## The SICP Picture Language

This package provides support for the picture language used in SICP. The non-standard primitives cons-stream and amb are also provided.

### 1` `Introduction

The SICP Picture Language is a small language for drawing pictures. It shows the power of data abstraction and closure. The picture language stems from Peter Henderson’s 1982 paper "Functional Geometry" and was included by Hal Abelson in "Structure and Interpretation of Computer Programs".

Before using this package, read section 2.2.4 of SICP, which is an excellent introduction to the ideas of the picture language. This manual meant as a reference guide.

Peter Henderson has written an updated version of "Functional Geometry", which explains how to construct the Escher fish image.

Note: The primitives cons-stream and amb needed in other chapters of SICP are also provided.

### 2` `Reference

The basic concept of the picture language is a *painter*. A painter draws
it’s image (shifted and scaled) within a frame given by a parallelogram.
Painters can be combined to construct new painters.

### 3` `Example

> (paint (number->painter 0)) |

(rotate90 diagonal-shading)) |

(rotate180 diagonal-shading)))) |

### 4` `Vectors

A mathematical vector is called a *vect* here, in order
to avoid confusion with the builtin vectors of Scheme.

(make-vect x y) → vect? |

x : number? |

y : number? |

Constructs a vect with the given coordinates.

(vector-xcor v) → number? |

v : vect? |

Returns the x-coordinate.

(vector-ycor v) → number? |

v : vect? |

Returns the y-coordinate.

(vector-add v w) → vect? |

v : vect? |

w : vect? |

Adds the two vects by adding their coordinates pairwise.

(vector-sub v w) → vect? |

v : vect? |

w : vect? |

Subtracts the two vects by subtracting their coordinates pairwise.

(vector-scale s v) → vect? |

s : number? |

v : vect? |

Scales the vect by multiplying each coordinate of "v" with the number "s".

### 5` `Frames

A *frame* is descibed by three vectors.

^ |

| frame edge2 vector |

| |

_|__________> |

/| frame edge1 vector |

/ |

/ |

/ frame origin pointer |

(make-frame origin edge1 edge2) → frame? |

origin : vect? |

edge1 : vect? |

edge2 : vect |

Constructs a frame from a frame origin vector and two frame edge vectors.

(frame-origin f) → vect? |

f : frame? |

(frame-edge1 f) → vect? |

f : frame? |

(frame-edge2 f) → vect? |

f : frame? |

Extracts the origin, first edge or second edge from a frame.

(make-relative-frame origin corner1 corner2) → (frame? -> frame?) |

origin : vect? |

corner1 : vect? |

corner2 : vect? |

The function make-relative-frame provides a convenient way to transform frames. Given a frame and three points : origin, corner1, and corner2 (expressed in frame coordinates), it returns a new frame with those corners.

(frame-coord-map f) → (vect? -> vect?) |

f : frame? |

Each frame determines a system of "frame coordinates" (x,y) where (0,0) is the origin of the frame, x represents the displacement along the first edge (as a fraction of the length of the edge) and y is the displacement along the second edge.

The frame coordinate map is returned by frame-coord-map. E.g. these expression return the same value:

((frame-coord-map a-frame) (make-vect 0 0))

(frame-origin a-frame)

### 6` `Segments

A pair of vectors determines a directed line segment - the segment running from the endpoint of the first vector to the endpoint of the second vector.

(make-segment from to) → segment? |

from : vect? |

to : vect? |

(segment-start s) → vect? |

s : segment? |

(segment-end s) → vect? |

s : segment? |

### 7` `Primitive Painters

Painters take a frame and draw an image, transformed to fit inside the frame.

There are four ways to create painters:

from a constant: number->painter

from a list of line segments: segment->painter

form a procedure: procedure->painter

from a picture: picture->painter

(number->painter color) → painter? |

color : 0..255 |

Constructs a painter that fills the frame with a gray color indicated by the number. 0 is black and 255 is white.

(segments->painter los) → painter? |

los : list-of-segment? |

Constructs a painter that draws a stick figure given by the segments (wrt the unit square).

(procedure->painter p) → painter? |

p : procedure? |

Creates painters from procedures. We assume that the procedure f is defined on the unit square.

Then to plot a point p in the target frame, we find the inverse image T^-1(p) of p under the transformation that maps the unit square to the target, and find the value of f at T-1(p).

(picture->painter p) → painter? |

p : picture |

The picture p is defined on some frame.

Given a point p in the target frame, we compute T^-1(p) where T is the transformation that takes the picture frame to the target frame, and find the picture value at the closest integer point.

(load-painter filename) → painter? |

filename : path? |

Uses the image file given by filename to create a painter.

### 8` `Higher Order Painters

(transform-painter origin corner1 corner2) |

→ (painter? -> painter?) |

origin : vect? |

corner1 : vect? |

corner2 : vect? |

A painter can be transformed to produce a new painter which, when given a frame, calls the original painter on the transformed frame.

Transform-painter will given an origin and two corners, return a function that takes a painter as argument and returns a transformed painter.

(flip-horiz p) → painter? |

p : painter |

Returns a painter that flips the image horizontally.

(flip-vert p) → painter? |

p : painter |

Returns a painter that flips the image vertically.

(rotate90 p) → painter? |

p : painter |

(rotate180 p) → painter? |

p : painter |

(rotate270 p) → painter? |

p : painter |

Returns a painter that rotates the image.

(beside p1 p2) → painter? |

p1 : painter |

p2 : painter |

Constructs a painter that paints the images side-by-side.

(below p1 p2) → painter? |

p1 : painter |

p2 : painter |

Constructs a painter that paints the second image below the first.

(superpose p1 p2) → painter? |

p1 : painter |

p2 : painter |

Constructs a painter that paints the two images on top of each other.

### 9` `Simple Builtin Painters

The following painter values are buitin:

black, white and gray Fills the frame with black (0), white (255) or gray (150).

diagonal-shading Fills the frame with a shades of gray. The color transition goes from black in the upper left corner is black, to gray in the bottom right corner.

einstein Draws an image of Einstein.

### 10` `Painting

The procedures paint and paint-hi-res takes a painter as input and return a snip containing the painter’s image. A snip is an image that DrScheme can display automatically.

(paint p) → snip? |

p : painter? |

(paint-hi-res p) → snip? |

p : painter? |

### 11` `Authors

Abelson & Sussman: Structure and Interpretation of Computer Programs.

Daniel Coore: Original MIT Scheme code.

Mike Sperber: PLT port.

Jens Axel Søgaard: Documentation.

### 12` `Other

See also the readme.html from the SICP web-site for more documentation and exercises.

Peter Henderson’s "Functional Geometry".