Spline path computation using integer addition

A while back, I posted to the RepRap orgs with an idea for using spline paths instead of linear paths on an extruder.

Whether this is a good idea or not, I wanted to put down the algorithm for performing the integer spline computations.

Given:
X0, X1, dXn, dXn as the endpoints, and derivatives of a function x(t) for t=0 and t=n.

Desired:
A set of functions to compute x(t+1) give x(t), using only addition.

Solution:
Using algebra, and solving for the various functions needed:

Cubic spline basis:
x(t) = at^3 + bt^2 + ct + d
x'(t) = 3at^2 + 2bt + c

Endpoints and derivitives at endpoints yields the following:

a = ( n * dX0 + 2 * X0 - 2 * X1 + n * dXn ) / ( n^3 )
b = -( 2 * n * dX0 + 3 * X0 - 3 * X1 + n * dX1 ) / ( n^2 )
c = dX0
d = X0

To solve using only addition, functions p(t), q(t), and r(t) are defined as solutions to:

x(t+1) = x(t) + p(t)
p(t+1) = p(t) + q(t)
q(t+1) = q(t) + r(t)
r(t+1) = r(t)

The solution to this system is:

p(t) = 3at^2 + 3at + 2bt + a + b + c
q(t) = 6at + 6a + 2b
r(t) = 6a

So, to get the points x(0), x(1), x(2), ..., x(n-1), x(n) using only integer addition:

x(0) = X0
p(0) = a + b + c
q(0) = 6a + 2b
r(0) = 6a

x(t+1) = x(t) + p(t)
p(t+1) = p(t) + q(t)
q(t+1) = q(t) + r(t)
r(t+1) = r(t)

To use only integer addition, all numbers above are coded using a fixed point integer:

POS(t) = x(t) / 65536

The accuracy is relatively good for 64 steps. The cumulative errors in the calculation require greater accuracy for more steps than this.

For N dimensions, you need 4 values per dimension per spline path. Note that for continuous spline cubics, the x(0) point does not need to be computed for any except the first because it will be the x(n) of the previous cubic.

Dirt and Roots

The daily sketch group topic is NATURAL FEATURE OF DIRT AND ROOTS RISES OUT OF WINDSWEPT PLAIN.

Done in Corel Painter.

Drop Ship

The daily sketch group today was "HULKING MILITARY ROBOT IS UNLOADED FROM DROP-SHIP AT DOCK PLATFORM". Wanted something slightly organic, while still seeming somewhat mechanical. This is what I ended up with.

This is a rough sketch of my son and Daisy Sniffwiggen. I used Corel Painter for this.

Corel Painter seems reasonable. I used Paintshop for the images I made yesterday. This one was done using a chalk medium, then smeared to remove most of the textures, then a paper surface applied. I think this one came out fairly well.

Been a while

It's been a while since I created this blog. Still not quite sure what I'm going to do with it. Just got a new Wacom tablet, and have been playing around with it.

To try and get a hang of it, I've been doing a few sketches and such. Since I've been getting ideas from a few places around the net, thought I'd post them here so I can post them to the forums.
This picture is from one ConceptArt.org's daily sketch group articles. The topic was "Mechanical Soldier Worms Made Trench Warfare That Much Worse."

I did one for yesterdays topic too, but decided not to post it to the forum. This was titled "Mouse Suits-Up For The Long Journey".

First Post

This is my first blog. I'll be adding more content later, once I figure out what I want to do here.

For now, this is just a placeholder.