Degree of Curves: Is there any good tutorials?


#1

Hi,

I’ve been trying to understand better about degree of curves in general.

The points being used to draw the curve minus 1 equals to degree, but
when selecting to draw a curve with degree of 3, but have more points than 4,
using curvaturegraph the curvature is discontinued at the third point
yet the whole curve description is degree 3.

I was expecting to be curvature discontinuous up to 4th point then in the 5th point
the curvature is calculated from point 2~ point 5.(if 5 points were used to draw the curve.)

If there’s any good tutorials to better understand degree/curvature of curve and perhaps degree of surfaces please post a link! or can someone explain it?


How to control surface with rails and profiles?
#2

Often AutoDesk has good explanations. I don’t know if this document fully covers your case, but perhaps you can start from here [ Evaluation 1: Curve Curvature ]:

// Rolf


(John Brock) #3

This is a good reference too:
http://www.rhino3d.com/nurbs


(David Cockey) #4

The curvatures in the image above are all continuous. The height of the curvature comb is proportional to the curvature.

The rate of change of the curvature is discontinuous at the kinks in the curvature.


#5

david summed it up pretty nice in one sentence nothing to add, only that you can combat discontinuity quite well with the command fair, sometimes hitting it a few times of course.

do you have any idea how i can switch to english? the page seems to autodetect my system language and i have no chance reading it in english.


#6

@RIL Thank you. The drawing helps in understanding better.


#7

@John_Brock Thank you for the link.
I need to go through it couple of times…
May come back to you for some questions later…


#8

@davidcockey For the curves of deg3 drawn with
6control points, is it so that where the discontinuity of the curve is, so kink of the rate of change in curvature, is where two curves are connected at a tangent?
In that case, is the curve after that kink a deg2 curve as it is drawn using 3 control points?


#9

@RichardZ In the screenshot attached,
does the curve on the right still have a difference in the rate of curvature or is it slightly off if you zoom in?
I’ve yet to try the fair command…


(David Cockey) #10

At a kink in the curvature graph the curvature is continuous, but the rate at which the curvature changes as you move along the curve is discontinuous at the kink.

A discontinuity in curvature results in a discontinuity in the curvature graph

A degree 3 curves has continuous curvature unless two or more control points are stacked on top of each other. Two control points on top of each other may result in a discontinuity in curvature. Three control points on top of each other may result in a kink in the curve.

A degree 3 curve with n control points and no control points stacked has n-3 “spans”. Each span in a degree 3 curve has continuous curvature and continuous rate of change of curvature. Curvature is continuous between spans, and rate of change of curvature may be discontinuous between spans.

A degree 2 curve has continuous tangency unless control points are stacked on top of each other. Two or more control points stacked may result in a discontinuity in tangency.

A degree 2 curve with n control points and no control points stacked has n-2 “spans”. Each span in a degree 2 curve has continuous tangency and continuous curvature. Tangency is continuous between spans, and curvature may be discontinuous between spans.


#11

@davidcockey
Thank you for your detail explanation.
I’m starting to understand, but seems I need to study further.
So deg 2 crv promise tangency, deg 3 crv tangency& curvature continuous unless stacked.
Deg 4 crv will I suppose add continuous rate of curvature curve?

Seems I need to go study curve point, curve edit point, span, and knots and their relations.
And after that, pros and cons for different degrees and their practical uses.

Thank you again, I might need to come back to this post later.
For now I’ll use keywords to search and find some materials.


(John Brock) #12

Years ago, there was a really good beginning tutorial from one of the marine designers, that started with an introduction to NURBS curves that described all of this very well.

Sadly, I have lost track of it.


(Scott Davidson) #13

Is this the document? fairing.pdf (165.6 KB)


(Scott Davidson) #14

Gerard wrote this one that has some great information: Rapid_Hull_Modeling_in_Rhinoceros.pdf (308.7 KB)


(John Brock) #15

That’s the one; Rapid Hull Modeling.


What is the best way to connect line and curve with G2 or higher continuity?
(John Brock) #16

@Toshiaki_Takano
This is the one I was remembering:


#17

@scottd @John_Brock
Thank you both!
I’ll have a look through it. Thanks again!


#18

for my rhino university basic courses i use the attached pdf to explain curve degree, single-span and multi-span curves, because it has simple explanations and good illustrations. i downloaded it some years ago from pivot.no.
hope, it helps.
rhino_basics.pdf (2.5 MB)


#19

You can see the explain instructions in vimeo.com
… Rhino3D tutorials chanel.
(Curves curvature and tang.)

Greetings


#20

@brt1
Thanks for the material. It’s a good summary.