# Quadratic Interpolation

Robert P. Munafo, 2023 Jun 18.

Quadratic interpolation is a patch iteration method qualitatively similar to the earlier synchronous orbit iteration (SOI) method, but using more key points. It was first implemented in the AlmondBread program in 1997.

As in the synchronous orbit algorithm, a rectangular grid of points is iterated in parallel by iterating just a few "key" points, and using interpolation to get answers for all the other pixels. But whereas SOI uses four key points (the corners of the rectangle), quadratic interpolation uses nine key points, by adding the midpoint of each edge plus the point at the center of the rectangle. All the other pixels are then determined using quadratic interpolation.

In Apr 1997, Michael Ganss wrote:

[...] AlmondBread uses 3 key points per dimension (9 total) and interpolates between these points using Newton interpolation of 2nd degree. Approximation is a method to find the (approximate) value of a known function.

"Newton interpolation" suggests that they were using a Newton polynomial technique to generate coefficients.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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