2.2 Image Guide
This section introduces the 2htdp/image library
through a series of increasingly complex image constructions
and discusses some subtle details of cropping and outline
images. To build a simple-looking house, we can place a triangle above
a rectangle. We can give the house two roofs by putting two triangles next to
each other. But if we want the new roof to be a little smaller, then they do not line
up properly. Instead, we can use beside/align to line up the two triangles
along their bottoms instead of along the middles (which is what
beside does). To add a door to the house, we can overlay a brown rectangle,
aligning it with the center bottom of the rest of the house. We can use a similar technique to put a doorknob on the door, but instead of
overlaying the doorknob on the entire house, we can overlay it just on the
door. A rotary phone dial can be built by from a black disk and 10 little white ones
by placing the white disks, one at a time, at the top of the black disk and
then rotating the entire black disk. To get started, lets define a function
to make little white disks with numbers on them: We’ll use place-and-turn to put the numbers onto the disk: For example: We can write a single function to put all of the numbers together into the dial: That definition is long and tedious to write. We can shorten it using
foldl: To finish off the dial, we need to rotate it a little bit to its natural
position and put a white disk in the center of it. Here’s the inner dial: and here’s a function to build the entire rotary dial, with an argument
that scales the dial: Looking at the image, it feels like the numbers are too close to the edge of
the dial. So we can adjust the place-and-turn function to put a little
black rectangle on top of each number. The rectangle is invisible because it
ends up on top of the black dial, but it does serve to push the digits down
a little. With shapes that have opaque colors like "red" and "blue",
overlaying one on top completely blots out the one one the bottom. For example, the green rectangle here completely covers the blue one
where the two rectangles overlap. But 2htdp/image also supports colors that are not
completely opaque, via the (optional) fourth argument to color. In this example, the color (color 0 255 0 127) looks just
like the color (color 127 255 127) when the background
is white. Since white is (color 255 255 255), we end up
getting 1/2 of 255 for the red and blue components
and 255 for the green one. We can also use alpha blending to make some interesting effects.
For example, the function spin-alot takes an image argument
and repeatedly places it on top of itself, rotating it each time by
1 degree. Here are some uses of spin-alot, first showing the original
shape and then the spun shape. It is also possible to make interesting looking shapes with little recursive functions.
For example, this function repeatedly puts white circles that grow, evenly spaced around
the edge of the given shape: More conventional fractal shapes can also be written using the image
library, e.g.: We can adjust the carpet to add a little color: The Koch curve can be constructed by simply placing four
curves next to each other, rotated appropriately: And then put three of them together to form the Koch snowflake. When rotating an image, some times the image looks best when it
rotates around a point that is not the center of the image. The
rotate function, however, just rotates the image as
a whole, effectively rotating it around the center of its
bounding box. images are snip% objects, so can
be inserted into text%
and pasteboard% objects they implement the convert protocol for 'png-bytes they implement the pict-convert protocol, and there is a low-level interface for drawing directly into
a dc<%> object: render-image. The image library treats coordinates as if they are in the upper-left corner
of each pixel, and infinitesimally small (unlike pixels, which have some area). Thus, when drawing a solid square of whose side-length is 10, the image library
colors in all of the pixels enclosed by the square starting at the upper
left corner of (0,0) and going down to the upper left corner of (10,10),
so the pixel whose upper left at (9,9) is colored in, but the pixel
at (10,10) is not. All told, 100 pixels get colored in, just as expected for
a square with a side length of 10. When drawing lines, however, things get a bit more complex. Specifically,
imagine drawing the outline of that rectangle. Since the border is
between the pixels, there really isn’t a natural pixel to draw to indicate
the border. Accordingly, when drawing an outline square (without a
pen specification, but just a color as the last argument),
the image library uses a pen whose width is 1 pixel, but draws a line
centered at the point (0.5,0.5) that goes down and around to the point (10.5,10.5).
This means that the outline slightly exceeds the bounding box of the shape.
Specifically, the upper and left-hand lines around the square are within
the bounding box, but the lower and right-hand lines are just outside. If you are reading along with this section using
DrRacket,
note that DrRacket clips images to their bounding boxes when rendering them
in the interactions window; read on for the ramifications but know for now that
what you see in the example results here will not be exactly the same as
what you see in the interactions window for that reason. The reason interior lines in this grid are the same thickness as the lines around the edge
is because the rectangles overlap with each other.
That is, the upper-left rectangle’s right edge is right on top of the
next rectangle’s left edge. The special case of adding 0.5 to each coordinate when drawing the square
applies to all outline polygon-based shapes that just pass color,
but does not apply when a pen
is passed as the last argument to create the shape.
For example, if using a pen of thickness 2 to draw a rectangle, we get a
shape that has a border drawing the row of pixels just inside and just outside
the shape. One might imagine that a pen of thickness 1 would draw an outline around the shape with
a 1 pixel thick line, but this would require 1/2 of each pixel to be illuminated, something
that is not possible. Instead, the same pixels are lit up as with the 2 pixel wide pen, but
with only 1/2 of the intensity of the color. So a 1 pixel wide black pen object draws
a 2 pixel wide outline, but in gray. > (rectangle 20 20 "outline" p1) When combining pens and cropping, we can make a rectangle that has a line that is one pixel
wide, but where the line is drawn entirely within the rectangle. This rectangle has a two-pixel wide
black pen, but we can crop out the outer portion of the pen. Using that we can build a grid now too, but this grid has doubled lines on the
interior. While this kind of rectangle is not useful for building grids, it
is important to be able to build rectangles whose drawing does not
exceed its bounding box. Specifically, this kind of drawing is used
by frame and empty-scene so that the extra drawn pixels
are not lost if the image is later clipped to its bounding box. When using image->color-list with outline shapes, the results
can be surprising for the same reasons. For example, a
2x2 black, outline rectangle consists of nine black pixels, as discussed above,
but since image->color-list only returns the pixels that are
within the bounding box, we see only three black pixels and one white one. (list (color 0 0 0 255) (color 0 0 0 255) (color 0 0 0 255) (color 255 255 255 0)) The black pixels are (most of) the upper and left edge of the outline shape,
and the one white pixel is the pixel in the middle of the shape. Alpha blending can cause imprecision in color comparisons resulting in
shapes that appear equal? even though they were created with
different colors. This section explains how that happens. #t (color 0 0 0 50) (color 0 0 0 50) Going back to the two example rectangles,
the drawing library multiplies 50/255 by 1 for the first
shape and multiplies 50/255 by 2 for the second shape (since they
are both drawn on a white background). Then it rounds them to integers, which
results in 0 for both colors, making the images the same.2.2.1 Overlaying, Above, and Beside: A House
> (above (triangle 40 "solid" "red") (rectangle 40 30 "solid" "black")) 
> (above (beside (triangle 40 "solid" "red") (triangle 40 "solid" "red")) (rectangle 80 40 "solid" "black")) 
> (above (beside (triangle 40 "solid" "red") (triangle 30 "solid" "red")) (rectangle 70 40 "solid" "black")) 
> (define victorian (above (beside/align "bottom" (triangle 40 "solid" "red") (triangle 30 "solid" "red")) (rectangle 70 40 "solid" "black"))) > victorian 
> (define door (rectangle 15 25 "solid" "brown")) > (overlay/align "center" "bottom" door victorian) 
> (define door-with-knob (overlay/align "right" "center" (circle 3 "solid" "yellow") door)) > (overlay/align "center" "bottom" door-with-knob victorian) 
2.2.2 Rotating and Overlaying: A Rotary Phone Dial
> (define (a-number digit) (overlay (text (number->string digit) 12 "black") (circle 10 "solid" "white"))) > (define (place-and-turn digit dial) (rotate 30 (overlay/align "center" "top" (a-number digit) dial))) > (place-and-turn 0 (circle 60 "solid" "black")) 
> (place-and-turn 8 (place-and-turn 9 (place-and-turn 0 (circle 60 "solid" "black")))) 
> (define (place-all-numbers dial) (place-and-turn 1 (place-and-turn 2 (place-and-turn 3 (place-and-turn 4 (place-and-turn 5 (place-and-turn 6 (place-and-turn 7 (place-and-turn 8 (place-and-turn 9 (place-and-turn 0 dial))))))))))) > (place-all-numbers (circle 60 "solid" "black")) 
> (define (place-all-numbers dial) (foldl place-and-turn dial '(0 9 8 7 6 5 4 3 2 1))) > (place-all-numbers (circle 60 "solid" "black")) 
> (define inner-dial (overlay (text "555-1234" 9 "black") (circle 30 "solid" "white"))) > (define (rotary-dial f) (scale f (overlay inner-dial (rotate -90 (place-all-numbers (circle 60 "solid" "black")))))) > (rotary-dial 2) 
> (define (place-and-turn digit dial) (rotate 30 (overlay/align "center" "top" (above (rectangle 1 5 "solid" "black") (a-number digit)) dial))) > (rotary-dial 2) 
2.2.3 Alpha Blending
> (overlay (rectangle 60 100 "solid" (color 127 255 127)) (rectangle 100 60 "solid" (color 127 127 255))) 
> (overlay (rectangle 60 100 "solid" (color 0 255 0 127)) (rectangle 100 60 "solid" (color 0 0 255 127))) 
> (define (spin-alot t) (local [(define (spin-more i θ) (cond [(= θ 360) i] [else (spin-more (overlay i (rotate θ t)) (+ θ 1))]))] (spin-more t 0))) > (rectangle 12 120 "solid" (color 0 0 255)) 
> (spin-alot (rectangle 12 120 "solid" (color 0 0 255 1))) 
> (triangle 120 "solid" (color 0 0 255)) 
> (spin-alot (triangle 120 "solid" (color 0 0 255 1))) 
> (isosceles-triangle 120 30 "solid" (color 0 0 255)) 
> (spin-alot (isosceles-triangle 120 30 "solid" (color 0 0 255 1))) 
2.2.4 Recursive Image Functions
> (define (swoosh image s) (cond [(zero? s) image] [else (swoosh (overlay/align "center" "top" (circle (* s 1/2) "solid" "white") (rotate 4 image)) (- s 1))])) > (swoosh (circle 100 "solid" "black") 94) 
> (define (sierpinski-carpet n) (cond [(zero? n) (square 1 "solid" "black")] [else (local [(define c (sierpinski-carpet (- n 1))) (define i (square (image-width c) "solid" "white"))] (above (beside c c c) (beside c i c) (beside c c c)))])) > (sierpinski-carpet 5) 
> (define (colored-carpet colors) (cond [(empty? (rest colors)) (square 1 "solid" (first colors))] [else (local [(define c (colored-carpet (rest colors))) (define i (square (image-width c) "solid" (car colors)))] (above (beside c c c) (beside c i c) (beside c c c)))])) > (colored-carpet (list (color 51 0 255) (color 102 0 255) (color 153 0 255) (color 204 0 255) (color 255 0 255) (color 255 204 0))) 
> (define (koch-curve n) (cond [(zero? n) (square 1 "solid" "black")] [else (local [(define smaller (koch-curve (- n 1)))] (beside/align "bottom" smaller (rotate 60 smaller) (rotate -60 smaller) smaller))])) > (koch-curve 5) 
> (above (beside (rotate 60 (koch-curve 5)) (rotate -60 (koch-curve 5))) (flip-vertical (koch-curve 5))) 
2.2.5 Rotating and Image Centers
> (overlay (rotate 0 (hero 255)) (rotate 10 (hero 125)) (rotate 20 (hero 100)) (rotate 30 (hero 75)) (rotate 40 (hero 50)) (rotate 50 (hero 25))) 
> (define (hero-on-blank α) (define the-hero (hero α)) (define w (image-width the-hero)) (define h (image-height the-hero)) (define d (max w h)) (define dx (/ w 2)) ; centroid x offset (define dy (* 2/3 h)) ; centroid y offset (define blank (circle d "solid" (color 255 255 255 0))) (place-image/align the-hero (- d dx) (- d dy) "left" "top" blank)) 2.2.6 Image Interoperability
2.2.7 The Nitty Gritty of Pixels, Pens, and Lines
> (define s1 (square 20 'outline 'black)) > (define r1 (beside s1 s1 s1 s1 s1 s1)) > (above r1 r1 r1 r1 r1 r1) 
> (define p2 (make-pen "black" 2 "solid" "round" "round")) > (define s2 (crop 0 0 20 20 (rectangle 20 20 "outline" p2))) > s2 
> (define r2 (beside s2 s2 s2 s2 s2 s2)) > (above r2 r2 r2 r2 r2 r2) 
> (image->color-list (rectangle 2 2 "outline" "black")) 2.2.8 The Nitty Gritty of Alpha Blending
> (rectangle 100 100 "solid" (make-color 1 1 1 50)) 
> (overlay (rectangle 100 100 "solid" (make-color 1 1 1 50)) (rectangle 200 200 "solid" "green")) 
> (equal? (rectangle 100 100 'solid (make-color 1 1 1 50)) (rectangle 100 100 'solid (make-color 2 2 2 50))) > (first (image->color-list (rectangle 100 100 'solid (make-color 1 1 1 50)))) > (first (image->color-list (rectangle 100 100 'solid (make-color 2 2 2 50))))
