Re: What to do wanting a 4th order Bézier?

[David Kastrup]

Hans Åberg <address@hidden> writes:

>> On 18 Sep 2016, at 14:41, Simon Albrecht <address@hidden> wrote:
>> 
>> On 18.09.2016 13:54, Andrew Bernard wrote:
>
>>> What is it exactly that you are expecting a quartic to give you?
>> 
>> Oh, I think you’re quite overestimating the amount of in-depth
>> mathematical background I had – I just thought: ‘A 3rd order Bézier
>> curve can have one turning point, but I need two turning points, so
>> I’d need a 4th order Bézier’. Which I now see is wrong, after some
>> experimenting with the interactive fields in that article you
>> linked: it requires a 5th order Bézier for that, and then it already
>> gets quite unhandy.
>
> Curves of higher polynomial order did not catch on, because they are
> not stable: small changes in the input variables can sometimes cause
> dramatic effects in the output curve.

Well, it depends on the input variables.  Bézier polynomials are, after
everything is said and done, polynomials of an independent parameter.
Expressing them in terms of control points makes them more tangible and
makes for a numerically robust evaluation scheme.  That also holds for
higher order Bézier polynomials: moving the control points does not have
"dramatic" effects in the output curve with respect to the x and y
coordinates as a function of the moving parameter t.  Still the
probability of cusps as a result of independent curvature changes in the
separate coordinates increases.

It would be my guess that the hands-on manipulative features of control
points have made cubic Beziers the go-to curve approximation and design
tool.  Even though you need only quadratic NURBS to represent conic
sections and their higher-dimensional equivalents perfectly, you don't
have a similarly straightforward handle on them like control points
provide.

-- 
David Kastrup