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Ph Gibone wrote:
>> You did not mention if it was a dos or windows program.
> WIndow 32 bits.
This I can use.
> > One question I have is how you handle infinite objects with your
> >program. Several of the examples in the shapesq.inc file are infinite
> >in one plane. If you make a mesh for one of these types the program
> >would not stop making triangles for as long as hard drive would have
> >room to hold them. Do you automaticaly clip the object at some boundry
> >or did you simply make it refuse to handle shapes of this type ?
>
> It is a program to render parametrized functions, not implicit ones so, you
> have to choose the range of the 2 parameters, and so the program knows where
> to stop !
Ok. I believe I understand what you are saying.
> I will write a little help and send it to you wwith the program in a few
> days (everything written in french on the screen).
A very big help - thank you.
> Philippe
>
> PS : I just tried the triaxial tritorus and I wonder if there is an error in
> you equation, I also tried the tear drop : great ! is it a parametrized
> equation of the "well known" piriform or someting else ?
The only thing I see that may be wrong is it looks like I might have
dropped a ")" from the end of the z: coordinates line. See arrow below.
x = sin(u) (1+cos(v))
y = sin(u+2 PI/3) (1+cos(v+2 PI/3))
z = sin(u+4 PI/3) (1+cos(v+4 PI/3) <-------
If that does not fix it I will try to locate the site where I copied
the functions from originaly. I am not sure where it was from but I
have a couple of links that might aim me in the right direction.
As far as the tear drop is concerned I am not sure who did the work on
the parameterazation for that but I can asure you that it was not me.
I use a calculator for counting the number of fingers on my hands :)
--
Ken Tyler
mailto://tylereng@pacbell.net
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On Sun, 18 Apr 1999 22:19:03 -0700, Ken <tyl### [at] pacbell net> wrote:
> A am aware that they are greek symbols. I simply didn't have a
> handy reference to their decriptions, english names, and mathmatical
> equivelent functions they represent when I copied them to my
> "Do it someday" file.
The '8' is a theta, and the 'Note:1' is a phi, if I had to guess.
> I don't have the iso patch here and I still haven't gotten the hang of
> the Superpatch yet, assuming it has that covered. Perhaps some day...
It does indeed have that covered, though I'm not sure how well it will
do on polar coordinates; I need to look at the new iso patch and merge
in the new functionality it has added.
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On 19 Apr 1999 10:41:36 -0500, par### [at] my-dejanewscom (Ron Parker)
wrote:
>It does indeed have that covered, though I'm not sure how well it will
>do on polar coordinates; I need to look at the new iso patch and merge
>in the new functionality it has added.
I rendered the objects with parametric and posted a thumbnail in
povray.binaries.images. It worked fine. Since these are parametric
equations, the new polar coordinate machinery isn't needed to render
them. Just set the u,v limits correctly.
Jerry Anning
clem "at" dhol "dot" com
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Ken wrote:
> This describes the parametric equations which approximatly model a
> drop of water, for example, a tear drop.
...
> The equations as functions of longitute phi and lattitude theta are:
>
> x = 0.5 *(1-cos(8)) sin(8) cos(Note:1)
> y = 0.5 *(1-cos(8)) sin(8) sin(Note:1)
> z = cos(8)
An implicit equation for the aforementioned tear drop is:
1 - 4x^2 - 4y^2 - 2z + 2z^3 - z^4 = 0,
or, it simplifies a bit as 4(x^2+y^2)=(1+z)(1-z)^3, which is a surface
of revolution bounded by -1 <= z <= 1. The POV command is:
quartic{ <
0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0,
-1, 2, 0, -2, 1> }
Sometimes, a parametric map involving trig functions can be converted
to an algebraic implicit equation by using new variables:
u=cos(t),
v=sin(t),
and introducing a new implicit relation:
u^2+v^2=1.
The idea is to make the parametric equations into polynomials, at the
cost of increasing the number of variables and equations. Then u and
v can (sometimes) be eliminated from the system of polynomial
equations to get an implicit equation in x,y,z.
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Adam Coffman wrote:
>
> An implicit equation for the aforementioned tear drop is:
> 1 - 4x^2 - 4y^2 - 2z + 2z^3 - z^4 = 0,
> or, it simplifies a bit as 4(x^2+y^2)=(1+z)(1-z)^3, which is a surface
> of revolution bounded by -1 <= z <= 1. The POV command is:
>
> quartic{ <
> 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0,
> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0,
> -1, 2, 0, -2, 1> }
>
> Sometimes, a parametric map involving trig functions can be converted
> to an algebraic implicit equation by using new variables:
> u=cos(t),
> v=sin(t),
> and introducing a new implicit relation:
> u^2+v^2=1.
> The idea is to make the parametric equations into polynomials, at the
> cost of increasing the number of variables and equations. Then u and
> v can (sometimes) be eliminated from the system of polynomial
> equations to get an implicit equation in x,y,z.
Thank you for taking the time to comment on this and I appreciate
you expanding on the brief description to give it meaning. I wish I
could follow more of what you said but some of it did sink in. The Pov
representation of the tear drop shape you provided is a real treasure
among the mathematically challenged, like myself, and is likewise very
much appreciated.
--
Ken Tyler
mailto://tylereng@pacbell.net
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Ken wrote:
> ... have nice ray-tracing properties. ???
>
> I would be interested to hear an elaboration on that particular statement.
>
>
To address your request for a few more remarks on Steiner surfaces andray
tracing, I'll only be able to make a few comments about geometry.
I'll refer to the literature to let the computer scientists explain
for themselves why they're interested in Steiner surfaces.
1. For quartics in general, a line (such as a light ray) meets the
surface in at most four points, or else it is completely contained in
the surface. Lines intersecting one of the Steiner surface's double
lines meet the surface in at most two other points, and lines passing
through the triple point can meet the surface in at most one other
point.
2. Let P be a point on a Steiner surface, and let L be a line tangent
to the surface at P. Then there's a conic curve (ellipse, hyperbola,
parabola, or union of lines) lying on the surface, so that L is
tangent to the conic at P. (I am skipping a few details, for example,
what about the singular points...) This property is called having a
"double infinity" of conics, and the Steiner surfaces are known to be
the only surfaces with this property. The idea is that every line in
the parametric domain is mapped to a conic curve by the quadratic
rational parametrization.
3. The intersection of a parametric Steiner surface with one of its
tangent planes is, in general, a union of two conic curves. (possibly
complex or singular)
4. The simple form of the parametric and implicit equations means the
normal vector at a point P is easy to calculate, and there are
formulas for the normal vector in terms of the coefficients of the
equations. Presumably these formulas are computationally faster than
calculating derivatives.
...
If you happen to be near a math/CS library, here are a few more
references to Steiner surfaces, in addition to the Apery book I
mentioned in a previous post.
F. Aries and R. Senoussi, "Approximation de surfaces parametriques par
des carreaux rationnels du second degre en lancer de rayons," Revue de
CFAO et d'informatique graphique (6) 12 (1997), 627--645.
W. L. F. Degen, "The types of triangular Bezier surfaces," in The
Mathematics of Surfaces, VI (Brunel Univ., 1994), 153--170,
Inst. Math. Appl. Conf. Ser. New Ser., 58, G. Mullineux, ed., Oxford
Univ. Press, New York, 1996.
W. Boehm and H. Prautzsch, "Geometric Concepts for Geometric Design,"
published by A. K. Peters, 1994.
T. Sederberg and D. Anderson, "Steiner surface patches," IEEE
Comput. Graph. Appl. 5 (1985) 23--36.
T. Sederberg, "Ray tracing of Steiner patches," Computer Graphics 18
(1984) 159--164.
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Ken wrote:
[ ... ]
> This message reminds me of something I have been meaning to ask for a few
> months now but always seem to forget. Below are three Parametric equations
> that describe shapes that are either interesting, useful, or both. Alas my
> math skills are prehistoric compared to what it takes to plug them into the
> quartic matrix that Pov uses to display these types of advanced shapes.
Of course, you don't need to convert to quartics, use patched
versions of POV-Ray, or other programs, if you get POV to make
the triangles for the parametric surface. With arrays and macros,
the process goes much faster and easier than the 'old days'
As an example, I will post a rendering of your 'Klein's Cycloid' in
povray.binaries.images under subject 'cycloid', and the pov file in,
i guess, povray.text.scene-files.
My question is: what the hell is it?
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david sharp wrote:
>
> Ken wrote:
> [ ... ]
> > This message reminds me of something I have been meaning to ask for a few
> > months now but always seem to forget. Below are three Parametric equations
> > that describe shapes that are either interesting, useful, or both. Alas my
> > math skills are prehistoric compared to what it takes to plug them into the
> > quartic matrix that Pov uses to display these types of advanced shapes.
>
> Of course, you don't need to convert to quartics, use patched
> versions of POV-Ray, or other programs, if you get POV to make
> the triangles for the parametric surface. With arrays and macros,
> the process goes much faster and easier than the 'old days'
>
> As an example, I will post a rendering of your 'Klein's Cycloid' in
> povray.binaries.images under subject 'cycloid', and the pov file in,
> i guess, povray.text.scene-files.
Thank you for the donation. I am interested to seewhat you did.
> My question is: what the hell is it?
And a good question that is too !
See my response to this question in the povray.binaries.images group
posted to your image thread.
--
Ken Tyler
mailto://tylereng@pacbell.net
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