pp: cantor set problem.

Very hard. My original solution was able to draw the Cantor
Set just as the one in their solution. Still, I had completely
misunderstood the other two problems and I copied what they
did.

09a-fractals/pp02-cantor-set.rkt

@@ -0,0 +1,177 @@
+#lang htdp/isl
+(require 2htdp/image)
+(require 2htdp/universe)
+
+
+;
+; PROBLEM:
+;
+; A Cantor Set is another fractal with a nice simple geometry.
+; The idea of a Cantor set is to have a bar (or rectangle) of
+; a certain width w, then below that are two recursive calls each
+; of 1/3 the width, separated by a whitespace of 1/3 the width.
+;
+; So this means that the
+;   width of the whitespace   wc  is  (/ w 3)
+;   width of recursive calls  wr  is  (/ (- w wc) 2)
+;
+; To make it look better a little extra whitespace is put between
+; the bars.
+;
+;
+; Here are a couple of examples (assuming a reasonable CUTOFF)
+;
+
+(define IMG-CANTOR-SET-EXAMPLES (bitmap "../imgs/026-cantor-set.png"))
+
+;
+;
+; PROBLEM A:
+;
+; Design a function that consumes a width and produces a cantor set of
+; the given width.
+;
+;
+; PROBLEM B:
+;
+; Add a second parameter to your function that controls the percentage
+; of the recursive call that is white each time. Calling your new function
+; with a second argument of 1/3 would produce the same images as the old
+; function.
+;
+; PROBLEM C:
+;
+; Now you can make a fun world program that works this way:
+;   The world state should simply be the most recent x coordinate of the mouse.
+;
+;   The to-draw handler should just call your new cantor function with the
+;   width of your MTS as its first argument and the last x coordinate of
+;   the mouse divided by that width as its second argument.
+;
+
+
+;; A world program for running a Cantor Set.
+;; The mouse position controls parameters of the fractal.
+
+;; =========================================
+;; CONSTANTS:
+
+(define W 400)
+(define H 600)
+(define LIM 4)
+(define BH 20) ; bar height.
+(define SPC-H (/ BH 2)) ; spacing bar height.
+
+
+;; =========================================
+;; DATA DEFINITIONS:
+
+;; WorldState is Number
+;; interp. the last x-coordinate of the mouse
+(define WS1 100)
+(define WS2 4)
+#;
+(define (fn-for-ws ws)
+  (... ws))
+
+
+;; =========================================
+;; FUNCTIONS:
+
+;; WorldState -> WorldState
+;; Run the interactive Cantor Set generator;
+;; Call: (main 0)
+;; <no testes for the main function>
+(define (main ws)
+  (big-bang ws
+            (on-draw render)
+            (on-mouse handle-mouse)))
+
+;; WorldState -> Image
+;; Render Cantor Set based on current mouse position.
+(check-expect (render 100) (cantor W (/ 100 W)))
+(check-expect (render 300) (cantor W (/ 300 W)))
+
+(define (render ws)
+  (cantor W (/ ws W)))
+
+;; WorldState Integer Integer MouseEvent -> WorldState
+;; update the current ws according to mouse position
+(check-expect (handle-mouse 0   0 10 "move") 0)
+(check-expect (handle-mouse 100 0 10 "move") 0)
+(check-expect (handle-mouse  20 0 10 "button-down") 20)
+
+; <template according to MouseEvent>
+(define (handle-mouse ws x y mevt)
+  (cond [(mouse=? mevt "move") x]
+        [else ws]))
+
+;; Natural Natural -> Image
+;; Render Cantor Set:
+;; -> first line has width w;
+;; -> second line center white band has width (* w r)
+(check-expect (cantor 0 0.5)
+              (mkb 0))
+
+(check-expect (cantor 32 0)
+              (above (mkb 32)
+                     (mks 32)
+                     (mkb 32)
+                     (mks 32)
+                     (mkb 32)
+                     (mks 32)
+                     (mkb 32)))
+
+(check-expect (cantor 32 1)
+              (above (mkb 32)
+                     (mks 32)
+                     (mkspc 32)))
+
+(check-expect (cantor 32 .5)
+              (above (mkb 32)
+                     (mks 32)
+                     (beside
+                      (above (mkb 8)
+                             (mks 8)
+                             (beside
+                              (mkb 2)
+                              (mkspc 4)
+                              (mkb 2)))
+                      (rectangle 16 BH "solid" "white")
+                      (above (mkb 8)
+                             (mks 8)
+                             (beside
+                              (mkb 2)
+                              (mkspc 4)
+                              (mkb 2))))))
+
+;
+; TERMINATION ARGUMENT:
+; --------------------
+;   trivial case: w <= CUTOFF
+;   reduction step: (/ (- w (* r w)) 2)
+;   argument: as long as CUTOFF is > 0, and 0 <= r < 1, repeatedly multiplying w by r
+;   (the reduction step) will reduce w to eventually reach the base case.
+;
+
+; <template from generative recursion>
+(define (cantor w r)
+  (if (<= w LIM)
+      (rectangle w BH "solid" "blue")
+      (local [(define wc  (* w r))            ;width of center white band
+              (define ws  (/ (- w wc) 2))     ;width of side blue bands
+              (define ctr (rectangle wc BH "solid" "white"))
+              (define l/r (cantor ws r))]
+        (above (mkb w)
+               (mks w)
+               (beside l/r ctr l/r)))))
+
+(define (mkb w)
+  (rectangle w BH "solid" "blue"))
+
+(define (mkspc w)
+  (rectangle w BH "solid" "white"))
+
+(define (mks w)
+  (rectangle w SPC-H "solid" "white"))
+