[George Peter Staplin] April 1, 2006 - The code that follows performs 2D rotations by using scaling and a precomputed table.  The table is generated using sin() and cos(), but it could easily be generated once, and the need for floating point completely eliminated.

[Miguel Sofer] responded to my question about how to do this, and quite graciously volunteered to write the majority of this code.  I wrote the graphical part.
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[http://www.xmission.com/~georgeps/images/wiki/2D_integer-based_rotations.png]

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#
# This is a joint project --
# by Miguel Sofer (primarily) and George Peter Staplin
#

package require Tk

#
# This is used by the test procedure to compare the irot result with drot.
#
proc drot {x y a} {
    set a [expr {($a*acos(0))/90}];# convert to radians
    set c [expr {cos($a)}]
    set s [expr {sin($a)}]
    list [expr {int(round($x*$c-$y*$s))}] [expr {int(round($x*$s+$y*$c))}]
}
 
 
# create the table
set Cos {}; set Sin {}
set M [expr {pow(2,30)}]
set Coeff [expr {acos(0)/90}]
for {set a 0} {$a <= 45} {incr a} {
    set A [expr {$Coeff*$a}]
    lappend Cos [expr {int($M*cos($A))}]
    lappend Sin [expr {int($M*sin($A))}]
}
set M [expr {int($M)}]
 
proc ifun a {
    global Cos Sin
 
    set a [expr {$a%360}]
    # sign of sin and sign of cos
    set ss 1; set sc 1
 
    # Insure -180<$a<=180
    if {$a > 180} {
	set a [expr {$a-360}]
    }
 
    # Consider negative angles; after this 0<=$a<=180
    if {$a < 0} {
	set ss [expr {-$ss}]
	set a [expr {-$a}]
    }
 
    # Convert to first quadrant
    if {$a > 90} {
	set sc [expr {-$sc}]
	set a [expr {180-$a}]
    }
 
    # Lookup only the first 45 degrees
    if {$a <= 45} {
	set cos [expr {$sc*[lindex $Cos $a]}]
	set sin [expr {$ss*[lindex $Sin $a]}]
    } else {
	set a [expr {90-$a}]
	set cos [expr {$sc*[lindex $Sin $a]}]
	set sin [expr {$ss*[lindex $Cos $a]}]
    }
 
    list $cos $sin
}
 
proc irot {x y a} {
    global M
    foreach {c s} [ifun $a] break
 
    set c [expr {wide($c)}]
    set s [expr {wide($s)}]
 
    set xx [expr {int(($x*$c-$y*$s)/$M)}]
    set yy [expr {int(($x*$s+$y*$c)/$M)}]
 
    list $xx $yy
}
 
proc test {x y a} {
    list [irot $x $y $a] [drot $x $y $a]
}

proc point {win x y} {
    $win create rectangle $x $y [expr {$x + 4}] [expr {$y + 4}] -fill white
    $win create text [expr {$x + 5}] [expr {$y - 5}] -text "$x,$y" -fill white
}

proc draw.circle {win radius centerx centery} {
    set lastx [expr {$centerx + $radius}]
    set lasty [expr {$centery + $radius}]

    for {set d 1} {$d < 360} {incr d} {
	set rot [irot $radius $radius $d]
	set x [lindex $rot 0] ; set y [lindex $rot 1]

	set x [expr {$x + $centerx}] ; set y [expr {$y + $centery}]

	$win create line $lastx $lasty $x $y -fill white
	set lastx $x ; set lasty $y
    }

    point $win $centerx $centery
}

proc main {} {
    wm title . "Miguel and George's Excellent Adventure!"
    pack [canvas .c] -fill both -expand 1

    draw.circle .c 50 100 100
    draw.circle .c 25 200 200
    draw.circle .c 50 300 100
}
main

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[Category Image Processing]