Matrix

Matrix is a row-major 3x3 matrix used by image transformations in MuPDF (which complies with the respective concepts laid down in the Adobe PDF References). With matrices you can manipulate the rendered image of a page in a variety of ways: (parts of) the page can be rotated, zoomed, flipped, sheared and shifted by setting some or all of just six float values.

Since all points or pixels live in a two-dimensional space, one column vector of that matrix is a constant unit vector, and only the remaining six elements are used for manipulations. These six elements are usually represented by [a, b, c, d, e, f]. Here is how they are positioned in the matrix:

matrix

Please note:

  • the below methods are just convenience functions – everything they do, can also be achieved by directly manipulating the six numerical values
  • all manipulations can be combined – you can construct a matrix that rotates and shears and scales and shifts, etc. in one go. If you however choose to do this, do have a look at the remarks further down or at the Adobe PDF References.
Method / Attribute Description
Matrix.prerotate() perform a rotation
Matrix.prescale() perform a scaling
Matrix.preshear() perform a shearing (skewing)
Matrix.pretranslate() perform a translation (shifting)
Matrix.concat() perform a matrix multiplication
Matrix.invert() calculate the inverted matrix
Matrix.norm() the Euclidean norm
Matrix.a zoom factor X direction
Matrix.b shearing effect Y direction
Matrix.c shearing effect X direction
Matrix.d zoom factor Y direction
Matrix.e horizontal shift
Matrix.f vertical shift
Matrix.is_rectilinear true if rect corners will remain rect corners

Class API

class Matrix
__init__(self)
__init__(self, zoom-x, zoom-y)
__init__(self, shear-x, shear-y, 1)
__init__(self, a, b, c, d, e, f)
__init__(self, matrix)
__init__(self, degree)
__init__(self, sequence)

Overloaded constructors.

Without parameters, the zero matrix Matrix(0.0, 0.0, 0.0, 0.0, 0.0, 0.0) will be created.

zoom-* and shear-* specify zoom or shear values (float) and create a zoom or shear matrix, respectively.

For “matrix” a new copy of another matrix will be made.

Float value “degree” specifies the creation of a rotation matrix which rotates anit-clockwise.

A “sequence” must be any Python sequence object with exactly 6 float entries (see Using Python Sequences as Arguments in PyMuPDF).

fitz.Matrix(1, 1), fitz.Matrix(0.0) and fitz.Matrix(fitz.Identity) create modifyable versions of the Identity matrix, which looks like [1, 0, 0, 1, 0, 0].

norm()

(New in version 1.16.0)

Return the Euclidean norm of the matrix as a vector.

prerotate(deg)

Modify the matrix to perform a counter-clockwise rotation for positive deg degrees, else clockwise. The matrix elements of an identity matrix will change in the following way:

[1, 0, 0, 1, 0, 0] -> [cos(deg), sin(deg), -sin(deg), cos(deg), 0, 0].

Parameters:deg (float) – The rotation angle in degrees (use conventional notation based on Pi = 180 degrees).
prescale(sx, sy)

Modify the matrix to scale by the zoom factors sx and sy. Has effects on attributes a thru d only: [a, b, c, d, e, f] -> [a*sx, b*sx, c*sy, d*sy, e, f].

Parameters:
  • sx (float) – Zoom factor in X direction. For the effect see description of attribute a.
  • sy (float) – Zoom factor in Y direction. For the effect see description of attribute d.
preshear(sx, sy)

Modify the matrix to perform a shearing, i.e. transformation of rectangles into parallelograms (rhomboids). Has effects on attributes a thru d only: [a, b, c, d, e, f] -> [c*sy, d*sy, a*sx, b*sx, e, f].

Parameters:
  • sx (float) – Shearing effect in X direction. See attribute c.
  • sy (float) – Shearing effect in Y direction. See attribute b.
pretranslate(tx, ty)

Modify the matrix to perform a shifting / translation operation along the x and / or y axis. Has effects on attributes e and f only: [a, b, c, d, e, f] -> [a, b, c, d, tx*a + ty*c, tx*b + ty*d].

Parameters:
  • tx (float) – Translation effect in X direction. See attribute e.
  • ty (float) – Translation effect in Y direction. See attribute f.
concat(m1, m2)

Calculate the matrix product m1 * m2 and store the result in the current matrix. Any of m1 or m2 may be the current matrix. Be aware that matrix multiplication is not commutative. So the sequence of m1, m2 is important.

Parameters:
  • m1 (Matrix) – First (left) matrix.
  • m2 (Matrix) – Second (right) matrix.
invert(m = None)

Calculate the matrix inverse of m and store the result in the current matrix. Returns 1 if m is not invertible (“degenerate”). In this case the current matrix will not change. Returns 0 if m is invertible, and the current matrix is replaced with the inverted m.

Parameters:m (Matrix) – Matrix to be inverted. If not provided, the current matrix will be used.
Return type:int
a

Scaling in X-direction (width). For example, a value of 0.5 performs a shrink of the width by a factor of 2. If a < 0, a left-right flip will (additionally) occur.

Type:float
b

Causes a shearing effect: each Point(x, y) will become Point(x, y - b*x). Therefore, looking from left to right, e.g. horizontal lines will be “tilt” – downwards if b > 0, upwards otherwise (b is the tangens of the tilting angle).

Type:float
c

Causes a shearing effect: each Point(x, y) will become Point(x - c*y, y). Therefore, looking upwards, vertical lines will be “tilt” – to the left if c > 0, to the right otherwise (c ist the tangens of the tilting angle).

Type:float
d

Scaling in Y-direction (height). For example, a value of 1.5 performs a stretch of the height by 50%. If d < 0, an up-down flip will (additionally) occur.

Type:float
e

Causes a horizontal shift effect: Each Point(x, y) will become Point(x + e, y). Positive (negative) values of e will shift right (left).

Type:float
f

Causes a vertical shift effect: Each Point(x, y) will become Point(x, y - f). Positive (negative) values of f will shift down (up).

Type:float
is_rectilinear

Rectilinear means that no shearing is present and that any rotations are integer multiples of 90 degrees. Usually this is used to confirm that (axis-aligned) rectangles before the transformation are still axis-aligned rectangles afterwards.

Type:bool

Note

  • This class adheres to the Python sequence protocol, so components can be accessed via their index, too. Also refer to Using Python Sequences as Arguments in PyMuPDF.
  • A matrix can be used with arithmetic operators – see chapter Operator Algebra for Geometry Objects.
  • Changes of matrix properties and execution of matrix methods can be executed consecutively. This is the same as multiplying the respective matrices.
  • Matrix multiplication is not commutative – changing the execution sequence in general changes the result. So it can quickly become unclear which result a transformation will yield.

Examples

Here are examples to illustrate some of the effects achievable. The following pictures start with a page of the PDF version of this help file. We show what happens when a matrix is being applied (though always full pages are created, only parts are displayed here to save space).

This is the original page image:

original

Shifting

We transform it with a matrix where e = 100 (right shift by 100 pixels).

e100

Next we do a down shift by 100 pixels: f = 100.

f100

Flipping

Flip the page left-right (a = -1).

aminus1

Flip up-down (d = -1).

dminus1

Shearing

First a shear in Y direction (b = 0.5).

bnull5

Second a shear in X direction (c = 0.5).

cnull5

Rotating

Finally a rotation by 30 clockwise degrees (prerotate(-30)).

rot60