1. Coordinate Systems

PostScript uses a flexible coordinate system that separates logical drawing coordinates from physical device coordinates. Understanding this system is crucial for creating device-independent documents.

1.1. Overview

PostScript employs multiple coordinate spaces that transform between user-defined coordinates and device pixels:

  • User Space - The coordinate system you work in (default: points, 1/72 inch)

  • Device Space - The actual pixel coordinates of the output device

  • Current Transformation Matrix (CTM) - Converts user space to device space

This separation allows the same PostScript code to work correctly on different devices (screens, printers, plotters) without modification.

1.2. User Space Basics

1.2.1. Default User Space

By default, PostScript user space has these properties:

  • Origin (0,0) at the bottom-left corner

  • X-axis increases to the right

  • Y-axis increases upward

  • Units are in points (1 point = 1/72 inch)

  • Positive angles rotate counter-clockwise

% Draw a simple line from origin
0 0 moveto      % Start at bottom-left (0,0)
100 100 lineto  % Draw to (100, 100) - up and right
stroke

1.2.2. Coordinate Units

The default unit is the PostScript point:

  • 1 point = 1/72 inch = 0.3528 mm

  • 72 points = 1 inch

  • 28.35 points ≈ 1 centimeter

% A 1-inch square
0 0 moveto
72 0 lineto      % 1 inch right
72 72 lineto     % 1 inch up
0 72 lineto      % 1 inch left
closepath
stroke

1.2.3. Page Coordinates

Standard paper sizes in points:

% US Letter: 8.5" × 11"
% Width: 8.5 × 72 = 612 points
% Height: 11 × 72 = 792 points

% A4: 210mm × 297mm
% Width: 210 / 0.3528 ≈ 595 points
% Height: 297 / 0.3528 ≈ 842 points

% Center of US Letter page
306 396  % (612/2, 792/2)

1.3. Coordinate Transformations

1.3.1. The translate Operator

Moves the origin to a new location:

% Move origin to center of US Letter page
306 396 translate

% Now (0,0) is at page center
0 0 moveto
100 0 rlineto    % Draw 100 points to the right from center
stroke

Practical example - drawing multiple shapes:

% Draw three circles at different positions
gsave
  100 100 translate
  0 0 50 0 360 arc stroke
grestore

gsave
  300 100 translate
  0 0 50 0 360 arc stroke
grestore

gsave
  200 300 translate
  0 0 50 0 360 arc stroke
grestore

1.3.2. The scale Operator

Changes the size of the coordinate system:

% Scale by 2 in both directions
2 2 scale

% Now 1 unit = 2 points
0 0 moveto
50 0 lineto   % Actually draws 100 points
stroke

Non-uniform scaling:

% Scale x by 2, y by 3
2 3 scale

0 0 moveto
50 50 lineto  % Actually draws 100 points right, 150 points up
stroke

Converting units with scale:

% Work in millimeters instead of points
% 1 mm = 72/25.4 points ≈ 2.835 points
72 25.4 div dup scale

% Now coordinates are in millimeters
0 0 moveto
100 0 lineto  % 100 mm line
stroke

1.3.3. The rotate Operator

Rotates the coordinate system (angle in degrees, counter-clockwise):

% Rotate 45 degrees
45 rotate

0 0 moveto
100 0 lineto  % Line at 45-degree angle
stroke

Rotating around a point other than origin:

% Rotate around point (200, 300)
200 300 translate    % Move origin to rotation center
45 rotate            % Rotate
-200 -300 translate  % Move origin back

% Or more simply:
% Draw at (200, 300), rotated 45 degrees
gsave
  200 300 translate
  45 rotate
  % Draw here with (0,0) at the rotation point
  0 0 50 0 360 arc
  stroke
grestore

1.3.4. Combining Transformations

Transformations are cumulative and applied in order:

% Order matters!
gsave
  100 100 translate  % Move first
  45 rotate          % Then rotate
  2 2 scale          % Then scale

  % Draw a square (will be rotated and scaled)
  0 0 moveto
  50 0 lineto
  50 50 lineto
  0 50 lineto
  closepath
  stroke
grestore

Different order gives different results:

% Compare: rotate then translate
gsave
  45 rotate          % Rotate first
  100 100 translate  % Then translate (in rotated space!)

  0 0 50 0 360 arc
  stroke
grestore
% Circle appears in different location!

1.4. The Current Transformation Matrix (CTM)

1.4.1. Understanding the CTM

The CTM is a 6-element matrix that transforms user space to device space:

[ a b c d tx ty ]

Where:
a, d = scaling factors
b, c = rotation/skew factors
tx, ty = translation values

Transformation formula:

x' = a*x + c*y + tx
y' = b*x + d*y + ty

1.4.2. Accessing the CTM 

% Get current matrix
matrix currentmatrix  % Returns: [a b c d tx ty]

% Example: print the default matrix
matrix currentmatrix ==
% Outputs: [1 0 0 1 0 0] (identity matrix)

1.4.3. Setting a Custom CTM 

% Create a custom transformation matrix
% [sx 0 0 sy tx ty] - scale and translate
[2 0 0 2 100 100] concat

% Or create matrix and apply it
matrix
dup 0 2 put      % a = 2 (x scale)
dup 3 2 put      % d = 2 (y scale)
dup 4 100 put    % tx = 100
dup 5 100 put    % ty = 100
concat

1.4.4. Resetting the CTM 

% Reset to default (identity matrix)
initmatrix

% Or reset to device default
initgraphics  % Also resets other graphics state

1.5. Saving and Restoring Coordinate Systems

1.5.1. Using gsave and grestore

The graphics state stack preserves the CTM:

% Save current state
gsave
  % Modify coordinate system
  100 100 translate
  2 2 scale
  45 rotate

  % Draw something
  0 0 50 0 360 arc stroke

% Restore original state
grestore

% Back to original coordinates
0 0 moveto
100 0 lineto
stroke

1.5.2. Nested Transformations 

% Main scene transformation
gsave
  300 400 translate

  % Draw multiple objects with individual transforms
  gsave
    0 0 translate
    0 0 30 0 360 arc stroke
  grestore

  gsave
    100 0 translate
    0 0 30 0 360 arc stroke
  grestore

  gsave
    50 100 translate
    0 0 30 0 360 arc stroke
  grestore

grestore

1.6. Device Space

1.6.1. Understanding Device Space

Device space represents the actual output device:

  • Origin typically at bottom-left of page/screen

  • Units are device pixels or dots

  • Resolution varies by device (72 dpi, 300 dpi, 600 dpi, etc.)

1.6.2. Transforming Between Spaces 

% Transform user coordinates to device coordinates
/userX 100 def
/userY 200 def

matrix currentmatrix
userX userY
dtransform        % Device transform

% Stack now has: deviceX deviceY

Inverse transformation (device to user):

% Transform device coordinates to user coordinates
/deviceX 500 def
/deviceY 600 def

matrix currentmatrix
deviceX deviceY
idtransform       % Inverse device transform

% Stack now has: userX userY

1.6.3. Device-Independent Drawing

Key principle: Always work in user space coordinates:

% Good: device-independent
/drawCircle {  % x y radius -> -
  0 360 arc stroke
} def

100 100 50 drawCircle  % Works on any device

% Bad: device-dependent
/drawCircle {
  % Assumes specific device resolution
  500 500 moveto  % Hardcoded device coordinates
  % ...
} def

1.7. Practical Coordinate Patterns

1.7.1. Centering Content 

% Center a drawing on the page
% For US Letter (612 × 792)
306 396 translate  % Move to center

% Draw centered content
-50 -50 moveto
100 0 rlineto
0 100 rlineto
-100 0 rlineto
closepath
stroke

1.7.2. Creating a Grid System 

% Define a 12-column grid (like Bootstrap)
/pageWidth 612 def
/columns 12 def
/gutterWidth 20 def

/columnWidth {
  pageWidth gutterWidth columns 1 sub mul sub
  columns div
} def

% Position element in column 3
/col3X {
  columnWidth 2 mul      % 2 full columns
  gutterWidth 2 mul add  % 2 gutters
} def

col3X 100 translate
% Draw content here

1.7.3. Aspect Ratio Preservation 

% Scale to fit while preserving aspect ratio
/srcWidth 800 def
/srcHeight 600 def
/dstWidth 400 def
/dstHeight 400 def

% Calculate scale factor (use minimum to fit)
/scaleX dstWidth srcWidth div def
/scaleY dstHeight srcHeight div def
/scale scaleX scaleY lt { scaleX } { scaleY } ifelse def

% Center and scale
dstWidth 2 div dstHeight 2 div translate
scale dup scale
srcWidth -2 div srcHeight -2 div translate

% Draw content (will be scaled and centered)
0 0 srcWidth srcHeight rectfill

1.7.4. Margin System 

% Define page margins (in points)
/marginLeft 72 def    % 1 inch
/marginRight 72 def
/marginTop 72 def
/marginBottom 72 def

% Calculate content area
/pageWidth 612 def
/pageHeight 792 def
/contentWidth pageWidth marginLeft sub marginRight sub def
/contentHeight pageHeight marginTop sub marginBottom sub def

% Set up coordinate system for content
marginLeft marginBottom translate

% Now (0, 0) is at content area origin
% And (contentWidth, contentHeight) is top-right of content area

1.8. Common Coordinate System Patterns

1.8.1. Pattern 1: Local Coordinate System

Define objects in their own coordinate system:

% Define a star in its own coordinate system (centered at origin)
/star {
  newpath
  0 50 moveto
  14.7 15.4 lineto
  47.6 15.4 lineto
  18.1 -5.9 lineto
  29.4 -40.5 lineto
  0 -18.2 lineto
  -29.4 -40.5 lineto
  -18.1 -5.9 lineto
  -47.6 15.4 lineto
  -14.7 15.4 lineto
  closepath
} def

% Use it anywhere
gsave
  100 100 translate
  star fill
grestore

gsave
  300 300 translate
  2 2 scale
  star fill
grestore

1.8.2. Pattern 2: Percentage-Based Layout

Work in percentages instead of absolute coordinates:

% Set up percentage-based coordinate system
/pageWidth 612 def
/pageHeight 792 def

% Scale so 100 = 100% of page
100 pageWidth div 100 pageHeight div scale

% Now work in percentages
50 50 translate  % Center (50% of page width and height)
10 10 moveto     % 10% from left, 10% from bottom

1.8.3. Pattern 3: Viewport System

Create multiple viewports on one page:

% Define viewport (x, y, width, height as percentages)
/viewport {  % x y w h -> -
  4 dict begin
    /h exch 792 mul 100 div def
    /w exch 612 mul 100 div def
    /y exch 792 mul 100 div def
    /x exch 612 mul 100 div def

    gsave
      x y translate

      % Clip to viewport
      newpath
      0 0 moveto
      w 0 rlineto
      0 h rlineto
      w neg 0 rlineto
      closepath
      clip

      % Set up coordinate system within viewport
      % (will be restored by grestore)
  end
} def

/endviewport {
  grestore
} def

% Use viewports
0 0 50 50 viewport
  % Draw in bottom-left quadrant
endviewport

50 0 50 50 viewport
  % Draw in bottom-right quadrant
endviewport

1.9. Coordinate System Best Practices

1.9.1. Always Use Graphics State Stack 

% Good: isolated transformations
gsave
  100 100 translate
  % ...
grestore

% Bad: permanent transformations
100 100 translate
% ... transformations accumulate!

1.9.2. Document Your Coordinate System 

% At the start of your document
% This file uses:
% - User space units: points (1/72 inch)
% - Origin: bottom-left
% - Page size: US Letter (612 × 792 points)
% - Margin: 1 inch (72 points) on all sides

1.9.3. Use Relative Coordinates for Shapes 

% Good: relative positioning
/drawSquare {  % size -> -
  dup 0 rlineto
  dup 0 exch rlineto
  neg 0 rlineto
  closepath
} def

% Use absolute positioning only for initial moveto
100 100 moveto
50 drawSquare

% Bad: absolute coordinates in shape
/drawSquare {
  100 100 lineto  % Hardcoded - not reusable
  150 100 lineto
  % ...
} def

1.9.4. Create Coordinate Utilities 

% Convert inches to points
/inch { 72 mul } def

% Convert mm to points
/mm { 72 mul 25.4 div } def

% Usage
2 inch 3 inch translate   % 2" right, 3" up
100 mm 150 mm moveto      % Position in mm

1.10. Debugging Coordinates

1.10.1. Visualize Coordinate System

Draw a coordinate grid to understand current space:

/drawGrid {
  gsave
    0.1 setlinewidth
    0.8 setgray

    % Draw vertical lines
    -500 10 500 {
      dup -500 moveto 500 lineto stroke
    } for

    % Draw horizontal lines
    -500 10 500 {
      dup -500 exch moveto 500 exch lineto stroke
    } for

    % Draw axes
    0 setgray
    1 setlinewidth
    -500 0 moveto 500 0 lineto stroke
    0 -500 moveto 0 500 lineto stroke
  grestore
} def

% Use it to see your coordinate system
gsave
  100 100 translate
  2 2 scale
  45 rotate
  drawGrid
grestore

1.10.2. Show Current Transformation 

/showCTM {
  (Current Transformation Matrix:) print
  matrix currentmatrix {
    ( ) print
    =string cvs print
  } forall
  () print
} def

% Usage
100 100 translate
showCTM
% Prints: Current Transformation Matrix: 1 0 0 1 100 100

1.10.3. Coordinate Debugging Function 

/debugPoint {  % x y label -> -
  gsave
    % Draw point
    0 setgray
    1 setlinewidth
    newpath
    2 copy 3 0 360 arc
    fill

    % Show label
    /Helvetica findfont 10 scalefont setfont
    moveto
    5 5 rmoveto
    show
  grestore
} def

% Usage
100 200 (Point A) debugPoint
150 250 (Point B) debugPoint

1.11. Common Pitfalls

1.11.1. Forgetting to Save/Restore State 

% Wrong: transformations accumulate
100 100 translate
% Draw
100 100 translate  % Now at (200, 200) total!
% Draw

% Correct: use gsave/grestore
gsave
  100 100 translate
  % Draw
grestore

gsave
  100 100 translate  % Back to base, then +100, +100
  % Draw
grestore

1.11.2. Incorrect Transformation Order 

% Want to rotate around point (100, 100)

% Wrong order:
100 100 translate
45 rotate
% Rotates around new origin, not (100,100) in original space

% Correct order:
45 rotate
100 100 rotate 45 neg mul rotate  % Complex!

% Better approach:
100 100 translate  % Move origin to rotation center
45 rotate           % Rotate
% Draw (0,0) is now at (100,100) in original space

1.11.3. Mixing Coordinate Systems 

% Wrong: mixing user and device coordinates
matrix currentmatrix
100 200 dtransform  % Get device coordinates
moveto              % But moveto expects user coordinates!

% Correct: stay in one system
100 200 moveto      % User coordinates

1.12. See Also

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