1. rotate

Rotates the coordinate system axes.

1.1. Syntax 

angle rotate → -
angle matrix rotate → matrix

1.2. Stack Effects

Table 1. Before
Level Object

0

angle (number, in degrees)
Table 2. After (first form)
Level Object

(empty)

No results
Table 3. After (second form)
Level Object

0

matrix (modified matrix)

1.3. Description

With no matrix operand, rotate builds a temporary transformation matrix R and concatenates it with the current transformation matrix (CTM). Precisely, rotate replaces the CTM by R × CTM.

The effect is to rotate the user coordinate system axes about their origin by angle degrees. Positive angles rotate counterclockwise; negative angles rotate clockwise. The position of the user coordinate origin and the sizes of the x and y units are unchanged.

If the matrix operand is supplied, rotate replaces the value of matrix by R and pushes the modified matrix back on the operand stack. In this case, rotate does not affect the CTM.

The rotation matrix R for angle θ (in degrees) has the form:

[cos(θ) sin(θ) -sin(θ) cos(θ) 0 0]

This transforms coordinates according to:

x' = x×cos(θ) - y×sin(θ)
y' = x×sin(θ) + y×cos(θ)

1.4. PostScript Level

Level 1 and later

1.5. Examples

Simple rotation
45 rotate         % Rotate 45 degrees counterclockwise
0 0 moveto
100 0 lineto
stroke            % Line at 45-degree angle
Rotating text
/Helvetica findfont 24 scalefont setfont
100 100 translate
45 rotate
0 0 moveto
(Rotated Text) show
Multiple rotations
% Draw 12 lines radiating from origin (like clock)
/drawLine {
  newpath
  0 0 moveto
  100 0 lineto
  stroke
} def

0 1 11 {
  drawLine
  30 rotate       % Rotate 30 degrees for next line
} for

1.6. Common Use Cases

1.6.1. Creating Radial Patterns 

/petal {
  newpath
  0 0 moveto
  50 0 lineto
  40 10 lineto
  closepath
  fill
} def

% Draw flower with 8 petals
0 1 7 {
  gsave
    pop           % Discard loop counter
    petal
  grestore
  45 rotate       % 360/8 = 45 degrees
} for

1.6.2. Rotating Around a Point 

% Rotate 45 degrees around point (100, 100)
100 100 translate  % Move origin to rotation center
45 rotate          % Rotate
-100 -100 translate % Move origin back
% Now coordinate system is rotated around (100, 100)

1.6.3. Angled Text Labels 

% Label at an angle
gsave
  200 300 translate
  -30 rotate      % Clockwise 30 degrees
  0 0 moveto
  (Label) show
grestore

1.6.4. Creating Spirals 

/spiral {
  0 0 moveto
  0 1 100 {
    /i exch def
    i 0 lineto
    5 rotate
    i 1 add 0 translate
  } for
  stroke
} def

1.7. Common Pitfalls

Rotation Center - rotate always rotates around the current origin, not around drawn content. 
% This rotates around (0,0), not around the rectangle
100 100 translate
0 0 50 50 rectfill
45 rotate         % Rectangle stays at same position!

% To rotate rectangle around its center:
gsave
  100 100 translate  % Move to rectangle center
  45 rotate          % Rotate
  -25 -25 translate  % Offset by half size
  0 0 50 50 rectfill
grestore
Angles in Degrees - PostScript uses degrees, not radians. Common mistake with mathematical constants. 
% Wrong:
3.14159 rotate    % Tiny rotation (3.14 degrees)

% Right:
180 rotate        % Half circle rotation
Cumulative Rotations - Multiple rotations accumulate. 
45 rotate
45 rotate         % Total rotation is 90 degrees

% Use gsave/grestore to isolate:
gsave
  45 rotate
  % Draw content
grestore
% Rotation undone
Clockwise vs Counterclockwise - Remember: positive is counterclockwise, negative is clockwise. 
90 rotate         % Quarter turn left
-90 rotate        % Quarter turn right
Order Matters - Rotation before translation is different from translation before rotation. 
% These produce different results:
45 rotate 100 100 translate
100 100 translate 45 rotate

1.8. Error Conditions

Error Condition

[rangecheck]

Resulting matrix values exceed implementation limits

[stackunderflow]

Fewer than 1 operand on stack (first form) or fewer than 2 (second form)

[typecheck]

Operand is not a number, or matrix operand is not an array

1.9. Implementation Notes

  • Angles greater than 360 or less than -360 are valid and work correctly

  • Very large angles may experience precision loss due to floating-point arithmetic

  • Common angles (0, 90, 180, 270, 360) may be optimized internally

  • Rotation is applied during path construction, not during painting

  • The sin and cos functions use the same angle interpretation

1.10. Matrix Mathematics

The rotation matrix for rotate by angle θ is:

R = [cos(θ)  sin(θ)  -sin(θ)  cos(θ)  0  0]

Concatenating with the CTM:

CTM' = R × CTM
     = [cos(θ)  sin(θ)  -sin(θ)  cos(θ)  0  0] × [a  b  c  d  e  f]
     = [a×cos(θ)-c×sin(θ)  b×cos(θ)-d×sin(θ)
        a×sin(θ)+c×cos(θ)  b×sin(θ)+d×cos(θ)  e  f]

1.11. Graphics State Effects

Rotation affects:

  • Path coordinates: Rotated around origin

  • Line width: Direction-dependent if CTM has non-uniform scaling

  • Dash pattern: May appear different at different angles

  • Text: Rotated according to the transformation

  • Images: Rotated if within transformed coordinate system

1.12. Special Angles

Angle Effect Matrix

0°

No rotation

[1 0 0 1 0 0]

90°

Quarter turn left

[0 1 -1 0 0 0]

180°

Half turn

[-1 0 0 -1 0 0]

270° or -90°

Quarter turn right

[0 -1 1 0 0 0]

360°

Full circle

[1 0 0 1 0 0]

1.13. Performance Considerations

  • Rotation is slightly more expensive than translation or scaling

  • Common angles may be optimized

  • Very large or very small angles don’t impact performance significantly

  • Combining transformations (translate, scale, rotate) into a single matrix can improve performance

1.14. Combining Transformations 

% Create complex transformation
gsave
  100 100 translate  % Position
  45 rotate          % Orientation
  2 2 scale          % Size
  % Draw unit-sized content
  0 0 50 50 rectfill
grestore

% Equivalent using matrix:
matrix
100 100 translate
45 rotate
2 2 scale
concat
0 0 50 50 rectfill

1.15. See Also

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