I just noticed that sometimes Bézier curves (or B-splines or 
polynomials) are referred to as 2nd or 3rd order, and sometimes as 2nd 
or 3rd degree.  Are these terms interchangeable?

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Cousin Ricky <ric### [at] yahoocom> wrote:
> I just noticed that sometimes Bézier curves (or B-splines or
> polynomials) are referred to as 2nd or 3rd order, and sometimes as 2nd
> or 3rd degree.  Are these terms interchangeable?


I've seen people claim they are the same, but I also found this:

"In B-spline terminology, order means degree + 1."
https://math.stackexchange.com/questions/1730596/n-order-b-splines-interpolation

So maybe order is the number of control points, and degree is the highest power
of the polynomial?

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On 2020-09-26 3:50 PM (-4), Bald Eagle wrote:
> Cousin Ricky <ric### [at] yahoocom> wrote:
>> I just noticed that sometimes Bézier curves (or B-splines or
>> polynomials) are referred to as 2nd or 3rd order, and sometimes as 2nd
>> or 3rd degree.  Are these terms interchangeable?
> 
> 
> I've seen people claim they are the same, but I also found this:
> 
> "In B-spline terminology, order means degree + 1."
> https://math.stackexchange.com/questions/1730596/n-order-b-splines-interpolation
> 
> So maybe order is the number of control points, and degree is the highest power
> of the polynomial?

Thanks, this gave me a springboard for further searching.  It seems that 
the term "order" not only has multiple meanings, it can also be 
ambiguous within the same meaning!  See conflicting definitions of 
"order" just between these Wikipedia articles:
   https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Higher-order_curves
   https://en.wikipedia.org/wiki/Degree_of_a_polynomial
   https://en.wikipedia.org/wiki/Order_of_a_polynomial

My takeaway is that I should purge the word "order" from my code and 
documentation, and stick to "degree."  Also, the release of quadratic 
Bézier splines in SphereSweep will be delayed a day or two because of this.

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On 9/26/20 4:50 PM, Cousin Ricky wrote:
> On 2020-09-26 3:50 PM (-4), Bald Eagle wrote:
>> Cousin Ricky <ric### [at] yahoocom> wrote:
>>> I just noticed that sometimes Bézier curves (or B-splines or
>>> polynomials) are referred to as 2nd or 3rd order, and sometimes as 2nd
>>> or 3rd degree.  Are these terms interchangeable?
...
> 
> My takeaway is that I should purge the word "order" from my code and 
> documentation, and stick to "degree."  Also, the release of quadratic 
> Bézier splines in SphereSweep will be delayed a day or two because of this.

Expect reasonable.

The day to day use is foggy - me as guilty as any I expect. In doing the 
solver work a couple years back, I came to the following understanding:

The degree of a polynomial equation is that of the largest term by 
exponent with a non-zero coefficient.

While programming for polynomials the order is the maximum degree a 
given algorithm can handle. This might be greater than the degree of any 
given equation being evaluated, manipulated, solved for roots, or 
whatever - using that algorithm. The coefficient storage required in 
these domains then is usually the (order + 1) * coefficient_size.

When matrix math comes into play, folks tend to think in terms of matrix 
order which is degree + 1 or 'polynomial' order + 1.

Your kilometers may vary.

Bill P.

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