I have been experimenting for a while now on curve creation using ICE and I have created quite a few nodes.
Below is what I have done so far.

Reading various SIGRAPH papers and math books I notice that they are using matrices. So I improved on the ICE nodes and created various nodes in matrix form. The _m at the end of their names identify the curve as a matrix form.

I am still researching chordal / centripetal / uniform curves and NURBS curves.

I will then move to piece wise curves and then surfaces. That should be interesting.

More to come.

And ... just for the sake of it and if you want to join the adventure, here is two ICE nodes curves in matrix form:

BD_Bezier4_m
DB_RationalBezier4_m

Enjoy!

Dan

Could you perhaps show a screenshot of these compound in action,
i.e. where they go in an ICE tree and what they actually do?

Heres a quick screenshot of my tests of the compounds. Much like fabric Bezier compounds they construct the curves (or rather description for the curve since ICE doesnt support xsi curves yet) except they are constructed differently. Im using debugging tools to display them here.

Interesting work, Daniel.

Wish i would understand whats going on lol

Great compounds, thanks!

Thank you for all the good feedbacks.

A curve in matrix form is easier to experiment with and I have noted little difference in speed between the polynomial formula and the matrix form. The advantage with the matrix form is how easy it is to switch between curve definitions by just changing the matrix.

As an example, here is the rational bezier3 in matrix form and it's equivalent in polynomial form. The polynomial form is more daunting and can become a mess very quickly.

Here are two more nodes that I will use to demonstrate their usage:

DB_Bezier3_m
DB_RationalBezier3_m

If you try to create a circle with the normal bezier3, you will approximate it but it will not be a correct circle. With the rational bezier3 you can create a perfect circle.

A rational curve introduce the notion of weight on each point. Each points are therefore a 4D representation (x, y, z, w) where the w is how much influence the point will have on the curve (pull toward (w greater than 1) or push away (w less than 1) from the point, with 1 being neutral)

As an example, here is a perfect circle created with three rational bezier3 curves. The tip of the triangle has less weight at those points (0.7071) to create the circle. Every where else, the points have equal weight of 1.

Each rational bezier3 represent a segment of the circle (one third of it) with the end of one segment being the start of the next segment. To close the circle, we make the last point the first point of the circle. The red triangle is only to show a representation of the three rational bezier3, linked together (not normally needed). Each curve segments are colored differently.