Loren C. Carpenter

The A -buffer, an antialiased hidden surface method

The A-buffer (anti-aliased, area-averaged, accumulation buffer) is a general hidden surface mechanism suited to medium scale virtual memory computers. It resolves visibility among an arbitrary collection of opaque, transparent, and intersecting objects. Using an easy to compute Fourier window (box filter), it increases the effective image resolution many times over the Z-buffer, with a moderate increase in cost. The A-buffer is incorporated into the REYES 3-D rendering system at Lucasfilm and was used successfully in the “Genesis Demo” sequence in Star Trek II.

Scan line methods for displaying parametrically defined surfaces

This paper presents three scan line methods for drawing pictures of parametrically defined surfaces. A scan line algorithm is characterized by the order in which it generates the picture elements of the image. These are generated left to right, top to bottom in much the same way as a picture is scanned out on a TV screen. Parametrically defined surfaces are those generated by a set of bivariate functions defining the X, Y, and Z position of points on the surface. The primary driving mechanism behind such an algorithm is the inversion of the functions used to define the surface. In this paper, three different methods for doing the numerical inversion are presented along with an overview of scan line methods.

The Reyes image rendering architecture

An architecture is presented for fast high-quality rendering of complex images. All objects are reduced to common world-space geometric entities called micropolygons, and all of the shading and visibility calculations operate on these micropolygons. Each type of calculation is performed in a coordinate system that is natural for that type of calculation. Micropolygons are created and textured in the local coordinate sysem of the object, with the result that texture filtering is simplified and improved. Visibility is calculated in screen space using stochastic point sampling with a z buffer. There are no clipping or inverse perspective calculations. Geometric and texture locality are exploited to minimize paging and to support models that contain arbitrarily many primitives.

Computer rendering of fractal curves and surfaces

Fractals are a class of highly irregular shapes that have myriad counterparts in the real world, such as islands, river networks, turbulence, and snowflakes. Classic fractals include Brownian paths, Cantor sets, and plane-filling curves. Nearly all fractal sets are of fractional dimension and all are nowhere differentiable. Previously published procedures for calculating fractal curves employ shear displacement processes, modified Markov processes, and inverse Fourier transforms. They are either very expensive or very complex and do not easily generalize to surfaces. This paper presents a family of simple methods for generating and displaying a wide class of fractal curves and surfaces. In so doing, it introduces the concept of statistical subdivision in which a geometric entity is split into smaller entities while preserving certain statistical properties.

Computer rendering of fractal curves and surfaces Editorial Note: Page(s) are missing from this article's PDF file. We are attempting to locate an issue to correct

Fractals are a class of highly irregular shapes that have myriad counterparts in the real world, such as islands, river networks, turbulence, and snowflakes. Classic fractals include Brownian paths, Cantor sets, and plane-filling curves. Nearly all fractal sets are of fractional dimension and all are nowhere differentiable.Previously published procedures for calculating fractal curves employ shear displacement processes, modified Markov processes, and inverse Fourier transforms. They are either very expensive or very complex and do not easily generalize to surfaces. This paper presents a family of simple methods for generating and displaying a wide class of fractal curves and surfaces. In so doing, it introduces the concept of statistical subdivision in which a geometric entity is split into smaller entities while preserving certain statistical properties.

Technical correspondence: comment on computer rendering of fractal stochastic models. author's reply

Brownian motion, the initiators length must be Gaussian with variance twice that of the midpoint displacement. In Figure 7 of [1], to the contrary, the above ratio is much larger, and the probability of its occurring in a self-similar chance process that also generates the wiggles is infinitesimal. Thus, Figure 7 is grossly atypical of the self-similar Brownian motion. On the other hand, Figure 7 is a sensible sample of Brownian motion with a strong deterministic drift. Drifting Brown-ian motions are hybrids. When the initiator length is even longer than in Figure 7 and the limit fractal is inspected on scales of the order of the initiators length, the motion is in effect, nonrandom and rectilinear (D = 1). When the initiators length is like that in Figure 7 of[l], the motion on the visible scales is a non-self-similar hybrid of D = 1 and D = 2, and no useful D can be defined for it. In the approximation drawn in Figure 7, the loops typical of D = 2 and of nondrifting Brownian motion only become visible during the last stage of recursion. If the recursion were to continue, the loops would proliferate. Since, excluding rare exceptions , the curves that approximate rivers are self-avoiding, a non-drifting Brownian motion model is excluded a priori. Since rivers happen to be self-similar, a strongly drifting Brownian motion is not acceptable. The interpolation of the coast of Australia in Figures 9 to 13 of [1] also suffers from local loops. Incidentally , the idea that coastlines can be modeled by a fractional Brownian motion is tried out on page 205 of [3], and found wanting. To the contrary , my fractal model of coastlines, using the level curves of a fractal relief, not only accounts satisfactorily for the detail, but yields reasonable-looking overall shapes. In [1], the overall shape had to be entered separately. (4) The aesthetic importance of self-similarity and the visible scars when it is only approximate may come as a surprise. Indeed, the self-similarity of rivers, coastlines, and 583 relief has not, as of yet, been reduced to more traditional natural principles. When first postulated in my work, it was a matter of convenience, but it is well on the way to becoming a new natural principle on its own. We are certainly pleased that Benoit Mandelbrots attention has been attracted by our work on sto-chastic modeling, and …

Graphics in the large: is bigger better?

The world of display devices is expanding rapidly, both literally and figuratively. New commercial and research devices come in larger sizes (measured in meters, not inches) and different physical forms (e.g. rectangular surfaces, cylindrical segments, truncated spheres). Such expansion means that graphics and interactive techniques are becoming far more amenable to group activities and can display more and more data at once.