Higher order curves : Java-Games

Curves of degree more then 3 are not very well covered
(i cant get any reading material).

Are there (online, book, papers, anything) references
on 4,5,6(more?) degree Beziers, splines etc ?

Algorithms and/or source will also be helpful.

In article <1115896049.115331.138860@g49g2000cwa.googlegroups .com>,
stafygrahics@yahoo.com wrote:

>Curves of degree more then 3 are not very well covered

Perhaps because it gets increasingly complicated to specify and render
them.

In article <1115896049.115331.138860@g49g2000cwa.googlegroups .com>,
stafygrahics@yahoo.com wrote:

>Curves of degree more then 3 are not very well covered

Perhaps because it gets increasingly complicated to specify and render
them.

"Lawrence D¹Oliveiro" <ldo@geek-central.gen.new_zealand> wrote in message
news:ldo-3F9293.20052813052005@lust.ihug.co.nz...
> In article <1115896049.115331.138860@g49g2000cwa.googlegroups .com>,
> stafygrahics@yahoo.com wrote:
>
> >Curves of degree more then 3 are not very well covered
>
> Perhaps because it gets increasingly complicated to specify and render
> them.

In addition, in the case of interpolation splines, results do not improve
much after an order 3 curve, in some cases they are worse.

The principles of deriving quadratic and cubic splines can be found in
quite a few places (google is your friend!). Extending them to 4th order
etc.. will take some time but you'll be able to work it out if you know the
process for the lower order splines. Using linear algebra the process is
not quite as bad, but less intuitive (haven't gone quite that far myself
yet).

I can't comment extensively on Beziers, as they were not as easy to work
with (for interpolation) in my experience.

Rod

"Lawrence D¹Oliveiro" <ldo@geek-central.gen.new_zealand> wrote in message
news:ldo-3F9293.20052813052005@lust.ihug.co.nz...
> In article <1115896049.115331.138860@g49g2000cwa.googlegroups .com>,
> stafygrahics@yahoo.com wrote:
>
> >Curves of degree more then 3 are not very well covered
>
> Perhaps because it gets increasingly complicated to specify and render
> them.

In addition, in the case of interpolation splines, results do not improve
much after an order 3 curve, in some cases they are worse.

The principles of deriving quadratic and cubic splines can be found in
quite a few places (google is your friend!). Extending them to 4th order
etc.. will take some time but you'll be able to work it out if you know the
process for the lower order splines. Using linear algebra the process is
not quite as bad, but less intuitive (haven't gone quite that far myself
yet).

I can't comment extensively on Beziers, as they were not as easy to work
with (for interpolation) in my experience.

Rod

Rod Runnheim wrote:
>>
>>>Curves of degree more then 3 are not very well covered
>>

If nobody uses them it's for a reason....!

>>Perhaps because it gets increasingly complicated to specify and render
>>them.
>

Not only that, but they have many undesirable
properties, instabilities, non-localness, etc.

Piecewise cubic curves are the way to go.

> I can't comment extensively on Beziers, as they were not as easy to work
> with (for interpolation) in my experience.

You don't use Beziers directly for interpolation
but all piecewise curves reduce to a list of
Beziers for final calculations (which is nice).



--
<\___/>
/ O O \
\_____/ FTB. For email, remove my socks.

Rod Runnheim wrote:
>>
>>>Curves of degree more then 3 are not very well covered
>>

If nobody uses them it's for a reason....!

>>Perhaps because it gets increasingly complicated to specify and render
>>them.
>

Not only that, but they have many undesirable
properties, instabilities, non-localness, etc.

Piecewise cubic curves are the way to go.

> I can't comment extensively on Beziers, as they were not as easy to work
> with (for interpolation) in my experience.

You don't use Beziers directly for interpolation
but all piecewise curves reduce to a list of
Beziers for final calculations (which is nice).



--
<\___/>
/ O O \
\_____/ FTB. For email, remove my socks.

In article <Y54he.44395$US.5333@news.ono.com>,
fungus <openglMY@SOCKSartlum.com> wrote:

>Piecewise cubic curves are the way to go.

I'm not so sure about that. I must admit I lean more towards quadratics
than cubics. There seems to be no easy automatic way to tell when you
can stop rendering a cubic (which is why PostScript, for example, needs
an explicit flatness setting to tell it when to stop), unlike a
quadratic, where the De Casteljau rendering algorithm has a second-order
convergence towards an error of less than one pixel, at which point you
can stop.

In article <Y54he.44395$US.5333@news.ono.com>,
fungus <openglMY@SOCKSartlum.com> wrote:

>Piecewise cubic curves are the way to go.

I'm not so sure about that. I must admit I lean more towards quadratics
than cubics. There seems to be no easy automatic way to tell when you
can stop rendering a cubic (which is why PostScript, for example, needs
an explicit flatness setting to tell it when to stop), unlike a
quadratic, where the De Casteljau rendering algorithm has a second-order
convergence towards an error of less than one pixel, at which point you
can stop.

"fungus" <openglMY@SOCKSartlum.com> wrote in message
news:Y54he.44395$US.5333@news.ono.com...
> Rod Runnheim wrote:
>>>
>>>>Curves of degree more then 3 are not very well covered
>>>
>
> If nobody uses them it's for a reason....!

You do not have enough information to conclude that nobody
uses them. And the OP's statement about "not very well
covered" is not true. There are many well-known books on
B-spline curves that cover the general degree case (Farin's
books, for example).

>>>Perhaps because it gets increasingly complicated to specify and render
>>>them.
>>
>
> Not only that, but they have many undesirable
> properties, instabilities, non-localness, etc.

What ever is this supposed to mean? What "instability"?
What do you mean by "non-localness". For example,
B-spline curves of any degree have local control. The
evaluation is also robust and stable.

> Piecewise cubic curves are the way to go.

Except when quadratic curves are the way to go. Or
any other degree for that matter. Natural splines are
piecewise cubic, but if your data points are dynamic,
then updating a single data point involves solving a
large linear system (tridiagonal, so O(n) instead of
O(n^3) for Gaussian elimination). But a B-spline
curve has local control, so you have much less work
to update the curve.

>> I can't comment extensively on Beziers, as they were not as easy to work
>> with (for interpolation) in my experience.
>
> You don't use Beziers directly for interpolation

And another "what does this mean"?

> but all piecewise curves reduce to a list of
> Beziers for final calculations (which is nice).

No. Maybe you mean "piecewise polynomial".

--
Dave Eberly
http://www.geometrictools.com