Quadratic shading has been proposed as a technique giving better results than Gouraud shading, but which is substantially faster than Phong shading. Several techniques for fitting a second order surface to six points have been proposed. We show in this paper how an approximation of the mid-edge samples can be done in a very efficient way. An approximation of the mid-edge vectors are derived. Several advantages are apparent when these vectors are put into the original formulation. First of all it will only depend on the vertex vectors. Moreover, it will simplify the setup and no extra square roots are necessary for normalizing the mid-edge vectors. The setup will be about three times faster than previous approaches. This makes quadratic shading very fast for interpolation of both diffuse and specular light, which will make it suitable for near Phong quality software renderings.

In this paper, we show how spherical linear interpolation can be used to produce shading with a quality at least similar to Phong shading at a computational effort in the inner loop that is close to that of the Gouraud method. We show how to use the Chebyshev's recurrence relation in order to compute the shading very efficiently. Furthermore, it can also be used to interpolate vectors in such a way that normalization is not necessary, which will make the interpolation very fast. The somewhat larger setup effort required by this approach can be handled through table look up techniques.

We present a technique that can be used to obtain a series of connected minimal bending trigonometric splines that will intersect any number of predefined points in space. The minimal bending property is obtained by a least square minimization of the acceleration. Each curve segment between two consecutive points will be a trigonometric Hermite spline obtained from a Fourier series and its four first terms. The proposed method can be used for a number of points and predefined tangents. The tangent length will then be optimized to yield a minimal bending curve. We also show how both the tangent direction and length can be optimized to give as smooth curves as possible. It is also possible to obtain a closed loop of minimal bending curves. These types of curves can be useful tools for 3D modelling, etc.

This paper presents fast implementations of the inverse square root and arcsine, both in double precision. In single precision it is often possible to use a small table and one ordinary Newton-Raphson iteration to compute elementary functions such as the square root. In double precision a substantially larger table is necessary to obtain the desired precision, or, if a smaller table is used, the additional Newton-Raphson iterations required to obtain the precision often requires the evaluation of other expensive elementary functions. Furthermore, large tables use a lot of the cash memory that should have been used for the application code.

Obtaining the desired precision using a small table can instead be realised by using a higher order method than the second order Newton-Raphson method. A generalization of Newton's method to higher order is Householder's method, which unfortunately often results in very complicated expressions requiring many multiplications, additions, and even divisions.

We show how a high-order method can be used, which only requires a few extra additions and multiplications for each degree of higher order. The method starts from the Taylor expansion of the difference of the value of the elementary function and a starting guess value for each iteration. If the Taylor series is truncated after the second term, ordinary Newton iterations are obtained. In several cases it is possible to algebraically simplify the difference between the true value and the starting guess value. In those cases we show that it is advantageous to use the Taylor series to higher order to obtain the fast convergent method. Moreover, we will show how the coefficients of a Chebyshev polynomial can be fitted to give as little error as possible for the functions close to zero and in the same time reduce the terms in the Taylor expansion.

In the paper we benchmark two example implementations of the method on the x86_64 architecture. The first is the inverse square root, where the actual table (to 12 bit precision) is provided by the processor hardware. The inverse square root is important in many application programs, including computer graphics, and explicit particle simulation codes, for instance the Monte Carlo and Molecular Dynamics methods of statistical mechanics. The other example is the arcsine function, which has a slow converging Taylor expansion and where no tables are provided by the hardware. The vectorized versions of the implementations of the inverse square root are 3.5 times faster than compiled code on the Athlon64 and about 5 times faster on the Core 2. The scalar version of the arcsine function is, depending on order and table size, between 2 and 3 times faster than the compiled code, and the vectorized version is between 3 and 4 times faster on the Athlon64, while it is between 4 and 5 times faster than the compiled version on the Core 2.

This paper presents fast implementations of the inverse square root and arcsine, both in double precision. In single precision it is often possible to use a small table and one ordinary Newton-Raphson iteration to compute elementary functions such as the square root. In double precision a substantially larger table is necessary to obtain the desired precision, or, if a smaller table is used, the additional Newton-Raphson iterations required to obtain the precision often requires the evaluation of other expensive elementary functions. Furthermore, large tables use a lot of the cash memory that should have been used for the application code.

Obtaining the desired precision using a small table can instead be realised by using a higher order method than the second order Newton-Raphson method. A generalization of Newton's method to higher order is Householder's method, which unfortunately often results in very complicated expressions requiring many multiplications, additions, and even divisions.

We show how a high-order method can be used, which only requires a few extra additions and multiplications for each degree of higher order. The method starts from the Taylor expansion of the difference of the value of the elementary function and a starting guess value for each iteration. If the Taylor series is truncated after the second term, ordinary Newton iterations are obtained. In several cases it is possible to algebraically simplify the difference between the true value and the starting guess value. In those cases we show that it is advantageous to use the Taylor series to higher order to obtain the fast convergent method. Moreover, we will show how the coefficients of a Chebyshev polynomial can be fitted to give as little error as possible for the functions close to zero and in the same time reduce the terms in the Taylor expansion.

In the paper we benchmark two example implementations of the method on the x86_64 architecture. The first is the inverse square root, where the actual table (to 12 bit precision) is provided by the processor hardware. The inverse square root is important in many application programs, including computer graphics, and explicit particle simulation codes, for instance the Monte Carlo and Molecular Dynamics methods of statistical mechanics. The other example is the arcsine function, which has a slow converging Taylor expansion and where no tables are provided by the hardware. The vectorized versions of the implementations of the inverse square root are 3.5 times faster than compiled code on the Athlon64 and about 5 times faster on the Core 2. The scalar version of the arcsine function is, depending on order and table size, between 2 and 3 times faster than the compiled code, and the vectorized version is between 3 and 4 times faster on the Athlon64, while it is between 4 and 5 times faster than the compiled version on the Core 2.

Whenever real-time snowfall is animated in computer games, no snow accumulation is simulated, as far as we know. Instead, so-called zero thickness is used, which means that the blanket of snow does not grow when the snowflakes reach the ground. In this paper we present a method for simulation of snow accumulation, which simulates the different stages, starting with a snow free environment and ending with a totally snow covered scene, all in real-time. The main focus is not on the physical properties of snow but on speed and visual result.

Anisotropic shading techniques can be used for materials like hair and silk. Such materials have a surface consisting of very small fibers having a main direction. Geometrically each fiber can be viewed as a cylinder. Poulin and Fournier proposed an illumination model for such materials using a flat topology only. In the paper a model using a varying topology is described. The varying topology is a more correct model and visible differences are clearly apparent.

Spherical linear interpolation has got a number of important applications in computer graphics. We show how spherical interpolation can be performed efficiently even for the case when the angle vary quadratically over the interval. The computation will be fast since the implementation does not need to evaluate any trigonometric functions in the inner loop. Furthermore, no renormalization is necessary and therefore it is a true spherical interpolation. This type of interpolation, with non equal angle steps, should be useful for animation with accelerating or decelerating movements, or perhaps even in other types of applications.

The main idea presented herein is to use a multiscale texture synthesis approach in order to both colourize and upscale greyscale textures. Such textures can be vintage photos to be used in archaeological or urban 3D visualizations and obviously the colour needs to be reconstructed some way. Due to limited quality, walls etc in such 3D visualizations will appear either pixelized or blurry when the viewer approaches them on a close distance. The latter if some kind of interpolation technique is being used to reduce the pixelization. The low resolution greyscale texture and a high resolution coloured texture is used for the colourization and upscaling, which will produce a colour version of the greyscale texture with 4 times higher resolution in each upscale step. The novel idea is to use multiscale texture synthesis in HSV space for the first upscale in order to create a RGB colour image for subsequent upscaling, using either ordinary RGB multiscale texture synthesis or continue using HSV multiscale texture synthesis. These two main approaches will be compared and discussed.

One problem in the field of computational geometry is the triangulation of convex layers. The rotating caliper algorithm is an alternative to the constrained Delaunay triangulation method. We present an improved triangulation algorithm, which gives a mesh quality close to that of the Constrained Delaunay but substantially faster. Each layer will be connected to the neighboring layer by edges and from the two vertices constituting an edge the proposed algorithm will select the shortest diagonal to its next neighbors in the polygonal chain on the other side, i.e. from the outer layer to the inner layer or vice versa. We discuss quality issues regarding the rotating caliper method and some improvements to it, as well as how a Constrained Delaunay can be efficiently implemented for convex layers.