TITLE: Spirals
NAME: Wong Chee Fah
COUNTRY: Malaysia
EMAIL: wongcf@cyberway.com.sg
TOPIC: Physics & Math
COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT.
JPGFILE: spiral.jpg
ZIPFILE: spiral.zip
RENDERER USED: 
    POV-Ray 3.01 (MSDos)

TOOLS USED: 
    Adobe Photoshop 4 - for image compression

RENDER TIME: 
    2h 45m 23s (9923s)

HARDWARE USED: 
    Pentium 133MHz, 64MB EDO, S3 Virge 4MB EDO

IMAGE DESCRIPTION: 


A christmas tree made from spirals of spheres 


DESCRIPTION OF HOW THIS IMAGE WAS CREATED: 


This is one of the first few images I've created using POV-Ray. I was going
through the various modellers and utilities for POV-Ray when I came upon
SPIRAL v2.20 by Truman Brown. The output from the program uses CTDS also by
Truman Brown. I read through the documentation and decided to attempt an
implementation of the same program using POV-Ray alone. This saves the trouble
of having to switch between programs. I've not used SPIRAL at all so I'm not
sure if the implementation produces the same result (One thing I do know is 
that CTDS connects points using spheres and cones while mine omits the
cylinders). Since my source code includes parametric equations I feel that it
is of some relevance to the Math topic. Anyway, the code is included for you to
judge.

I used parametric equations to creat the spirals. The parameters for the
spirals are entered at the beginning using #declare statements. With While
loops and conditionals statements, the position and size of each sphere is
calculated. The equations for the <x,y,z> coordinate of a spiral are as
follows :-

x = R1-((R1-R2)*t)*sin(2pi*r*t)        0 <= t <= 1
y = H*t           0 <= t <= 1
z = R1-((R1-R2)*t)*cos(2pi*r*t)        0 <= t <= 1

These equations describe a spiral that rises from the x-z plane with a radius
of R1, to a height of H, and ends in r revolutions with a radius of R2, as t is
varied from 0 to 1.

Besides the obvious choices of spiral height, number of revolutions, starting
and ending radii, the user can define other parameters such as spiral direction
(clockwise or anti-clockwise), and number of spirals. In addition, minor
spirals are also available and are just as configurable. Basically, the minor
spirals coil around the major spiral. The users have a choice of showing both
the major and minor spirals or only one of either or none at all (although I 
can't see the purpose of the last choice).

The whole tree is built by first building a single instance of the major and
minor spirals. The position of each sphere in the major and minor spiral are
calculated. With the calculated positions, a minor group is formed by repeating
the translation and rotation of a minor sphere to its minor position. The
number of repetitions depends on the number of minor spirals defined. The group
is then translated to the major position. Note that even if the major is not to
be shown as it is in this case, the position of each sphere in the major spiral
still has to be calculated so that the spheres in minor spirals can be
translated to the correct position. To build the whole spiral, a union
operation is applied over a while loop that creats the groups. Once that is
done the spiral is placed in the scene and copied with rotation about the
y-axis. The number of copies depends on the number of major spiral defined. A
y-gradient color map texture is applied and the scene is then completed with a
camera and a light source.

PS. Thanks to Truman Brown for his SPIRAL utility - the origin of the idea for
    this scene.

By the way, I tried 'growing' the tree by building some 120-frame animation
clips. The tree can be 'grown' by using the clock variable. One example is
given below. Enjoy.

#declare Show_Major = 0         // Show Major Spiral ( Default = 1 )
           // 0 for No, 1 for Yes
#declare Dir_Major = 1         // Major Spiral Direction ( Default = 1 )
           // 0 for Anti-Clockwise, 1 for Clockwise
#declare R1 = 60*clock         // Start Radius of Major Spiral
#declare R2 = 0          // End Radius of Major Spiral
#declare H = 200*clock         // Major Spiral Height
#declare Rev = 3*clock         // Number of Major Revolutions
#declare Start_Rad_Maj = 5*clock// Starting Radius of Major Sphere
#declare End_Rad_Maj = clock        // Ending Radius of Major Sphere
#declare Num_Maj = 360*clock        // Number of Spheres per Major Spiral
#declare S_Num_Maj = 4         // Number of Major Spirals

#declare Show_Minor = 1         // Show Minor Spiral ( Default = 1 )
           // 0 for No, 1 for Yes
#declare Dir_Minor = 0         // Minor Spiral Direction ( Default = 1 )
           // 0 for Anti-Clockwise, 1 for Clockwise
#declare R3 = 15*clock         // Start Radius of Minor Spiral
#declare R4 = 0          // End Radius of Minor Spiral
#declare Turns = 2*clock        // Number of Turns per Major Revolution
#declare Start_Rad_Min = 5*clock// Starting Radius of Minor Sphere
#declare End_Rad_Min = 0        // Ending Radius of Minor Sphere
#declare Num_Min = 1         // Number of Spheres generated per Major Sphere
#declare S_Num_Min = 4*clock        // Number of Minor Spirals per Major Spiral