Version 8 of Three-dimensional shapes
Updated 2002-09-18 06:22:14
Arjen Markus (11 september 2002) I have had this idea for some time now: create a set of procedures that will allow me to draw mathematical shapes like a torus or even a knot, colour them in ingenious ways, in short, play with 3D space.
The script below is just a start. It is not perfect, it is not flexible, but it does give you some sense of what I want.
Notes:
- The sorting of the polygons must be improved
- The viewpoint is now hidden in the construction of the torus's polygons - this is a quick hack to avoid the work involved with the first note.
- The viewpoint is not easily changed, well, you can change the angle around the y-axis, but that is all.
Warning: the picture was rendered with an improved script. I will upload that as well, but the whole script is getting fairly large. You can find the full script at [L1 ] and an animation of the torus being wrung out of shape at 
. (Thanks to Reinhard Max for providing the space). Note that the full script saves each frame for the animation, so you may want to comment out the "exec" statement at the end.
# solids3d.tcl --
#
# Package for displaying 3D solid bodies
# (sample Workbench module)
#
# Notes:
# This package is a quick hack to get started only
#
# Version information:
# version 0.1: initial implementation, september 2002
package provide Solids3D 0.1
namespace eval ::Solids3D {
namespace export func deriv
variable display_options
# pointOnTorus --
# Return the coordinates of a point on a torus, as given by
# two parameters (angles)
#
# Arguments:
# phi Angle with respect to x-axis
# theta Angle with respect to "inner" axis of torus
#
# Result:
# Point {x,y,z}, the coordinates of the point
#
proc pointOnTorus {phi theta} {
set cosphi [expr {cos($phi)}]
set sinphi [expr {sin($phi)}]
set costh4 [expr {0.25*cos($theta)}]
set sinth4 [expr {0.25*sin($theta)}]
set x [expr {$cosphi*(1.0+$costh4)}]
set y [expr {$sinphi*(1.0+$costh4)}]
set z [expr {1.0+$sinth4}]
return [list POINT $x $y $z]
}
# constructTorus --
# Return a list of polygons, together forming an approximation to a
# torus
#
# Arguments:
# nophi Number of steps along main perimeter
# notheta Number of steps along secondary perimeter
#
# Result:
# List of polygons
#
proc constructTorus {nophi notheta} {
variable angle
set pi 3.1415926
set dphi [expr {2.0*$pi/double($nophi)}]
set dtheta [expr {2.0*$pi/double($notheta)}]
set polygons {POLYGON-LIST}
for { set iphi 0 } { $iphi < $nophi } { incr iphi } {
set phi1 [expr {$dphi*double($iphi)}]
set phi2 [expr {$dphi*double($iphi+1)}]
for { set itheta 0 } { $itheta < $notheta } { incr itheta } {
set theta1 [expr {$dtheta*double($itheta)}]
set theta2 [expr {$dtheta*double($itheta+1)}]
set point1 [pointOnTorus $phi1 $theta1]
set point2 [pointOnTorus $phi1 $theta2]
set point3 [pointOnTorus $phi2 $theta2]
set point4 [pointOnTorus $phi2 $theta1]
set red [expr {int(125*(1.0+cos($phi1)))}]
set green [expr {int(125*(1.0+sin($phi1)))}]
set blue [expr {int(125*(1.0+sin($theta1)))}]
set colour [format "#%2.2X%2.2X%2.2X" $red $green $blue]
set zdepth [lindex [rotateY $angle $point1] 3]
lappend polygons \
[list POLYGON $colour $zdepth $point1 $point2 $point3 $point4]
}
}
return $polygons
}
# rotateY --
# Rotate around the y axis
#
# Arguments:
# angle Angle over which to rotate
# point Point to rotate
#
# Result:
# New coordinates
#
proc rotateY {angle point} {
foreach {dummy x y z} $point break
set xr [expr {$x*cos($angle)-$z*sin($angle)}]
set yr $y
set zr [expr {$x*sin($angle)+$z*cos($angle)}]
return [list POINT $xr $yr $zr]
}
# projectOnYZ --
# Project a point on the YZ-plane (and scale as we do this)
#
# Arguments:
# point Point to rotate
#
# Result:
# XY coordinates (for the screen)
#
proc projectOnYZ {point} {
variable scale
variable yoffset
variable zoffset
foreach {dummy x y z} $point break
set xp [expr {int($scale*($y-$yoffset))}]
set yp [expr {int($scale*($zoffset-$z))}]
return [list $xp $yp]
}
# display --
# Quick and dirty implementation to display a set of polygons
#
# Arguments:
# polygons List of 3D polygons
#
# Result:
# None
#
# Side effect:
# Display of polygons in the canvas
#
proc display {polygons} {
variable angle
#
# Sort the polygons first
# Note:
# The comparison is too simple, but for now it should work
#
set plane_polygons [lrange $polygons 1 end]
set sorted_polygons [lsort -real -index 2 $plane_polygons]
foreach polygon $sorted_polygons {
set colour [lindex $polygon 1]
set points [lrange $polygon 3 end]
set coords {}
foreach point $points {
set coords [concat $coords [projectOnYZ [rotateY $angle $point]]]
}
.cnv create polygon $coords -fill $colour -outline black
}
}
#
# Initialise the variables
#
variable angle [expr {0.25*3.1415926}]
variable scale 100.0
variable yoffset -1.5
variable zoffset 2.0
} ;# End of namespace
#
# Run the program
#
canvas .cnv -width 300 -height 300 -background white
pack .cnv -fill both
set torus [::Solids3D::constructTorus 16 16]
::Solids3D::display $torus