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Through Points spline, uniqueness

Genius
Genius
To my knowledge, three poles together produce a unique degree-2 bezier spline.
 
I just find that, as shown below, with three defining points, there can be a lot of degree-2 bezier splines, although NX Studio Spline gives only one.
 
I wonder whether this might lead to some issues in using Through Points splines, and what rule need be follow to avoid them.
 
Thanks for your comments!
 
Image.png
9 REPLIES 9

Re: Through Points spline, uniqueness

Siemens Phenom Siemens Phenom
Siemens Phenom

Your observation is correct. A list of data points to be interpolated (passed through) does not fully determine a spline curve, or even a Bezier curve.

 

To properly define a spline, you also need to assign parameter values to the data points. You can't do this in interactive NX, but you can in the SNAP API (and in NX/Open, too, I think).

 

Problems will arise if you have a workflow that assumes that the data points fully define the spline. For example, if you construct a spline curve through given points in two different systems, you should not expect to get the same curve.

山田
yamada

Re: Through Points spline, uniqueness

Genius
Genius

Hi @Yamada   Very clear. Thank you!

 

Re: Through Points spline, uniqueness

Siemens Phenom Siemens Phenom
Siemens Phenom

I have never seen this as a problem.
Typically one uses the through points method to solve a shape where the points somehow are predefined and must be passed by the spline, the specific shape is maybe less important. 
and the by poles method when the shape of the spline is maybe more important than the absolute position of the spline.

Regards,
Tomas

Re: Through Points spline, uniqueness

Genius
Genius

Hi @1u7obd   Thank you for your comments!

Re: Through Points spline, uniqueness

Honored Contributor
Honored Contributor

I have found that the engineering mind trends toward points, they like to define things precisely.  The styilng mind, tends towards poles, for asthetic reasons.

 

I've worked with both, and use poles to define my splines, as I find they behave better Smiley Happy

-Dave
NX1867(if it had versions) | Teamcenter 11.6 | Windows 10

Re: Through Points spline, uniqueness

Genius
Genius

Hi @DaveK   Thanks for sharing!

Re: Through Points spline, uniqueness

Genius
Genius
I have a further question regarding this topic.
 
As shown in the original post,
 
  • 3 poles together are enough to give a unique degree-2 B-spline.
  • through-points (two end points + a middle one without parameter values assigned) together are not enough to give a unique degree-2 B-spline.
 
Is it possible to know how many through-points  (two end points + middle ones without parameter values assigned) are enough to give a unique degree-2 B-spline?
 
Thanks! @Yamada

Re: Through Points spline, uniqueness

Siemens Phenom Siemens Phenom
Siemens Phenom
Four (though they have to be coplanar, of course).

For details, you can try googling for “parabola through four points”.

This doesn’t quite give you the uniqueness you want, because there are often *two* parabolas passing through 4 given points.

Of course, if you have 5 coplanar points, these uniquely determine a conic, which is a rational quadratic Bézier curve.
山田
yamada

Re: Through Points spline, uniqueness

Genius
Genius

Hi @Yamada   Thank you for your explanation!

 

I know that there's simple equation for poles: number of poles = number of segments + degrees. I had been thinking about whether there's a similar equation for through points. The situation seems to be more complicated than I imagined. 

 

I try to get some understanding about through points, because a through point is similar to an internal control curve in Studio Surface.  A difference is that internal control curves for Studio Surfaces are already assigned parameter values by the boudary curves. I'm just thinking about one qustion: for a degee m×n B-surface, how many internal control curves can be input without causing the surface segmentation.