Fractal Landscapes Without Creases and with Rivers

Appendix A Fractal landscapes without creases and with rivers Benoit B.Mandelbrot This text tackles three related but somewhat independent issues. The most ba sic defect of past fractal forgeries of landscape is that every one of them fails to include river networks. This is one reason why these forgeries look best when viewed from a low angle above the horizon, and are worst when examined from the zenith. This is also why achieving afully random combined model of rivers and of mountains will be a major advance for computer graphics (and also per haps for the science of geomorphology). The main goal of this text is to sketch two partly random solutions that I have developed very recently and Ken Mus grave has rendered. A second very different irritating defect is specific to the method of land scapes design (called midpoint displacement or recursive subdivision) which is described in [40]. It has gained wide exposure in "Star Trek II," and is fast, compact, and easy to implement. Its main shortcoming is that it generates "fea tures" that catch the eye and, by their length and straightness, make the rendered scene look "unnatural." This defect, which has come to be called "the creasing problem," is illustrated in the body of the present book. It was first pointed out in [69], and its causes are deep, so that it cannot be eliminated by cosmetic treat ment. This text sketches several methods that avoid creasing from the outset. (We were not aware of the fact that [76] had addressed this problem before us. His method is somewhat similar to ours, but there is no face-to-face overlap.) 243 244 Appendix A A third defect present in most forgeries is that the valleys and the mountains are symmetric because of the symmetry of the Gaussian distribution. Voss's pic tures in [69] have shown that this undesirably condition is greatly improved by after-the-fact non linear processing. Furthermore, our "mountains with rivers" involve extreme asymmetry, meaning that this paper tackles this issue head on. But, in addition, this text sketches a different and much simpler path towards the same end. It shows that it suffices to use the familiar midpoint displacement method with displacements whose distribution is itself far from being Gaussian, in fact are extremely asymmetric. In the search for algorithms for landscape design, hence in this text, a cen tral issue is that of context dependence. The familiar midpoint displacements method is context independent, which is the source both of its most striking virtue (simplicity) and of its most striking defect (creasing). In real landscapes, however, the very fact that rivers flow down introduces very marked context dependence; it is theoretically infinite, and practically it extends to the overall scale of the rendering. This introduction has listed our goals in the order of decreasing importance, but this is also the order of increasing complexity, and of increasing divergence from what is already familiar to computer graphics experts. Therefore, the dis cussion that follows is made easier by moving towards increasing complexity. It is hoped that the new procedures described in this text will inject fresh vi tality into the seemingly stabilized topic of fractal landscape design. In any event, landscape design deserves investigation well beyond what is already in the record, and beyond the relatively casual remarks in this paper. A.I Non-Gaussian and non-random variants of midpoint displacement A.I.I Midpoint displacement constructions for the paraboloids One of the purposes of this text is to explain midpoint displacement better, by placing it in its historical context. The Archimedes construction for the parabola, as explained in the Foreword, can be expanded to the surface S closest to it, which is the isotropic paraboloid z = P( x, y) = a - bx2 - by2. Here, the simplest initiator is a triangle instead of the interval, the section of S along each side of the triangle being a parabola. A second stage is to systematically in terpolate at the nodes of an increasingly tight lattice made of equilateral trian gles. This procedure, not known to Archimedes, is a retrofit to non-random A.I Variants ofmidpoint displacement 245 Fig. A.I: Mount Archimedes, constructed by positive midpoint displacements. Fig. A.2: Archimedes Saddle, constructed by midpoint displacements. classical geometry of the method Fournier et al. use to generate fractal surfaces. When all the interpolations along the cross lines go up, because b > 0 (resp., down, because b < 0), we converge to Mount Archimedes (Figure A.I) (resp., to the Archimedes Cup, which is the Mount turned upside down.) Suitable slight changes in this construction yield the other paraboloids. In reduced coordinates, a paraboloid's equation is z = P( x, y) = a - bx2 - cy2 , with b > 0 in all cases, and with c > 0 if the paraboloid is elliptic (a cap), but c < 0 if it is hyperbolic (a saddle). By suitable "tuning" of the up and down midpoint displacements, one can achieve either result. The Archimedes saddle is shown on Figure A.2. 246 Appendix A Fig. A.3: Takagi profile, constructed by positive midpoint displacements. A.1.2 Midpoint displacement and systematic fractals: The Takagi fractal curve, its kin, and the related surfaces Long after Archimedes, around 1900, the number theorist Teiji Takagi resur rected the midpoint displacement method, but with a different rule for the rela tive size (divided by 8) of the vertical displacements: Instead of their following the sequence 1 (once), 4 -1 (twice), 4 -2 (2 2 times) ... 4 -k (2 k times), etc., he made the relative sizes be 1 (once), 2 -1 (twice), 2 -2 (2 2 times) ... 2-k (2 k times), etc .. The effect is drastic, since the smooth parabola is then replaced by the very unsmooth Figure A.3, which is an early example of a function that is continuous but non differentiable anywhere. Then a mathematician named Landsberg picked w arbitrarily in the interval 1/2 < w < 1 and performed midpoint displacements of I (once), w (twice), ... wk (2 k times), etc .... Fig ure AA corresponds to w = .6. These curves have the fractal dimension 2 - Ilog2 wi. As w ranges from 1 /2 to 1, this dimension ranges from 1 (a borderline curve of dimension 1, but of logarithmically infinite length) to 2. A.I Variants ofmidpoint displacement 247 Fig. A.4: Landsberg profile, constructed by positive midpoint displacements. (When 0 < w < 1/2 but w =I 1/4, the curves are rectifiable, with a fractal dimension saturated at the value 1. Left and right derivatives exist at all points. The two derivatives differ when the abscissa is dyadic, that is, of the form p2 -k, with an integer p, but there is a well defined derivative when the abscissa is non-dyadic.) Again, let us retrofit the Takagi curve ala Fournier et aI.. Its surface coun terpart, which we call Mount Takagi, is shown as Figure A.5. A remarkable fact is revealed when we examine this surface from a low enough angle, rela tive to the vertical scale in the picture, and from a horizontal angle that avoids the lattice directions. At ridiculously low intellectual cost, we have delivered a realistic first-order illusion of a mountain terrain (other viewers may be more strongly taken by botanic illusions). However, seen from a high vertical angle, Mount Takagi reveals a pattern of valleys and of "cups" without outlets, which is totally unrealistic. From a horizontal angle along the lattice directions, the il lusion also collapses altogether: the original triangular grid used for simulation overwhelms our perception. Inspecting Mount Takagi from a randomly chosen low angle yields a surprising variety of terrains that do not immediately seem to be all the same. 248 Appendix A The lesson we just learned happens to be of much wider consequence. Every one of the fractal landscapes explored thus far is best when viewed at low angle. (The question of what is a high or a low angle, compared to the vertical, is re lated to the countetpart of the quantity 8 in the definition of Mount Archimedes, hence to the fact that our various reliefs are self-affine, rather than self-similar.) A.l.3 Random midpoint displacements with a sharply non-Gaussian displacements' distribution In random midpoint displacement (Fournier et al.), the structure of valleys and ridges is less accentuated than it is with Mount Takagi, yet it remains apparent and bothersome. It can be attenuated by modifying the probability distribution of the random midpoint displacements. The Gaussian distribution is symmetric, hence a fractional Brownian surface can be flipped upside down without chang ing statistically. Actual valley bottoms and mountain crests are not symmetric. This difficulty had led Voss to introduce non linear after the fact transforma tions. The resulting non Gaussian landscapes (plates 8 and 10) are stunning, but contrived. If one abandons the straight and true fractional Brown ideal, one may as well replace the Gaussian by something else. The first suitable non symmetric distribution we have played with is the asymmetric binomial of parameter p: X = 1 with the probability p and A.I Variants ofmidpoint displacement 249 Fig. A.6: First fractal landscapes constructed by random midpoint displacements with a very skew distribution. Fig. A.7: Second fractal landscapes constructed by random midpoint displacements with a very skew distribution. 250 Appendix A x = -pie 1 - p) with the probability 1 - p. Values p < l/2 yield flat valleys and peaky mountains, and values p > l/2 yield flat mountains and deep valleys. Replacing p by 1 - p flips a relief upside down.

Load more