Categories and Subject Descriptors (according to ACM CCS): Applications [Computer Graphics]: Visualization in Mathematics, Illustrative Visualization, Complex Functions, Riemann Surfaces