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**C**ontituity (intenal continuity?) = *m* (degree) - *r* (multiplicity)

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07-31-2018 02:46 PM

In a previous thread, @Yamada and @StevenVickers gave very helpful comments on the topic of continuity within a multi-segment spline.

For a multi-segment spline, there is:

In most splines without shrinked segments, the multiplicity of middle knots should be 2. Therefore it's easy to judge the internal continuities C1, C2, C3 ... with the equation above.

If a spline is divided at its middle knots, the continuity between the reulting B-curves is often like G1, G2, G3 ...

Is there any definite relationship between these two types of continuity Cx vs. Gx?

Thanks!

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Re: continuity within a multi-segment spline-II

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07-31-2018 03:02 PM

Re: continuity within a multi-segment spline-II

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07-31-2018 04:28 PM

Hi,

A single-segment spline requires one more pole that its degree to define it.

So a 5 degree spline requires 6 poles.

A 5 degree spline with 7 poles has one-too-many to be single segment, so you now have two segments (both degree 5). If you were to increase the degree to 6, segments would reduce accordingly.

8 poles in yours gives you two segments.

Continuity is an indication of "smoothness" at the knot point, it's not related to poles and degree.

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07-31-2018 08:23 PM - edited 07-31-2018 09:26 PM

It is indeed correct that

Continuity (at a given knot) = degree - multiplicity of the knot.

Another important relationship is

Number of knots = number of poles + degree + 1

And the final useful fact is that, in most systems, the start and end knots of a degree m spline will have multiplicity m+1.

Your spline of degree 5 has 8 poles, so it must have 8 + 5 + 1 = 14 knots. The start and end knots will each have multiplicity 6 (=5+1), so that's 12 knots accounted for. The other two knots look like they are at roughly t = 0.3 and t =0.7. Those are the parameter values at which the inter-segment joins occur. Each of these knots has multiplicity 1, so the continuity there is 5 - 1 = 4. In other words, your spline is C4 at each internal knot.

OK, so why does NX say the joints are C2, rather than C4. I think the answer is that the NX Info function assumes you won't care about higher levels of continuity (most people don't). So, when it says "C2", I suspect it means "C2 or better".

Interactive NX doesn't give you any direct control over knot multiplicities (as far as I know). So, the easiest way to play with these concepts is to write little programs that use SNAP functions.

yamada

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07-31-2018 09:10 PM - edited 07-31-2018 10:55 PM

> In most splines without shrinked segments, the multiplicity of middle knots should be 2.

That's not true. It's hard to generalize, but I'd say that the most common multiplicity for interior knots is 1, not 2. Interactive NX doesn't give you any direct control over knot multiplicities (as far as I know), and that makes sense because most people don't care about them. So, the easiest way to play with these concepts is to write little programs that use SNAP functions.

> Is there any definite relationship between these two types of continuity Cx vs. Gx?

Yes, but it involves a lot of mathematics. The basic idea is that Cx continuity is related to derivatives of order x. So, when we say that a spline is C1 at some point, we mean that there is no abrupt change in the first derivative at the point. Gx continuity is related to intrinsic geometric shape properties like tangent direction, curvature, torsion, etc.

Derivatives depend on how the curve is parameterised, not just on its shape. Most NX users don't know anything about parameterization (and don't need to), so Gx continuity is what's important to them. In some sense, Gx is an engineering/manufacturing concept, whereas Cx is for mathematicians.

Another explanation ... think of a car driving along a trajectory across a snow-covered parking lot. Cx continuity tells us how smoothly the car moves, as a function of time. So, if the car's motion is C1, this means that there are no abrupt changes in its **velocity, **(first derivative) and C2 means no abrupt changes in **acceleration **(second derivative). On the other hand, Gx continuity tells us nothing about the motion of the car, it only tells us about the track that the car left in the snow. G1 continuity says that the track has no abrupt changes in **direction**, and G2 says there are no abrupt changes in **curvature**.

To confuse things further, consider the Bezier curve with poles (0,0), (1,1), (0,1), (1,0). This curve is certainly C-infinity (it's C1, C2, C3, C4, ...). But it has a sharp corner in it, so it is not even G1. However, this curve is an exception (because it has a place where its first derivative vector has zero length); it's usually true that any curve that's Cx is also Gx.

There's more in section 18.2 of the attached document.

yamada

Re: continuity within a multi-segment spline-II

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07-31-2018 10:38 PM - edited 07-31-2018 10:40 PM

Hi @Yamada Thank you for your thorough treatment of the topic in your both posts!

Every sentence in them is helpful, expecialy this one "it's usually true that any curve that's Cx is also Gx". As you said, "Gx continuity is what's important", and this is why I want to know how to judge Gx from Cx. Thanks again!

Re: continuity within a multi-segment spline-II

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07-31-2018 10:39 PM

Hi @carl_harrison Thanks for your response!

Re: continuity within a multi-segment spline-II

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07-31-2018 10:55 PM

Very sorry for a further question.

There seems to be two types of symbols for spline knots. Is the guess about them always correct?

- hollow rectangle ≤ C1
- solid circle > C1

Thanks!

Re: continuity within a multi-segment spline-II

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07-31-2018 11:03 PM

The output listing from Info-->Spline says:

Display symbols: C0= filled square with 2 rings C1= filled square with 1 ring C2= filled square Poles=o Defining Points=+

But, as before, "C2" probably means "C2 or better".

yamada

Re: continuity within a multi-segment spline-II

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07-31-2018 11:26 PM

Hi @Yamada Thank you!

It seems that I can't find info you provided in my NX. And there seems to be some problems with the info. Might be due to diffrent versions. Anyway, I know the symbols are linked to the Cx contunity. Thanks!

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