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Recently i've been messing around with interpolation methods on scalar fields because im trying to make the scalar field of discrete points, continuous.

I think I understand how bi-linear interpolation works. To my knowledge,In order to interpolate a value(using bilinear interpolation) you have to do three "lerps" using four discrete neighbor points on the field. But when it comes to higher order interpolation methods, I'm lost. I know how to use quadratic interpolation  to interpolate a value using 3 discrete points on a function. How would I use quadratic interpolation on a scalar field?

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I suspect that cubic interpolation rather than quadratic interpolation is the correct tool for this job. This website (http://freespace.virgin.net/hugo.elias/models/m_perlin.htm) describes 1D cubic interpolation in the function Cubic_Interpolate.

To extend this to 2D, I think you would need to do 4 cubic interpolations on the horizontal, then feed in those 4 results for one final cubic interpolation for the vertical.

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You might want to take a look at the generalization of Bézier Curves : http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Generalization

You can do interpolation to any degree.

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