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Thanks to the recent posting by Paul Bourke regarding super shapes
extended to 3D via spherical products,
http://news.povray.org/povray.general/32398/, and also to Ingo Janssen’s
“mesmerizing mesh maker macros” (including param.inc),
http://members.home.nl/seedseven/, I put together something for my (fall
semester) multi-variable calculus students, generalizing from super
shapes to other polar curves. I thought some POVRay users might be
interested, at least in the results. A fun evening can be had by
digging out an old analytic geometry book, or other source of polar
curves, and crossing different famous curves in different orders (at
least I found it to be fun.) (This description’s a bit of overkill for
POVRay users, but my students will be learning POVRay as we go, and thus
be pretty much as ignorant as I am, regarding POVRay. I did, at least,
clip out the reminders about how to use .inc files and such.)
Spherical Products and Polar Functions:
The spherical product of two parametrically-defined planar curves,
H(h_x(u), h_y(u)) and M(m_x(v), m_y(v)), is defined by (H * M)(u, v) via:
x(u, v) = h_x(u)m_x(v)
y(u, v) = h_y(u)m_x(v)
z(u, v) = m_y(v)
(I use underscores, when I can't subscript.)
Note that about half the sources I looked up defined the product in the
opposite order, and, since this product is not commutative, it’s
important to know which order a particular source is using!
Since every planar polar function can also be described via the
canonical parameterization : x = r(t)cos(t), y = r(t)sin(t), one can
form the spherical product of any two polar functions, r_1(u) and r_2(v):
r_1 * r_2 (or r_2 * r_1 for the alternate definition) via:
x(u, v) = r_1(u)cos(u)r_2(v)cos(v)
y(u, v) = r_1(u)sin(u)r_2(v)cos(v)
z(u, v) = r_2(v)sin(v)
To render these objects in a left-hand coordinate system (such as
POVRay) and achieve the same pictures as with Mathematica or Maple,
interchange the roles of y and z (as below, where F3 applies to the y
component.)
Here’s a sample of the call to Ingo Janssen’s Parametric macro, where
r_1(u) = cos(3u) and r_2(v) = sin(2v):
#declare F1= function(u,v){cos(3*u)*cos(u)*sin(2*v)*cos(v)}
#declare F2= function(u,v){cos(3*u)*sin(u)*sin (2*v)*cos(v)}
#declare F3= function(u,v){sin(2*v)*sin(v)}
object{
Parametric(
F1, F3, F2,
<FromU(0), FromV(0)>, <ToU(2*pi), ToV(2*pi)>,
150,150,"C_3UxS_2V.inc"
)
[ object_modifiers ]
}
The parameters in "Parametric" are described in the #include files.
Here are 3 examples:
(Truth-In-Advertising Disclaimer: The stem on the psychedelic flower is
a sphere_sweep, and not a part of the product. The captions were added
in Photoshop.)
Post a reply to this message
Attachments:
Download 'polarspsjpg.jpg' (28 KB)
Preview of image 'polarspsjpg.jpg'
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Wasn't it David Matthews who wrote:
>Thanks to the recent posting by Paul Bourke regarding super shapes
>extended to 3D via spherical products,
>http://news.povray.org/povray.general/32398/, and also to Ingo Janssen’s
>“mesmerizing mesh maker macros” (including param.inc),
>http://members.home.nl/seedseven/, I put together something for my (fall
>semester) multi-variable calculus students, generalizing from super
>shapes to other polar curves. I thought some POVRay users might be
>interested, at least in the results. A fun evening can be had by
>digging out an old analytic geometry book, or other source of polar
>curves, and crossing different famous curves in different orders (at
>least I found it to be fun.) (This description’s a bit of overkill for
>POVRay users, but my students will be learning POVRay as we go, and thus
>be pretty much as ignorant as I am, regarding POVRay. I did, at least,
>clip out the reminders about how to use .inc files and such.)
>
>Spherical Products and Polar Functions:
>
>The spherical product of two parametrically-defined planar curves,
>H(h_x(u), h_y(u)) and M(m_x(v), m_y(v)), is defined by (H * M)(u, v) via:
> x(u, v) = h_x(u)m_x(v)
> y(u, v) = h_y(u)m_x(v)
> z(u, v) = m_y(v)
>
>(I use underscores, when I can't subscript.)
>
>Note that about half the sources I looked up defined the product in the
>opposite order, and, since this product is not commutative, it’s
>important to know which order a particular source is using!
>
>Since every planar polar function can also be described via the
>canonical parameterization : x = r(t)cos(t), y = r(t)sin(t), one can
>form the spherical product of any two polar functions, r_1(u) and r_2(v):
>
>r_1 * r_2 (or r_2 * r_1 for the alternate definition) via:
>
>x(u, v) = r_1(u)cos(u)r_2(v)cos(v)
>y(u, v) = r_1(u)sin(u)r_2(v)cos(v)
>z(u, v) = r_2(v)sin(v)
>
>To render these objects in a left-hand coordinate system (such as
>POVRay) and achieve the same pictures as with Mathematica or Maple,
>interchange the roles of y and z (as below, where F3 applies to the y
>component.)
>
>Here’s a sample of the call to Ingo Janssen’s Parametric macro, where
>
>r_1(u) = cos(3u) and r_2(v) = sin(2v):
>
>#declare F1= function(u,v){cos(3*u)*cos(u)*sin(2*v)*cos(v)}
>#declare F2= function(u,v){cos(3*u)*sin(u)*sin (2*v)*cos(v)}
>#declare F3= function(u,v){sin(2*v)*sin(v)}
>
>object{
> Parametric(
> F1, F3, F2,
> <FromU(0), FromV(0)>, <ToU(2*pi), ToV(2*pi)>,
> 150,150,"C_3UxS_2V.inc"
> )
> [ object_modifiers ]
>}
>
>The parameters in "Parametric" are described in the #include files.
Nice.
One tiny change to make life a little easier would be to #declare the
r_1 and r_2 functions, so that when you want to experiment with
different functions you only need to change them once.
// Change these lines to try different shapes
#declare r_1 = function(u,v){cos(3*u)}
#declare r_2 = function(u,v){sin(2*v)}
// Leave these lines alone
#declare F1= function(u,v){r_1(u,v)*cos(u) *r_2(u,v)*cos(v)}
#declare F2= function(u,v){r_1(u,v)*sin(u) *r_2(u,v)*cos(v)}
#declare F3= function(u,v){r_2(u,v)*sin(v)}
--
Mike Williams
Gentleman of Leisure
Post a reply to this message
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Mike Williams wrote:
>Nice.
>
>One tiny change to make life a little easier would be to #declare the
>r_1 and r_2 functions, so that when you want to experiment with
>different functions you only need to change them once.
>
>// Change these lines to try different shapes
>#declare r_1 = function(u,v){cos(3*u)}
>#declare r_2 = function(u,v){sin(2*v)}
>
>// Leave these lines alone
>#declare F1= function(u,v){r_1(u,v)*cos(u) *r_2(u,v)*cos(v)}
>#declare F2= function(u,v){r_1(u,v)*sin(u) *r_2(u,v)*cos(v)}
>#declare F3= function(u,v){r_2(u,v)*sin(v)}
>
>Mike Williams
>Gentleman of Leisure
>
Thanks.
In fact, I tried to do that, but I had something goofed up on my syntax. I
still have a mental block about declaring and using functions, and I don't
know why, since they're just like they are in math, and I use them there
every day.
A second thing I forgot to mention, is that if you are using uv_mapping, you
must make sure to only create the surface once. In my example, for
example, u should only go from 0 to pi, while v goes from 0 to 2pi.
This determination is a bit more complicated than it first appears.
If r_1 and r_2 were non-negative, -pi/2 < v < pi/2 would always suffice.
But the above example shows this is not the case in general.
Since r_1(t) = cos ( 3t ) is created from 0 to pi, then repeated (on the
plane) and r_2(t) = sin ( 2t ) is created from 0 to 2pi, you might guess
that you simply limit your domain to what would create one copy of each, in
the plane. This works, in the above example, and in the example:
r_1(u) = cos(u) + 0.5 and r_2(v) = sin (v) + 0.5. Here each (u and v) must
go from 0 to 2pi.
It also works with:
r_1= r_2=cos(3t), which is created by letting u and v each go from 0 to pi.
However, due to symmetry, if you let
r_1 = r_2 = cos(2t),
the correct domain is 0 < u < pi, 0 < v < 2pi.
It is true that you need never increase the domains of u and v beyond what
would create one copy of the plane curve (since you can think of the surface
as being created by applying r_2 to the generator r_1(u) and vice-versa.)
I had to make animations to figure out how the complications arose (it also
showed me where the "extra" 3 petals come from in what is, on the plane, a
3 - petal rose, r = cos (3t), when r_1 =cos(3u) and r_2=cos (2v), and why
they disappear when the product is reversed. . . .)
Dave Matthews
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