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Barron Gillon wrote in message <3cacf454@news.povray.org>...
>Is there any built in way to find the exact length of a spline in POV 3.5?
Mark Wagner <mar### [at] gte net> replied:
> POV-Ray doesn't know or care about the length of a spline. In fact, I
don't
> know if it's even possible to calculate the exact length of a cubic or
> natural spline.
I don't believe it is - from my studies of splines a while back it seems the
integration of the relevant functions just isn't possible. The easiest way
to find a spline's length, then, is probably sampling linear distances at
steps along the spline. Also, a rough approximation can be calculated using
the bezier hull length. My own Spline Macro File
(http://www.geocities.com/ccolefax) uses both methods to allow returning the
length of splines and other features like constant-speed traversing of
spline paths.
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Chris Colefax <chr### [at] tagpovrayorg> wrote:
> The easiest way
> to find a spline's length, then, is probably sampling linear distances at
> steps along the spline.
Isn't the main problem with this that it always returns a value which
is too small? (Although usually it doesn't matter as long as it's close
enough to the true value...)
--
#macro M(A,N,D,L)plane{-z,-9pigment{mandel L*9translate N color_map{[0rgb x]
[1rgb 9]}scale<D,D*3D>*1e3}rotate y*A*8}#end M(-3<1.206434.28623>70,7)M(
-1<.7438.1795>1,20)M(1<.77595.13699>30,20)M(3<.75923.07145>80,99)// - Warp -
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"Warp" <war### [at] tagpovrayorg> wrote:
> Isn't the main problem with this that it always returns a value which
> is too small? (Although usually it doesn't matter as long as it's close
> enough to the true value...)
probably ...
because, if you do a linear-interpolation between the two points,
it is quite uncertain, that the way-on-spline is shorter ... No?
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Jan Walzer <jan### [at] lzernet> wrote:
> because, if you do a linear-interpolation between the two points,
> it is quite uncertain, that the way-on-spline is shorter ... No?
A line is the shortest distance between two points. This means that any
other path is longer. Thus the spline can never be shorter than the linear
estimate. As a spline seldom consists of straight lines (except for the
linear spline), it's always longer than the linear estimate.
--
#macro M(A,N,D,L)plane{-z,-9pigment{mandel L*9translate N color_map{[0rgb x]
[1rgb 9]}scale<D,D*3D>*1e3}rotate y*A*8}#end M(-3<1.206434.28623>70,7)M(
-1<.7438.1795>1,20)M(1<.77595.13699>30,20)M(3<.75923.07145>80,99)// - Warp -
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"Chris Colefax" <chr### [at] tagpovrayorg> wrote in message
news:3cad4420@news.povray.org...
> to find a spline's length, then, is probably sampling linear distances at
> steps along the spline. Also, a rough approximation can be calculated
using
> the bezier hull length.
Is it the same approximation method as mentioned in
http://research.microsoft.com/~hollasch/cgindex/curves/cbezarclen.html ?
Regards,
Gleb
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"Warp" <war### [at] tagpovrayorg> wrote:
> > because, if you do a linear-interpolation between the two points,
> > it is quite uncertain, that the way-on-spline is shorter ... No?
>
> A line is the shortest distance between two points. This means that any
> other path is longer. Thus the spline can never be shorter than the linear
> estimate. As a spline seldom consists of straight lines (except for the
> linear spline), it's always longer than the linear estimate.
seems we're telling each other to be right ...
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Chris Colefax wrote:
>
> I don't believe it is - from my studies of splines a while back it
> seems the integration of the relevant functions just isn't possible.
A quadratic spline
P(x) = Ax^2 + Bx + C
has the derivative in x of
dP(x)/dx = 2Ax + B
which has a magnitude of
sqrt( 4(A.A)x^2 + 4(A.B)x + (B.B) ),
which can be integrated over x (the integral is in my calc book).
A cubic spline,
P(x) = Ax^3 + Bx^2 + CX + D
Will have a quadratic derivative in x, namely
dP/dx = 3Ax^2 + 2Bx + C
The magnitude of which is
|dP/dx| = sqrt( 9(A.A)x^4 + 12(A.B)x^3 + 4(B.B)x^2 + 6(A.C)x^2
+ 4(B.C)x + (C.C) )
Dunno if this can be integrated over x. Taylor's expansion might
provide a clue to the integral.
Regards,
John
--
Rusty is rendering!
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Jan Walzer <jan### [at] lzernet> wrote:
> seems we're telling each other to be right ...
I wasn't saying "no, you are wrong, it's this way". I was just profundizing
in the subject. :)
--
#macro N(D)#if(D>99)cylinder{M()#local D=div(D,104);M().5,2pigment{rgb M()}}
N(D)#end#end#macro M()<mod(D,13)-6mod(div(D,13)8)-3,10>#end blob{
N(11117333955)N(4254934330)N(3900569407)N(7382340)N(3358)N(970)}// - Warp -
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"Warp" <war### [at] tagpovrayorg> wrote:
> > seems we're telling each other to be right ...
>
> I wasn't saying "no, you are wrong, it's this way". I was just profundizing
> in the subject. :)
I didn't say that you're saying that, either ... ;)
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"Chris Colefax" <chr### [at] tagpovrayorg> wrote:
> to find a spline's length, then, is probably sampling linear distances at
> steps along the spline. Also, a rough approximation can be calculated
> using
> the bezier hull length.
Gleb <gk1### [at] sotonacuk> asked:
> Is it the same approximation method as mentioned in
> http://research.microsoft.com/~hollasch/cgindex/curves/cbezarclen.html ?
Yes, the average of the linear length between the end points and the linear
length of the bezier hull is the approximation I've used (in fact, I believe
I discovered this at the same URL you posted).
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