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Digging Deep: Parametric Objects
by Mike Kost
Introduction
Povray parametric objects provide a powerful way to create unusual
surfaces.
Unfortunately, there's very little information out on the Internet
describing parametric objects. Hopefully this Digging Deep article will
provide a springboard to help people get started using parametric
objects in Povray 3.6.
Quick Reading
Here's some reading material on topics that this article will cover
Before Getting Into It
This tutorial will not cover isosurface topics like gradients and
bounding boxes. If
you're unfamiliar with isosurfaces, try these websites for more
information:
Parametric Object Basics
Parametric objects require a bit of brain bending. With parametric
objects, two variables are created, u and v, that can be swept from any
range. From u and v, we calculate the x, y, and z coordinates
represented by every possible u and v value. This defines the surface
of the object.
Ummm... Huh?
Lets consider the situation where we say that u is the x coordinate and
v is the z coordinate. The y coordinate will be calculated as sin(5*u)
and we want a surface that goes from <-1,0,-1> to <1,0,1>.
Here's what the Povray code and result would look like:
#declare Fx = function { u
};
#declare Fy = function { sin(pi*u) };
#declare Fz = function { v };
#declare maxG = 2;
#declare startU = -1;
#declare startV = -1;
#declare endU = 1;
#declare endV = 1;
parametric {
function { Fx(u,v,0) },
function { Fy(u,v,0) },
function { Fz(u,v,0 },
<startU, startV>, <endU, endV>
contained_by { box {<-1,-1,-1>,<1,1,1>} }
max_gradient maxG
pigment { rgb 0.9 }
finish { phong 0.5 phong_size 10 }
}
Full source
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The parametric object, in essence, renders the object by sweeping x
from -1 to 1, z from -1 to 1, and calculating y = sin(x). The result
shown above is
be just what'd you expect from a 3d function with y =
sin(x).
Thinking Differently
This first example illustrated the basics for parametric surfaces, but
in no way explains why they're so powerful. The object could easily
have been made with a regular isosurface with less computation time.
That's because this object was constructed in the Cartesian coordinate
system. The Cartesian coordinate system represents positions in space
using the x, y, and z coordinates. Other coordinate systems, like polar
coordinates or cylindrical coordinates, uses different methodologies to
locate a point.
The polar and cylindrical coordinate systems are illustrated below
Cylindrical Coordinate System[1]
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Polar Coordinate System[2]
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Since this tutorial isn't dedicated to coordinate systems, refer to the
Wikipedia entries on coordinate systems if it's still looking
confusing. For this tutorial, it's most
important to understand how to convert between the systems.
Cartesian
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Cylindrical
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Polar
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x
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r * cos(theta)
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r * sin (phi) * cos(theta)
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y
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h
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r * cos(phi)
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z
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r * sin(theta)
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r * sin (phi) * sin(theta)
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Now, on to an example.
Cylindrical Coordinate Examples
Working in the cylindrical coordinate system, it's easy to generate
lathe-like surfaces from the simple to the bizarre. Notice that the Fx,
Fy, and Fz functions are written like the conversion from cylindrical
to Cartesian coordinates assuming theta = u and h = v.
#declare Fr = function {
0.8 };
#declare Fx = function { Fr(u,v,0)*cos(u) };
#declare Fy = function { v };
#declare Fz = function { Fr(u,v,0)*sin(u) };
#declare maxG = 1;
#declare startU = 0;
#declare startV = -1;
#declare endU = 2*pi;
#declare endV = 1;
parametric {
function { Fx(u,v,0) },
function { Fy(u,v,0) },
function { Fz(u,v,0) }
<startU, startV>, <endU, endV>
contained_by { box {<-1,-1,-1>,<1,1,1>} }
max_gradient maxG
pigment { rgb 0.9 }
finish { phong 0.5 phong_size 10 }
}
Full Source
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In the first example, h = v and theta = u. We specify the lathe
behavior by generating a radius, Fr, based on the current height. By
sweeping theta from 0 to 2*pi, the line specified by r = ( 0.5*(h+1))
is revolved around the y axis.
Now lets upgrade to a more complicated radius function, Fr.
#declare Fr = function {
0.2 + 0.2*sin(5*pi*v) };
#declare Fx = function { Fr(u,v,0)*cos(u) };
#declare Fy = function { v };
#declare Fz = function { Fr(u,v,0)*sin(u) };
#declare maxG = 2;
#declare startU = 0;
#declare startV = -1;
#declare endU = 2*pi;
#declare endV = 1;
parametric {
function { Fx(u,v,0) },
function { Fy(u,v,0) },
function { Fz(u,v,0) }
<startU, startV>, <endU, endV>
contained_by { box {<-1,-1,-1>,<1,1,1>} }
max_gradient maxG
pigment { rgb 0.9 }
finish { phong 0.5 phong_size 10 }
}
Full Source
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With an additional tweaks, we can vary the 'center' at a given height,
causing the object to revolve through space.
#declare Fr = function {
0.2 + 0.2*sin(5*pi*v) };
#declare Fx = function { Fr(u,v,0)*cos(u) + 0.4*cos(4*v) };
#declare Fy = function { v };
#declare Fz = function { Fr(u,v,0)*sin(u) + 0.4*sin(4*v)};
#declare maxG = 2;
#declare startU = 0;
#declare startV = -1;
#declare endU = 2*pi;
#declare endV = 1;
parametric {
function { Fx(u,v,0) },
function { Fy(u,v,0) },
function { Fz(u,v,0) }
<startU, startV>, <endU, endV>
contained_by { box {<-1,-1,-1>,<1,1,1>} }
max_gradient maxG
pigment { rgb 0.9 }
finish { phong 0.5 phong_size 10 }
}
Full Source
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Polar Coordinate Examples
The polar coordinate system works well when manipulating spherical
objects. Notice that Fx, Fy, and Fz were written like the conversion
function from polar to Cartesian coordinates.
#declare Fr = function {
0.8 };
#declare Fx = function { Fr(u,v,0)*cos(u)*sin(v) };
#declare Fy = function { Fr(u,v,0)*cos(v)};
#declare Fz = function { Fr(u,v,0)*sin(u)*sin(v) };
#declare maxG = 2;
#declare startU = 0;
#declare startV = 0;
#declare endU = 2*pi;
#declare endV = pi;
parametric {
function { Fx(u,v,0) },
function { Fy(u,v,0) },
function { Fz(u,v,0) }
<startU, startV>, <endU, endV>
contained_by { box {<-1,-1,-1>,<1,1,1>} }
max_gradient maxG
pigment { rgb 0.9 }
finish { phong 0.5 phong_size 10 }
}
Full Source
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And now lets repeat that last object with manipulations to the radius
function Fr. Just because the
equations are written in a spherical system does not mean that the
result will be spherical.
#declare Fr = function {
0.5 + 0.5*sin(10*v) };
#declare Fx = function { Fr(u,v,0)*cos(u)*sin(v) };
#declare Fy = function { Fr(u,v,0)*cos(v)};
#declare Fz = function { Fr(u,v,0)*sin(u)*sin(v) };
#declare maxG = 2;
#declare startU = 0;
#declare startV = 0;
#declare endU = 2*pi;
#declare endV = pi;
parametric {
function { Fx(u,v,0) },
function { Fy(u,v,0) },
function { Fz(u,v,0) }
<startU, startV>, <endU, endV>
contained_by { box {<-1,-1,-1>,<1,1,1>} }
max_gradient maxG
precompute 20 x,y,z
pigment { rgb 0.9 }
finish { phong 0.5 phong_size 10 }
}
Full Source
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And now, a really busy Fr.
#declare Fr = function {
0.2 + 0.4*cos(13*u) + 0.4*sin(10*v) };
#declare Fx = function { Fr(u,v,0)*cos(u)*sin(v) };
#declare Fy = function { Fr(u,v,0)*cos(v)};
#declare Fz = function { Fr(u,v,0)*sin(u)*sin(v) };
#declare maxG = 4;
#declare startU = 0;
#declare startV = 0;
#declare endU = 2*pi;
#declare endV = pi;
parametric {
function { Fx(u,v,0) },
function { Fy(u,v,0) },
function { Fz(u,v,0) }
<startU, startV>, <endU, endV>
contained_by { box {<-1,-1,-1>,<1,1,1>} }
max_gradient maxG
precompute 20 x,y,z
pigment { rgb 0.9 }
finish { phong 0.5 phong_size 10 }
}
Full Source
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With parametric objects, it's possible to produce some shapes that
would be near-impossible to create using isosurfaces.
Wrapping Up
This Digging Deep has really just touched the surface for parametric
objects, but hopefully it'll be a springboard to help expand knowledge
on the topic.
Want To Know More?
As previously mentioned, the Internet doesn't have much on parametric
objects, but there are a few good sites that talk more about what you
can do
with parametric objects:
Notes and Disclaimers
Last edited: 9/25/04
Copyright (C) 2004 Mike Kost
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