Comments on: Higher-dimensional, higher-order derivatives, functionally

By: Mark Wassell

Mark Wassell — Mon, 30 Jun 2008 08:43:07 +0000

Hi,

I am trying to write a Haskell function that will create a surface by extruding a (closed) curve f along a curve g. So given f: u -> (x,y) and g : v -> (x,y,z), I end up with h : (u,v) -> (x,y,z).

As I understand it the calculation for a given (u,v) is to calculate the slope of g at v and then to translate and tilt f u accordingly.

I think I need to have Double :~> (Double,Double) and Double :~> (Double,Double,Double) as types for f and g. Firstly I need the slope of g at v and secondly I need to pass on the means of obtaining the derivatives of the resulting surface (although thinking about the torus example, this can come for free?). I will be returning (Double,Double) :~> (Double,Double,Double).

Does this sound as though I am on the right track?

By: conal

conal — Fri, 23 May 2008 15:43:52 +0000

Yes, the notion of derivative towers in Beautiful differentiation seems to correspond to the Taylor series expanded at that point, with coefficients multiplied by progressive factorials. See The Music of Streams by Doug McIlroy, especially the conversions between Horner and Maclaurin form.

The towers in this current post ("Higher dimensional, ...") are much more general, and I imagine have an analogous relationship to a generalized notion of Taylor series and probably to section 2.4 in The Music of Streams.

By: Lemming

Lemming — Fri, 23 May 2008 14:45:07 +0000

I guess the derivative tower is a power series when neglecting factorial factors.

By: Dougal Stanton

Dougal Stanton — Wed, 21 May 2008 10:33:03 +0000

Fabulous stuff! You also helped me to realise that something I’d been playing with recently was a vector space.