physically based animation and rendering of .physically based animation and rendering of lightning
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Physically Based Animation and Rendering of Lightning
Theodore Kim Ming C. LinDepartment of Computer Science
University of North Carolina at Chapel Hillkim, lin @cs.unc.edu
We present a physically-based method for animating andrendering lightning and other electric arcs. For the simula-tion, we present the dielectric breakdown model, an elegantformulation of electrical pattern formation. We then extendthe model to animate a sustained, dancing electrical arc,by using a simplified Helmholtz equation for propagatingelectromagnatic waves. For rendering, we use a convolu-tion kernel to produce results competitive with Monte Carloray tracing. Lastly, we present user parameters for manipu-lation of the simulation patterns.
The forked tendrils of electrical discharge have a longhistory as a dramatic tool in the visual effects industry. Fromthe genesis of the monster in the 1931 movie Frankenstein,to the lightning from the Emperors fingers in Return of theJedi, to the demolition of the Coliseum by lightning in lastyears The Core, lightning is an ubiquitous effect in sciencefiction and fantasy films.
Despite the popularity of this effect, there has been rela-tively little research into physically-based modeling of thisphenomenon. The existing research is largely empirical, es-sentially generating a random tree-like structure that quali-tatively resembles lightning. The previous work is also lim-ited to brief flashes of lightning, and provides no method foranimating a dancing, sustained stream of electricity. How-ever, modeling the fractal geometry of electrical dischargeand similar patterns has attracted much attention in physics.To our best knowledge, our algorithm is the first rigorous,physically-based modeling of lightning in computer graph-ics. We also believe our approach is accurate enough that itsapplications extend beyond visual effects to more physicallydemanding applications, such as commercial flight simula-tion.
Main Contributions: In this paper, we present a
physically-based algorithm to simulate lightning, andpropose a novel extension for animation of continuous elec-trical streams. The simulation results are then renderedusing an efficient convolution technique. The result-ing image quality rivals that of Monte Carlo ray tracing.Lastly, we present user parameters for intuitive manipu-lation of the simulation. Our approach offers the follow-ing:
A physically-inspired approach based on the dielectricbreakdown model for electrical discharge;A novel animation technique for sustained electricalstreams that solves a simplified Helmholtz equation forpropagating electromagnetic waves;A fast, accurate rendering method that uses a convolu-tion kernel to describe light scattering in participatingmedia;A parameterization that enables simple artistic controlof the simulation.
Organization: The rest of the paper is organized as fol-lows. A brief survey of related work is presented in Sec. 2.In Sec. 3, we briefly summarize the physics of lightning for-mation. We present the original dielectric breakdown modelas well as our proposed extension in Sec. 4. A efficient ren-dering method is present in Sec. 5. User parameters are pre-sented in Sec. 6, followed by implementation details anddiscussion in Sec. 7. Finally, conclusions and possible di-rections for future work are given in Sec. 8.
2. Previous Work
Reed and Wyvill present a lightning model based on theempirical observation that most lightning branches deviateby an average of 16 degrees from parent branches . Aset of randomly rotated line segments are then generatedwith their angles normally distributed around 16 degrees.In subsequent work, modifications are made to this randomline segment model. Glassner  performs a second pass
on the segments to add tortuosity, and Kruszewski  re-places the normal distribution with a more easily controlledrandomized binary tree.
Notably, Sosorbaram et al.  use the dielectric break-down model (DBM) to guide the growth of a random linesegment tree with a local approximation of the potentialfield. But, their approach does not appear to implement fullDBM, as it does not solve the full Laplace equation.
Electric discharges are neither solid, liquid, or gas, butinstead are the fourth phase of matter, plasma. It is a lightsource with no resolvable surface, so traditional renderingtechniques are not directly applicable. To address this prob-lem, Reed and Wyvill  describe a ray tracing extensionfor both a lightning bolt and its surrounding glow. Alterna-tively,  proposes rendering 3D textures. Dobashi, Ya-mamoto, and Nishita  provide the most rigorous treat-ment of the problem by first presenting the associated vol-ume rendering integral, and then presenting an efficient, ap-proximate solution.
In electrical engineering, there are three popular mod-els of electric discharge: gas dynamics , electromagnet-ics , and distributed circuits . However, none of theseare directly applicable to visual simulation, as they respec-tively approximate the electricity as a cylinder of plasma, athin antenna, and two plates in a circuit.
3. The Physics of Electric Discharge
We classify the physics literature into two categories.The first deals with the physical, experimentally observedproperties of lightning and related electrical patterns. Agood survey of this approach is given by Rakov and Uman. The second is a more qualitative approach that char-acterizes the geometric, fractal properties of electric dis-charge. A good survey of this approach is given by Vicsek.
3.1. Physical Properties
Electrical discharge occurs when a large charge differ-ence exists between two objects. For lightning, the caseis usually that the bottom of a cloud has a strong nega-tive charge and the ground possesses a relatively positivecharge. Electrons possess negative charge, the charge differ-ence is then equalized when electrons are transferred fromthe cloud to the ground in the form of lightning. This caseis referred to as downward negative lightning. While othertypes can exist, downward negative lightning accounts for90 percent of all cloud-to-ground lightning. For illustrativepurposes, we will show here how to simulate this most com-mon type of lightning. But, it should be noted that we canhandle the other types of lightning by trivially manipulat-ing the charge configuration.
Lightning is actually composed of several bolts, orstrokes in rapid succession. The first stroke is referredto as the stepped leader. The subsequent strokes, calleddart leaders, tend to follow the general path of the pre-vious leaders, and do not exhibit as much branching asthe stepped leader. We note that the random line seg-ment approach of previous work in computer graphicsdoes not provide a clear method of simulating dart lead-ers. But, such a method is crucial for simulating sustainedelectric arcs, which are essentially stepped leaders fol-lowed by a large number of dart leaders.
Lightning is initiated in clouds by an event known as theinitial breakdown. During the initial breakdown, the con-ductivity in a small column of air jumps several orders ofmagnitude, effectively transforming the column from an in-sulator (or dielectric) to a conductor. Charge then flows intothe newly conductive air. Another breakdown then occurssomewhere along the perimeter of the newly charged air.This chain of events repeats, forming a thin, tortuous paththrough the air, until the charge reaches the ground.
3.2. Geometric Properties
The physical processes that give rise to the breakdownare still not well understood. However, a great deal ofprogress has been made in characterizing the geometricshape that the breakdown ultimately produces. Electric dis-charge has been observed to have a fractal dimension of ap-proximately 1.7 . Many disparate natural phenomenashare this same fractal dimension, including ice crystals,lichen, and fracture patterns. Collectively, all the patternsthat share these fractal properties are known as Laplaciangrowth phenomena.
There are three techniques for simulating Laplaciangrowth: Diffusion Limited Aggregation , the Dielec-tric Breakdown Model , and Hastings-Levitov confor-mal mapping . All three produce qualitatively similar re-sults. We elect to use the Dielectric Breakdown Model herebecause it gives the closest correspondence to the phys-ical system being simulated and allows the addition ofnatural, physically intuitive user controls.
4. The Dielectric Breakdown Model
The Dielectric Breakdown Model, or DBM, was first de-scribed by Niemeyer, Pietronero, and Wiesmann , and isalso sometimes referred to as the model. We first presentthe model described in the original paper, and then proposea modification to simulate dart leaders and sustained elec-tric arcs.
(a) Original configuration (b) Lightning configuration
Figure 1. Different charge configurations forsimulation. Grey: ; Black:
4.1. The Laplacian Growth Model
The original charge configuration from  is shown inFigure 1(a). Over a 2D grid, the quantity , the electricalpotential at each point, is tracked. First, a negative charge isplaced at the center by setting at the center grid cell.Then, a circle of positive charge is constructed around thecenter charge by setting a surrounding circle to . Thepotential at the remaining grid cells are then set by solv-ing the Laplace equation (Eqn. 1) over the grid, with thecenter charge and the surrounding circle treated as bound-ary conditions. The grid boundaries are also set to . (1)The Laplace equation produces a linear system that mustthen be solved. For information on solving the Laplaceequation and the related Poisson equation, the reader is re-fe