Curve Derivatives

Grasshopper
1

Hi all,

I have been searching in the forum for days, but couldn’t really find answer to my question about CDiv (Curve Derivatives) component in Grasshopper. I understand that the 1st derivative is tangent and the 2nd is curvature. It seems like the first in the component represents correct information-parallel to tangent vector at a given point, but I don’t understand why the 2nd differs from the direction of actual curvature vector. Could someone please shed some light on it?

image
image1158×863 22.1 KB

Thank you in advance,

W

1 Like
3

I had the same question too. Just posting to keep the question afloat.

4

This is a incorrect assumption based on school math. :slight_smile:

see this:

basically what you need to do is:

get a Frenet Frame, however this can flip normals in some situations. Better is a “Bishop Frame”
Its basically a comparision between the Frenet Frame at t and a Fernet Frame at t-tinystep.
Its nothing difficult. Just simple Vector operations.

A Frenet frame at t:
tangent= normalize(N’(t))
v= normalize(tangent+ N’’(t))
r = normalize(v × tangent)
normal = normalize(r × tangent)

Edit “Bishop Frames” are also know as “Rotation Minimising Frames”

2 Likes
5

This is a python implementation I made some years ago:

def getFrame(cps,t):
    ders = curveDerivatives(cps,t,3)
    dersOff = curveDerivatives(cps,t+0.001,2)
    pt = ders[0]
    tan = nor(ders[1])
    tanOff = nor(dersOff[1])
    r = cro(tan,tanOff)
    r = nor(r)
    nrm = cro(tan,r)
    return (pt,tan,nrm,r)

where "cro" is the "crossproduct" and "nor" is "normalising"
1 Like
6

And this is what R can offer:

Crv_DerivativesEntryLevel_V1.gh (123.4 KB)

1 Like
7

Tom,

Thank you for the good explanation. Please excuse my lack of knowledge on differential geometry, but is it fair to say that a curvature vector that we are given from a gh component is just an approximation or just for visualization purpose?

8

Peter,

Thank you for the example! What is the vector when derivative set to zero?

image
image1404×712 97.4 KB

9

A derivation of a NurbsCurve is a NurbsCurve of lower order.

Since you evaluate for a specific t in three dimensions you get a three dimensional vector back. This vector however doesn’t have anything to do with the normal of a curve, as well as the curvature. The curvature is calculated with the first and second derivative but its not just the second derivative.
Now, I’m not a mathematician, so I’m not able to prove it to you, especially in such a short comment. I’m just knowing that it is like this, from reading books and papers about Bezier and Nurbs.

1 Like
10

@woojsung this is how you calculate the curvature:

curvature
curvature.png1569×665 69.8 KB

CalculateCurvature.gh (11.7 KB)

3 Likes
11

@TomTom, I can’t thank you enough. I think it is getting really interesting at least to me. Could you please share any references so I can learn more about the formula? Ever since I became interested in the mathematical logic behind what I build in Rhino, I have been stumbling through books on topics like differential geometry, topology, but I feel like I am lost. Any books with lots of illustrations you could find useful on those subjects? - that would be much appreciated!!

1 Like
12

I’ll prepare a V1A updatte that explains all things

13

Oh, that is very nice of you! Thank you for your time for this.

14

@TomTom, I think I got the equation until it is devided by ||d1||. Very curious to know why…

15

But since there’s a MotoGP w/e (Brno) going on right now … in the mean time do some preparation: just read the 327 pages of this all times classic book and become a NURBS jedi master (Yoda and the likes):

Screen%20Shot%20066
Screen Shot 066.JPG806×727 154 KB

3 Likes
16

Haha sure no worries. Enjoy The race. Thank you for the reference.

17

Hero Valentino starts 2nd (cross fingers).

BTW: The Rhino Method (as is also the case in other Rhino Methods [trans matrices etc]) ouputs derivative vectors differently : C2 is Rhino’s C0, C1 is C1 and C3 is C2.

Screen%20Shot%20070
Screen%20Shot%200701339×746 203 KB

2 Likes
18

Will do. That is great! Now the mystery solved. Of course I still need more time to digest it though. Thank you very much!

19

BTW: Other than Master’s book (the best NURBS thingy ever) …
Screen%20Shot%20072

…and other than some basic (kinda) maths … I do hope that you speak some proper language (C, C++, C# etc).

Screen%20Shot%20074
Screen Shot 074.JPG1268×448 247 KB

see also:

http://www.nar-associates.com/nurbs/c_code.html

1 Like
20

Thank you @PeterFotiadis. A Primer on Bezier Curves, a friend of mine recommended as well, is one of my favorite materials - it seems like it is a live document, I just realized. Anyways lots of homeworks!

21

I do own that Book as well. Its however not that easy, intuitive and straightforward as the authors want it to be. mathbooks… tztz :dizzy_face:

1 Like