globals [it's-r/phi ;iterations of the r/phi calculation it's-lt/fd ; " " " lt/fd " it's-sh/lt/fd ; " " " sh/lt/fd " phi ;polar coordinate angle of rotation B-1 & B-2 ;angles in the pair of triangles for calculation of 'lt' and 'fd' ] to setup cp cg ct crt 1 set it's-r/phi 0 set it's-lt/fd 0 set it's-sh/lt/fd 0 set phi 0 end to go-r/phi ;converts from polar coordinates 'r' 'phi' ask turtles[ ;to cartesian for the display setxy 0 0 set heading 0 repeat 10000[ ifelse ((xcor >= 199) or (xcor <= -199) or ;stops calculation with trace (ycor >= 199) or (ycor <= -199))[stop] ;at edge of display [set phi 0 + (0.5 * it's-r/phi) ;sets angle increments of 0.5 degree set xcor r * cos phi ;calculates Cartesian set ycor r * sin phi ;coordinates stamp 45 ;colours trace if it's-r/phi = 2[type"r/phi xcor"print xcor type"r/phi ycor"print ycor type"heading"print heading];see comment in INFORMATION if plots = true[plot-run-r/phi] set it's-r/phi it's-r/phi + 1]]] end to-report r ;polar radius formulae ifelse spirals = 1 [report s * 1 * (2 * pi * phi / 360)] ;Archimedes [ifelse spirals = 2 [report s * 10 * sqrt(1 + (2 * pi * phi / 360) ^ 2)] ;evolute [ifelse spirals = 3 [report exp(s / 10 * (2 * pi * phi / 360))] ;logarithmic [report 0]]] end ;the lt/fd caculation uses the cosine formula etc and two triangles with a common side to go-lt/fd ;which are made from 3 consecutive values of the polar radius 'r', see INFORMATION PAGE ask turtles[ set heading 0 set color 15 pd ifelse spirals = 1[setxy 0 (s * -0.0001525) rt 90][ ;sets start conditions ifelse spirals = 2[setxy (s * 10) 0][ ; " " " if spirals = 3[setxy 1 0 rt(s * 5.35)]]] ; " " " (also see comment in INFORMATION) repeat 10000[ ifelse ((xcor >= 199) or (xcor <= -199) or ;stops calculation with trace (ycor >= 199) or (ycor <= -199))[stop][ ;at edge of display set B-1 asin (b1 * sin 0.5 / a1) ;the angle opposite the side b1 set B-2 asin (b2 * sin 0.5 / a2) ;the angle opposite the side b2 lt (B-2 - B-1) ;display requires lt 'angle' fd a2 ;display requires fd 'step' if it's-lt/fd = 1[type"lt/fd xcor"print xcor type"lt/fd ycor"print ycor type"heading"print heading];see comment in INFORMATION if plots = true[plot-run-lt/fd] set it's-lt/fd it's-lt/fd + 1]]] end to-report b1 ;a side of one of the triangles set phi 0.5 * it's-lt/fd ;and the radius 'r' at the previous iteration report r end to-report c ;the common side of the two triangles set phi 0.5 * (it's-lt/fd + 1) ;and the radius 'r' at the iteration now report r end to-report b2 ;a side of one the triangles set phi 0.5 * (it's-lt/fd + 2) ;and the radius 'r' at the next iteration report r end to-report a1 ;a side of one of the triangles and the report sqrt((b1 ^ 2) + (c ^ 2) - (2 * b1 * c * cos 0.5)) ;step forward made at the previous iteration end to-report a2 ;a side of one of the triangles and the report sqrt((b2 ^ 2) + (c ^ 2) - (2 * b2 * c * cos 0.5)) ;step forward at this iteration end ;The sh/lt/fd procedure is a shorter, approximate lt/fd code, derived by to go-sh/lt/fd ;matching sh/lt/fd plots of lt and fd with those of the lt/fd procedure ask turtles[ set heading 0 set color 65 pd ifelse spirals = 1[setxy 0 -0.0001525 rt 90][ ;sets start conditions ifelse spirals = 2[setxy (s * 10) 0][ ; " " " if spirals = 3[setxy 1 0 rt (s * 5.8)]]] ; " " " repeat 10000[ ifelse ((xcor >= 199) or (xcor <= -199) or ;stops calculation with trace (ycor >= 199) or (ycor <= -199))[stop][ ;at edge of display set phi 0.5 + (0.5 * it's-sh/lt/fd) lt beta ;see derived code below fd delta ;see derived code below if plots = true[plot-run-sh/lt/fd] set it's-sh/lt/fd it's-sh/lt/fd + 1]]] end to-report beta ifelse spirals = 1 [report 0.5 + (0.5 / (1 + (0.007 * it's-sh/lt/fd) ^ 3))] ;Archimedes [ifelse spirals = 2 [report 0.55 - (0.5 / (1 + (0.024 * it's-sh/lt/fd) ^ 2))] ;evolute [if spirals = 3 [report 0.5]]] ;logarithmic end to-report delta ifelse spirals = 1 [report (s * it's-sh/lt/fd * 0.000076)] ;Archimedes [ifelse spirals = 2 [report (s * (0.125 + (0.85 / 1157 * (it's-sh/lt/fd - 100))))] ;evolute [if spirals = 3 [report s * ((it's-sh/lt/fd / 3250) ^ 3.9)]]] ;logarithmic end to plot-run-r/phi set-current-plot "spirals using r/phi, lt/fd and sh/lt/fd code" set-current-plot-pen "r" plot r / 100 set-current-plot-pen "phi" plot phi / 5000 end to plot-run-lt/fd set-current-plot "spirals using r/phi, lt/fd and sh/lt/fd code" set-current-plot-pen "lt" plot (B-2 - B-1) set-current-plot-pen "fd" plot a2 end to plot-run-sh/lt/fd set-current-plot "spirals using r/phi, lt/fd and sh/lt/fd code" set-current-plot-pen "sh/lt" plot beta set-current-plot-pen "sh/fd" plot delta end @#$#@#$#@ GRAPHICS-WINDOW 367 10 778 442 200 200 1.0 1 10 1 1 CC-WINDOW 141 10 367 165 Command Center MONITOR 77 134 138 183 NIL it's-r/phi 0 1 BUTTON 2 144 70 177 run r/phi go-r/phi NIL 1 T OBSERVER BUTTON 141 207 209 282 NIL setup NIL 1 T OBSERVER SLIDER 146 170 358 203 s s 1 5 5.0 0.1 1 NIL CHOICE 5 10 122 55 spirals spirals 1 2 3 0 TEXTBOX 43 60 134 108 1 Archimedes\n2 evolute\n3 logarithmic\n BUTTON 3 192 72 225 run lt/fd go-lt/fd NIL 1 T OBSERVER MONITOR 76 184 138 233 NIL it's-lt/fd 0 1 BUTTON 3 240 71 273 run sh/lt/fd go-sh/lt/fd NIL 1 T OBSERVER MONITOR 75 234 138 283 NIL it's-sh/lt/fd 0 1 BUTTON 271 209 361 242 clear plots clear-plot NIL 1 T OBSERVER MONITOR 211 220 268 269 NIL phi 2 1 SWITCH 272 248 362 281 plots plots 1 1 -1000 PLOT 6 285 364 443 spirals using r/phi, lt/fd and sh/lt/fd code it's-r/phi or it's-lt/fd or it's-sh/lt/fd r,phi,lt,fd,sh/lt,sh/fd 0.0 1000.0 0.0 1.0 true true PENS "r" 1.0 0 -16777216 true "phi" 1.0 0 -44544 true "lt" 1.0 0 -16776961 true "fd" 1.0 0 -11352576 true "sh/lt" 1.0 0 -6524078 true "sh/fd" 1.0 0 -65281 true @#$#@#$#@ Model SPIRALS & CURVE MATCHING BACKGROUND ------------- Alternative procedures for displaying spirals are examined and a procedure for matching equations to curves is developed, which may have other applications. Anyone of three well known spirals can be displayed by selection at the CHOICE widget. The Archimedes spiral is generated by a rotation plus a constant velocity crossways displacement as in the case of a gramophone record, giving an equidistance spacing between the lines. Both the evolute and the logarithmic spirals are growth spirals in which the spacing between the lines increases. The original spirals were generated with the polar, r/phi procedure, the more natural from a mathematical point of view; but to display conversion to cartesian coordinates was used. The scale factor 's' can be adjusted to alter the spacing between the lines and therefore the "length" of the spirals displayed. Following a comment made by Seth Tisue at CCL, June 2003, lt/fd procedures were explored and added as follows: Using from the r/phi procedure, the cartesian coordinates of calculations of 3 iterations, previous, now and next was a possibility. However completing the right-angled triangles between these and solving for lt and fd was somewhat clumsy; so a similar method using the polar radii of iterations, previous 'b1', now 'c' and next 'b2', was chosen: where A is the constant 0.5° angular step of the polar r/phi code ___a2__ ___a1__ hence 'a1' becomes the previous step forward or 'fd' of the lt/fd code \C2 B2|B1 C1/ also 'a2' " " next " " " " " " " " \ | / and 'B2' - 'B1' becomes the angular step or 'lt' of the lt/fd code. b2\ c|c /b1 \ | / And the cosine formula a = sqroot[bxb + cxc - 2xbxcxcosA] \ | / also a/sinA = b/sinB = c/sinC are available. \A|A/ \|/ Overlaying the displays of the r/phi and lt/fd procedures shows that | the match is very good, including over the range of 's'. It is possible to obtain an approximate match with a shorter procedure,identified as sh/lt/fd, by comparing the PLOTS of the sh/lt/fd equations with those of the lt/fd equations and modifying the sh/lt/fd equations to obtain a close fit. The results for the Archimedes spiral, (the simplest), are acceptable. The other equations have been left half finished for the amusement of model viewers who wish to try their hand at "fitting an equation to a curve techniques" and then overlaying the spiral that is generated to see if the match of sh/lt/fd against r/phi is good. USING THE MODEL TO DISPLAY THE SPIRALS -------------------------------------- First CHOOSE a spiral, select a value for 's' and press SETUP. Then RUN one procedure followed by another if observing the match by overlaying; otherwise press SETUP in between RUNNING procedures. It runs quicker with PLOTS switched OFF. The overlay with close matching is difficult to see, so r/phi writes in yellow, lt/fd writes in red and sh/lt/fd writes in lime. It may be instructive to run the r/phi procedure with PLOTS switched ON but not when fitting an equation to a curve, when it could be confusing. USING THE MODEL TO FIT AN EQUATION TO A CURVE --------------------------------------------- First CHOOSE a spiral, select a value for 's' and press SETUP. With PLOTS switched ON the lt/fd procedure should be RUN to plot the curves of 'lt' and 'fd'; followed directly by the sh/lt/fd procedure to examine how closely its 'lt' and 'fd' plots follow those of lt/fd. The fit of these curves may suggest a change to the sh/lt/fd equations for 'lt'(beta) and 'fd'(delta) to obtain a better match when overlaying the spiral that is generated. If all else fails there is always the START conditions to give that last tweek to the match. COMMENT ------- The START conditions, 'xcor', 'ycor' and 'heading' are important in acheiving a good match between the spirals generated by the r/phi procedure and by the lt/fd procedures. To assist this process code has been included to print these values in the COMMAND CENTER. The heading value can change the xcor&ycor figures. Over a 5 : 1 range of values of 's' it does appear possible to obtain a near perfect match of the spirals drawn by procedure r/phi and procedure lt/fd. Perhaps a perfect match of the displayed spirals is not possible because steps in the display result from finite pixel size. The Author Derek Rush may be contacted by Email at derekrush@beeb.net October 2002 and July 2003. @#$#@#$#@ default true 0 Polygon -7566196 true true 150 5 40 250 150 205 260 250 ant true 0 Polygon -7566196 true true 136 61 129 46 144 30 119 45 124 60 114 82 97 37 132 10 93 36 111 84 127 105 172 105 189 84 208 35 171 11 202 35 204 37 186 82 177 60 180 44 159 32 170 44 165 60 Polygon -7566196 true true 150 95 135 103 139 117 125 149 137 180 135 196 150 204 166 195 161 180 174 150 158 116 164 102 Polygon -7566196 true true 149 186 128 197 114 232 134 270 149 282 166 270 185 232 171 195 149 186 149 186 Polygon -7566196 true true 225 66 230 107 159 122 161 127 234 111 236 106 Polygon -7566196 true true 78 58 99 116 139 123 137 128 95 119 Polygon -7566196 true true 48 103 90 147 129 147 130 151 86 151 Polygon -7566196 true true 65 224 92 171 134 160 135 164 95 175 Polygon -7566196 true true 235 222 210 170 163 162 161 166 208 174 Polygon -7566196 true true 249 107 211 147 168 147 168 150 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