Search for: All records

Sphere tracingis a fast and high-quality method for visualizing surfaces encoded by signed distance functions (SDFs). We introduce a similar method for a completely different class of surfaces encoded byharmonic functions, opening up rich new possibilities for visual computing. Our starting point is similar in spirit to sphere tracing: using conservativeHarnack boundson the growth of harmonic functions, we develop aHarnack tracingalgorithm for visualizing level sets of harmonic functions, including those that are angle-valued and exhibit singularities. The method takes much larger steps than naïve ray marching, avoids numerical issues common to generic root finding methods and, like sphere tracing, needs only perform pointwise evaluation of the function at each step. For many use cases, the method is fast enough to run real time in a shader program. We use it to visualize smooth surfaces directly from point clouds (via Poisson surface reconstruction) or polygon soup (via generalized winding numbers) without linear solves or mesh extraction. We also use it to visualize nonplanar polygons (possibly with holes), surfaces from architectural geometry, mesh exoskeletons, and key mathematical objects including knots, links, spherical harmonics, and Riemann surfaces. Finally we show that, at least in theory, Harnack tracing provides an alternative mechanism for visualizing arbitrary implicit surfaces.