Inferring transforms [computer graphics]

In simple two-dimensional texture mapping you take a 2D image and render it on the screen after some transformation or distortion. To accomplish this you will need to take each [X, Y] location on the screen and calculate a [U, V] texture coordinate to place there. A particularly common transformation is: U=(aX+bY+c)/(gX+hY+j), V=(dX+eY+f)/(gX+hY+j). By picking the proper values for the coefficients a…j, we can fly the 2D texture around to an arbitrary position, orientation, and perspective projection on the screen. You can, in fact, generate the coefficients by a concatenation of 3D rotation, translation, scale, and perspective matrices. However, the author discusses a more direct approach to finding a…j. It turns out that the 2D-to-2D mapping is completely specified if you give four arbitrary points in screen space and the four arbitrary points in texture space they must map to. The only restriction is that no three of the input or output points may be collinear. This method of transformation specification proves useful, for example, in taking flat objects digitized in perspective and processing them into orthographic views     

Polynomial discriminants. 2. Tensor diagrams

Recently I've been playing with diagrammatic ways of doing algebra and have come up with a lot of interesting results. This article presents the first of these - using tensor diagrams to compute discriminants of polynomials and to solve a related problem: line-curve tangency. I believe that the Einstein index notation can help us think about these and similar problems and allow us to come up with solutions that we wouldn't find any other way.     

Cubic curve update

The author considers the problem of cataloging all the shapes that can be generated by a cubic equation. In response to a letter commenting on an earlier column, he corrects an error in his listing of all the combinations of factorizations that could make degenerate curves, pointing out that all type 5's are really the same shape and all type 7's are really the same shape. He provides additional considerations on the possible shapes of nondegenerate curves     

Jim Blinn's corner-return of the Jaggy [high frequency filtering]

The antialiasing problem of filtering out high frequencies before sampling is considered. The techniques examined are simple filters, box filters, the triangle or tent filter, Gaussian and similarly shaped filters, and the ideal filter. The use of subsampling and the effect of D/A (digital-to-analog) converters are discussed     

The ultimate design tool

The author design tools from a tool user's point of view. The author considers the creative process and where design tools fit into that process. He considers pencil and paper as a tool and argues that they are better than computers as an ideation tool     

What's that deal with the DCT?

Discrete cosine transforms (DCTs) and discrete Fourier transforms (DFTs) are reviewed in order to determine why DCTs are more popular for image compression than the easier-to-compute DFTs. DCT-based image compression takes advantage of the fact that most images do not have much energy in the high-frequency coefficients. It is suggested that DCTs are more popular because fewer DCT coefficients than DFT coefficients are needed to get a good approximation to a typical signal, since the higher-frequency coefficients are small in magnitude and can be more crudely quantized than the low-frequency coefficients     

Three wrongs make a right

When dealing with graphics operations that must be fast (like the inner loops of rendering algorithms), I usually like to do calculations with fixed-point arithmetic (that is, scaled integers) rather than floating point arithmetic. The exact scaling factor used can have some interesting effects on the speed and errors in the calculation. In this article, I'll give some titbits I've discovered or picked up from others about this. In particular, I'll talk about some of the advantages of using odd numbers of the form 2ⁿ-1 as scaling factors. The motivation for this discussion is the desire to do arithmetic on pixel values: red, green, blue, or alpha. These values are in the range 0...1, so all numbers you see here are positive. In the discussion that follows, I'll use floating point as a testbed and as scaffolding to derive integer formulas. All final calculations take place using only integer arithmetic     

A trip down the graphics pipeline: the homogeneous perspectivetransform

The perspective transformation, which basically turns space inside out, is discussed. Some interesting topological properties of the space represented by homogeneous coordinates are reviewed. It is shown that the perspective transform has practical applications in selecting near and far clipping planes to avoid depth resolution problems encountered with many types of rendering algorithms