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More Fourier Noise Contours

Previously, I used single-frequency Fourier noise to do some halftoning. This time, I’ll use broad-frequency noise. I’ll filter 2D white noise with a gaussian with standard deviation . This reduces the high-frequency content. Let’s call the filtered noise , and the gradient-scaled version. Skipping all the tedious probability work, the pdf of the gradient-scaled noise […]

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Estimating Zero-Contour Length of Fourier Noise

One way to do stochastic halftoning is to draw contours of noise functions. For example, you can make 2D single-frequency Fourier noise, and draw a contour wherever the noise crosses the zero-level. If you do this with wavelength L, over an area of A, the total arc length turns out to be very nearly 2.2*A/L. […]

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Checkerboard Distortion Halftoning

In a previous post, I deformed a grid of lines to halftone an image. I wanted to do the same thing with a checkerboard, but I figured that it would be similarly difficult and would require optimization. So I used the same optimization framework with a checkerboard, applied it to a brightness ramp, and found: […]

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Grid Distortion Halftoning

Imagine that we have a screen door, and we want to push around the wires to make an image. It’s not a simple problem! There’s a fixed number of wires in each direction, so how do we properly distribute them? Let’s define a pair of phase maps for x and y, and . We’ll draw […]

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Joukowski Airfoil Performance

The Joukowski transformation is a great way to study basic airfoils using potential flow. It gives a nice way to derive the Kutta condition, as well as the classic lift slope . But in all those demonstrations, I never saw the lift and drag curves of the Joukowski airfoil! Luckily, I have XFOIL to remedy […]

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Color Mapping: Swapping Image Palettes

Sometimes you want one picture to have the color scheme of another. This is called color mapping or transfer. When I was making the rearranged images, I was concerned that I would get bad matches if I didn’t make sure the images were similar enough. It was helpful to ensure that their brightness histograms matched, […]

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Spiral Halftoning

Let’s halftone an image with some overlapping wiggly spirals! I want to draw N lines, spiraling out from the center of the image, and have them each slightly offset. With the proper offset, this will create the (approximately) proper darkness. With image darkness k, line width w, and line spacing L, we can get a […]

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Geometrically Approximating Pi

Archimedes found good estimates of pi by computing the perimeter of regular polygons that fit snugly inside and outside a circle. I’d like to take this basic approach and add some more modern twists. I say more modern, but I don’t mean cutting-edge. Taylor series are from the 1700s CE, compared to Archimedes living in […]

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The Sleekest Airfoil

What does a minimum force airfoil look like? I basically want to make a fairing that minimizes the total force over the range of 0-5 degrees. This mostly means reducing the lift generated at non-zero angles of attack, while keeping the drag manageable. Minimizing lift on an airfoil feels oddly perverse. I suppose that’s why […]

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Improving the NACA 0015 Airfoil

The four-digit NACA airfoil is perfectly decent, which is a bit surprising considering it was first published way back in 1933. Since it was designed before computers, I’m sure that it could be improved. The general form of the thickness equation is: This does not provide the proper thickness, but ensures a rounded leading edge […]