Math Types & Utilities (mathutils)¶

This module provides access to math operations.

Note

Classes, methods and attributes that accept vectors also accept other numeric sequences, such as tuples, lists.

Submodules:

The mathutils module provides the following classes:

import mathutils
from math import radians

vec = mathutils.Vector((1.0, 2.0, 3.0))

mat_rot = mathutils.Matrix.Rotation(radians(90.0), 4, 'X')
mat_trans = mathutils.Matrix.Translation(vec)

mat = mat_trans * mat_rot
mat.invert()

mat3 = mat.to_3x3()
quat1 = mat.to_quaternion()
quat2 = mat3.to_quaternion()

quat_diff = quat1.rotation_difference(quat2)

print(quat_diff.angle)

class mathutils.Color(rgb)¶

This object gives access to Colors in Blender.

Parameters:	rgb (3d vector) – (r, g, b) color values

import mathutils

# color values are represented as RGB values from 0 - 1, this is blue
col = mathutils.Color((0.0, 0.0, 1.0))

# as well as r/g/b attribute access you can adjust them by h/s/v
col.s *= 0.5

# you can access its components by attribute or index
print("Color R:", col.r)
print("Color G:", col[1])
print("Color B:", col[-1])
print("Color HSV: %.2f, %.2f, %.2f", col[:])


# components of an existing color can be set
col[:] = 0.0, 0.5, 1.0

# components of an existing color can use slice notation to get a tuple
print("Values: %f, %f, %f" % col[:])

# colors can be added and subtracted
col += mathutils.Color((0.25, 0.0, 0.0))

# Color can be multiplied, in this example color is scaled to 0-255
# can printed as integers
print("Color: %d, %d, %d" % (col * 255.0)[:])

# This example prints the color as hexidecimal
print("Hexidecimal: %.2x%.2x%.2x" % (col * 255.0)[:])

copy()¶

Returns a copy of this color.

Returns:	A copy of the color.
Return type:	`Color`

Note

use this to get a copy of a wrapped color with no reference to the original data.

freeze()¶

Make this object immutable.

After this the object can be hashed, used in dictionaries & sets.

Returns:	An instance of this object.

b¶

Blue color channel.

Type:	float

g¶

Green color channel.

Type:	float

h¶

HSV Hue component in [0, 1].

Type:	float

hsv¶

HSV Values in [0, 1].

Type:	float triplet

is_frozen¶

True when this object has been frozen (read-only).

Type:	boolean

is_wrapped¶

True when this object wraps external data (read-only).

Type:	boolean

owner¶: The item this is wrapping or None (read-only).

r¶

Red color channel.

Type:	float

s¶

HSV Saturation component in [0, 1].

Type:	float

v¶

HSV Value component in [0, 1].

Type:	float

class mathutils.Euler(angles, order='XYZ')¶

This object gives access to Eulers in Blender.

See also

Inverse matrix <https://en.wikipedia.org/wiki/Inverse_matrix> on Wikipedia.

invert_safe()¶: Set the matrix to its inverse, will never error. If degenerated (e.g. zero scale on an axis), add some epsilon to its diagonal, to get an invertible one. If tweaked matrix is still degenerated, set to the identity matrix instead.

See also

Inverse Matrix <https://en.wikipedia.org/wiki/Inverse_matrix> on Wikipedia.

inverted(fallback=None)¶

Return an inverted copy of the matrix.

Parameters:	fallback (any) – return this when the inverse can’t be calculated (instead of raising a `ValueError`).
Returns:	the inverted matrix or fallback when given.
Return type:	`Matrix`

inverted_safe()¶

Return an inverted copy of the matrix, will never error. If degenerated (e.g. zero scale on an axis), add some epsilon to its diagonal, to get an invertible one. If tweaked matrix is still degenerated, return the identity matrix instead.

Returns:	the inverted matrix.
Return type:	`Matrix`

lerp(other, factor)¶

Returns the interpolation of two matrices. Uses polar decomposition, see “Matrix Animation and Polar Decomposition”, Shoemake and Duff, 1992.

Parameters:	other (`Matrix`) – value to interpolate with. factor (float) – The interpolation value in [0.0, 1.0].
Returns:	The interpolated matrix.
Return type:	`Matrix`

normalize()¶: Normalize each of the matrix columns.

normalized()¶

Return a column normalized matrix

Returns:	a column normalized matrix
Return type:	`Matrix`

resize_4x4()¶: Resize the matrix to 4x4.

rotate(other)¶

Rotates the matrix by another mathutils value.

Parameters:	other (`Euler`, `Quaternion` or `Matrix`) – rotation component of mathutils value

Note

If any of the columns are not unit length this may not have desired results.

to_3x3()¶

Return a 3x3 copy of this matrix.

Returns:	a new matrix.
Return type:	`Matrix`

to_4x4()¶

Return a 4x4 copy of this matrix.

Returns:	a new matrix.
Return type:	`Matrix`

to_euler(order, euler_compat)¶

Return an Euler representation of the rotation matrix (3x3 or 4x4 matrix only).

Parameters:	order (string) – Optional rotation order argument in [‘XYZ’, ‘XZY’, ‘YXZ’, ‘YZX’, ‘ZXY’, ‘ZYX’]. euler_compat (`Euler`) – Optional euler argument the new euler will be made compatible with (no axis flipping between them). Useful for converting a series of matrices to animation curves.
Returns:	Euler representation of the matrix.
Return type:	`Euler`

to_quaternion()¶

Return a quaternion representation of the rotation matrix.

Returns:	Quaternion representation of the rotation matrix.
Return type:	`Quaternion`

to_scale()¶

Return the scale part of a 3x3 or 4x4 matrix.

Returns:	Return the scale of a matrix.
Return type:	`Vector`

Note

This method does not return a negative scale on any axis because it is not possible to obtain this data from the matrix alone.

to_translation()¶

Return the translation part of a 4 row matrix.

Returns:	Return the translation of a matrix.
Return type:	`Vector`

transpose()¶: Set the matrix to its transpose.

See also

Transpose <https://en.wikipedia.org/wiki/Transpose> on Wikipedia.

transposed()¶

Return a new, transposed matrix.

Returns:	a transposed matrix
Return type:	`Matrix`

zero()¶

Set all the matrix values to zero.

Return type:	`Matrix`

col¶

Access the matix by colums, 3x3 and 4x4 only, (read-only).

Type:	Matrix Access

is_frozen¶

True when this object has been frozen (read-only).

Type:	boolean

is_negative¶

True if this matrix results in a negative scale, 3x3 and 4x4 only, (read-only).

Type:	bool

is_orthogonal¶

True if this matrix is orthogonal, 3x3 and 4x4 only, (read-only).

Type:	bool

is_orthogonal_axis_vectors¶

True if this matrix has got orthogonal axis vectors, 3x3 and 4x4 only, (read-only).

Type:	bool

is_wrapped¶

True when this object wraps external data (read-only).

Type:	boolean

median_scale¶

The average scale applied to each axis (read-only).

Type:	float

owner¶: The item this is wrapping or None (read-only).

row¶

Access the matix by rows (default), (read-only).

Type:	Matrix Access

translation¶

The translation component of the matrix.

Type:	Vector

class mathutils.Quaternion([seq[, angle]])¶

This object gives access to Quaternions in Blender.

Parameters:	seq (`Vector`) – size 3 or 4 angle (float) – rotation angle, in radians

The constructor takes arguments in various forms:

(), no args: Create an identity quaternion
(wxyz): Create a quaternion from a (w, x, y, z) vector.
(exponential_map): Create a quaternion from a 3d exponential map vector.

See also

to_exponential_map()
(axis, angle): Create a quaternion representing a rotation of angle radians over axis.

See also

to_axis_angle()

import mathutils
import math

# a new rotation 90 degrees about the Y axis
quat_a = mathutils.Quaternion((0.7071068, 0.0, 0.7071068, 0.0))

# passing values to Quaternion's directly can be confusing so axis, angle
# is supported for initializing too
quat_b = mathutils.Quaternion((0.0, 1.0, 0.0), math.radians(90.0))

print("Check quaternions match", quat_a == quat_b)

# like matrices, quaternions can be multiplied to accumulate rotational values
quat_a = mathutils.Quaternion((0.0, 1.0, 0.0), math.radians(90.0))
quat_b = mathutils.Quaternion((0.0, 0.0, 1.0), math.radians(45.0))
quat_out = quat_a * quat_b

# print the quat, euler degrees for mear mortals and (axis, angle)
print("Final Rotation:")
print(quat_out)
print("%.2f, %.2f, %.2f" % tuple(math.degrees(a) for a in quat_out.to_euler()))
print("(%.2f, %.2f, %.2f), %.2f" % (quat_out.axis[:] +
                                    (math.degrees(quat_out.angle), )))

# multiple rotations can be interpolated using the exponential map
quat_c = mathutils.Quaternion((1.0, 0.0, 0.0), math.radians(15.0))
exp_avg = (quat_a.to_exponential_map() +
           quat_b.to_exponential_map() +
           quat_c.to_exponential_map()) / 3.0
quat_avg = mathutils.Quaternion(exp_avg)
print("Average rotation:")
print(quat_avg)

conjugate()¶: Set the quaternion to its conjugate (negate x, y, z).

conjugated()¶

Return a new conjugated quaternion.

Returns:	a new quaternion.
Return type:	`Quaternion`

copy()¶

Returns a copy of this quaternion.

Returns:	A copy of the quaternion.
Return type:	`Quaternion`

Note

use this to get a copy of a wrapped quaternion with no reference to the original data.

cross(other)¶

Return the cross product of this quaternion and another.

Parameters:	other (`Quaternion`) – The other quaternion to perform the cross product with.
Returns:	The cross product.
Return type:	`Quaternion`

dot(other)¶

Return the dot product of this quaternion and another.

Parameters:	other (`Quaternion`) – The other quaternion to perform the dot product with.
Returns:	The dot product.
Return type:	`Quaternion`

freeze()¶

Make this object immutable.

After this the object can be hashed, used in dictionaries & sets.

Returns:	An instance of this object.

identity()¶

Set the quaternion to an identity quaternion.

Return type:	`Quaternion`

invert()¶: Set the quaternion to its inverse.

inverted()¶

Return a new, inverted quaternion.

Returns:	the inverted value.
Return type:	`Quaternion`

negate()¶

Set the quaternion to its negative.

Return type:	`Quaternion`

normalize()¶: Normalize the quaternion.

normalized()¶

Return a new normalized quaternion.

Returns:	a normalized copy.
Return type:	`Quaternion`

rotate(other)¶

Rotates the quaternion by another mathutils value.

Parameters:	other (`Euler`, `Quaternion` or `Matrix`) – rotation component of mathutils value

rotation_difference(other)¶

Returns a quaternion representing the rotational difference.

Parameters:	other (`Quaternion`) – second quaternion.
Returns:	the rotational difference between the two quat rotations.
Return type:	`Quaternion`

slerp(other, factor)¶

Returns the interpolation of two quaternions.

Parameters:	other (`Quaternion`) – value to interpolate with. factor (float) – The interpolation value in [0.0, 1.0].
Returns:	The interpolated rotation.
Return type:	`Quaternion`

to_axis_angle()¶

Return the axis, angle representation of the quaternion.

Returns:	axis, angle.
Return type:	(`Vector`, float) pair

to_euler(order, euler_compat)¶

Return Euler representation of the quaternion.

Parameters:	order (string) – Optional rotation order argument in [‘XYZ’, ‘XZY’, ‘YXZ’, ‘YZX’, ‘ZXY’, ‘ZYX’]. euler_compat (`Euler`) – Optional euler argument the new euler will be made compatible with (no axis flipping between them). Useful for converting a series of matrices to animation curves.
Returns:	Euler representation of the quaternion.
Return type:	`Euler`

to_exponential_map()¶

Return the exponential map representation of the quaternion.

This representation consist of the rotation axis multiplied by the rotation angle. Such a representation is useful for interpolation between multiple orientations.

Returns:	exponential map.
Return type:	`Vector` of size 3

To convert back to a quaternion, pass it to the Quaternion constructor.

to_matrix()¶

Return a matrix representation of the quaternion.

Returns:	A 3x3 rotation matrix representation of the quaternion.
Return type:	`Matrix`

angle¶

Angle of the quaternion.

Type:	float

axis¶

Quaternion axis as a vector.

Type:	`Vector`

is_frozen¶

True when this object has been frozen (read-only).

Type:	boolean

is_wrapped¶

True when this object wraps external data (read-only).

Type:	boolean

magnitude¶

Size of the quaternion (read-only).

Type:	float

owner¶: The item this is wrapping or None (read-only).

w¶

Quaternion axis value.

Type:	float

x¶

Quaternion axis value.

Type:	float

y¶

Quaternion axis value.

Type:	float

z¶

Quaternion axis value.

Type:	float

class mathutils.Vector(seq)¶

This object gives access to Vectors in Blender.

Parameters:	seq (sequence of numbers) – Components of the vector, must be a sequence of at least two

import mathutils

# zero length vector
vec = mathutils.Vector((0.0, 0.0, 1.0))

# unit length vector
vec_a = vec.normalized()

vec_b = mathutils.Vector((0.0, 1.0, 2.0))

vec2d = mathutils.Vector((1.0, 2.0))
vec3d = mathutils.Vector((1.0, 0.0, 0.0))
vec4d = vec_a.to_4d()

# other mathutuls types
quat = mathutils.Quaternion()
matrix = mathutils.Matrix()

# Comparison operators can be done on Vector classes:

# (In)equality operators == and != test component values, e.g. 1,2,3 != 3,2,1
vec_a == vec_b
vec_a != vec_b

# Ordering operators >, >=, > and <= test vector length.
vec_a > vec_b
vec_a >= vec_b
vec_a < vec_b
vec_a <= vec_b


# Math can be performed on Vector classes
vec_a + vec_b
vec_a - vec_b
vec_a * vec_b
vec_a * 10.0
matrix * vec_a
quat * vec_a
vec_a * vec_b
-vec_a


# You can access a vector object like a sequence
x = vec_a[0]
len(vec)
vec_a[:] = vec_b
vec_a[:] = 1.0, 2.0, 3.0
vec2d[:] = vec3d[:2]


# Vectors support 'swizzle' operations
# See https://en.wikipedia.org/wiki/Swizzling_(computer_graphics)
vec.xyz = vec.zyx
vec.xy = vec4d.zw
vec.xyz = vec4d.wzz
vec4d.wxyz = vec.yxyx

classmethod Fill(size, fill=0.0)¶

Create a vector of length size with all values set to fill.

Parameters:	size (int) – The length of the vector to be created. fill (float) – The value used to fill the vector.

classmethod Linspace(start, stop, size)¶

Create a vector of the specified size which is filled with linearly spaced values between start and stop values.

Parameters:	start (int) – The start of the range used to fill the vector. stop (int) – The end of the range used to fill the vector. size (int) – The size of the vector to be created.

classmethod Range(start=0, stop, step=1)¶

Create a filled with a range of values.

Parameters:	start (int) – The start of the range used to fill the vector. stop (int) – The end of the range used to fill the vector. step (int) – The step between successive values in the vector.

classmethod Repeat(vector, size)¶

Create a vector by repeating the values in vector until the required size is reached.

Parameters:	tuple (`mathutils.Vector`) – The vector to draw values from. size (int) – The size of the vector to be created.

angle(other, fallback=None)¶

Return the angle between two vectors.

Parameters:	other (`Vector`) – another vector to compare the angle with fallback (any) – return this when the angle can’t be calculated (zero length vector), (instead of raising a `ValueError`).
Returns:	angle in radians or fallback when given
Return type:	float

angle_signed(other, fallback)¶

Return the signed angle between two 2D vectors (clockwise is positive).

Parameters:	other (`Vector`) – another vector to compare the angle with fallback (any) – return this when the angle can’t be calculated (zero length vector), (instead of raising a `ValueError`).
Returns:	angle in radians or fallback when given
Return type:	float

copy()¶

Returns a copy of this vector.

Returns:	A copy of the vector.
Return type:	`Vector`

Note

use this to get a copy of a wrapped vector with no reference to the original data.

cross(other)¶

Return the cross product of this vector and another.

Parameters:	other (`Vector`) – The other vector to perform the cross product with.
Returns:	The cross product.
Return type:	`Vector` or float when 2D vectors are used

Note

both vectors must be 2D or 3D

dot(other)¶

Return the dot product of this vector and another.

Parameters:	other (`Vector`) – The other vector to perform the dot product with.
Returns:	The dot product.
Return type:	`Vector`

freeze()¶

Make this object immutable.

After this the object can be hashed, used in dictionaries & sets.

Returns:	An instance of this object.

lerp(other, factor)¶

Returns the interpolation of two vectors.

Parameters:	other (`Vector`) – value to interpolate with. factor (float) – The interpolation value in [0.0, 1.0].
Returns:	The interpolated vector.
Return type:	`Vector`

negate()¶: Set all values to their negative.

normalize()¶: Normalize the vector, making the length of the vector always 1.0.

Warning

Normalizing a vector where all values are zero has no effect.

Note

Normalize works for vectors of all sizes, however 4D Vectors w axis is left untouched.

normalized()¶

Return a new, normalized vector.

Returns:	a normalized copy of the vector
Return type:	`Vector`

orthogonal()¶

Return a perpendicular vector.

Returns:	a new vector 90 degrees from this vector.
Return type:	`Vector`

Note

the axis is undefined, only use when any orthogonal vector is acceptable.

project(other)¶

Return the projection of this vector onto the other.

Parameters:	other (`Vector`) – second vector.
Returns:	the parallel projection vector
Return type:	`Vector`

reflect(mirror)¶

Return the reflection vector from the mirror argument.

Parameters:	mirror (`Vector`) – This vector could be a normal from the reflecting surface.
Returns:	The reflected vector matching the size of this vector.
Return type:	`Vector`

resize(size=3)¶: Resize the vector to have size number of elements.

resize_2d()¶: Resize the vector to 2D (x, y).

resize_3d()¶: Resize the vector to 3D (x, y, z).

resize_4d()¶: Resize the vector to 4D (x, y, z, w).

resized(size=3)¶

Return a resized copy of the vector with size number of elements.

Returns:	a new vector
Return type:	`Vector`

rotate(other)¶

Rotate the vector by a rotation value.

Parameters:	other (`Euler`, `Quaternion` or `Matrix`) – rotation component of mathutils value

rotation_difference(other)¶

Returns a quaternion representing the rotational difference between this vector and another.

Parameters:	other (`Vector`) – second vector.
Returns:	the rotational difference between the two vectors.
Return type:	`Quaternion`

Note

2D vectors raise an AttributeError.

slerp(other, factor, fallback=None)¶

Returns the interpolation of two non-zero vectors (spherical coordinates).

Parameters:	other (`Vector`) – value to interpolate with. factor (float) – The interpolation value typically in [0.0, 1.0]. fallback (any) – return this when the vector can’t be calculated (zero length vector or direct opposites), (instead of raising a `ValueError`).
Returns:	The interpolated vector.
Return type:	`Vector`

to_2d()¶

Return a 2d copy of the vector.

Returns:	a new vector
Return type:	`Vector`

to_3d()¶

Return a 3d copy of the vector.

Returns:	a new vector
Return type:	`Vector`

to_4d()¶

Return a 4d copy of the vector.

Returns:	a new vector
Return type:	`Vector`

to_track_quat(track, up)¶

Return a quaternion rotation from the vector and the track and up axis.

Parameters:	track (string) – Track axis in [‘X’, ‘Y’, ‘Z’, ‘-X’, ‘-Y’, ‘-Z’]. up (string) – Up axis in [‘X’, ‘Y’, ‘Z’].
Returns:	rotation from the vector and the track and up axis.
Return type:	`Quaternion`

to_tuple(precision=-1)¶

Return this vector as a tuple with.

Parameters:	precision (int) – The number to round the value to in [-1, 21].
Returns:	the values of the vector rounded by precision
Return type:	tuple

zero()¶: Set all values to zero.

is_frozen¶

True when this object has been frozen (read-only).

Type:	boolean

is_wrapped¶

True when this object wraps external data (read-only).

Type:	boolean

length¶

Vector Length.

Type:	float

length_squared¶

Vector length squared (v.dot(v)).

Type:	float

magnitude¶

Vector Length.

Type:	float

owner¶: The item this is wrapping or None (read-only).

w¶

Vector W axis (4D Vectors only).

Type:	float

ww¶: Undocumented

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x¶

Vector X axis.

Type:	float

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y¶

Vector Y axis.

Type:	float

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yxy¶: Undocumented

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z¶

Vector Z axis (3D Vectors only).

Type:	float

zw¶: Undocumented

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zzzz¶: Undocumented

Math Types & Utilities (mathutils)¶

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Returns:	A 3x3 roation matrix representation of the euler.
Return type:	`Matrix`

Returns:	Quaternion representation of the euler.
Return type:	`Quaternion`

Parameters:	size (int) – The size of the identity matrix to construct [2, 4].
Returns:	A new identity matrix.
Return type:	`Matrix`

Parameters:	axis (string or `Vector`) – Can be any of the following: [‘X’, ‘Y’, ‘XY’, ‘XZ’, ‘YZ’], where a single axis is for a 2D matrix. Or a vector for an arbitrary axis size (int) – The size of the projection matrix to construct [2, 4].
Returns:	A new projection matrix.
Return type:	`Matrix`

Parameters:	angle (float) – The angle of rotation desired, in radians. size (int) – The size of the rotation matrix to construct [2, 4]. axis (string or `Vector`) – a string in [‘X’, ‘Y’, ‘Z’] or a 3D Vector Object (optional when size is 2).
Returns:	A new rotation matrix.
Return type:	`Matrix`

Parameters:	factor (float) – The factor of scaling to apply. size (int) – The size of the scale matrix to construct [2, 4]. axis (`Vector`) – Direction to influence scale. (optional).
Returns:	A new scale matrix.
Return type:	`Matrix`

Parameters:	plane (string) – Can be any of the following: [‘X’, ‘Y’, ‘XY’, ‘XZ’, ‘YZ’], where a single axis is for a 2D matrix only. size (int) – The size of the shear matrix to construct [2, 4]. factor (float or float pair) – The factor of shear to apply. For a 3 or 4 size matrix pass a pair of floats corresponding with the plane axis.
Returns:	A new shear matrix.
Return type:	`Matrix`

Parameters:	vector (`Vector`) – The translation vector.
Returns:	An identity matrix with a translation.
Return type:	`Matrix`

Returns:	tuple of translation, rotation, and scale
Return type:	(`Vector`, `Quaternion`, `Vector`)

Returns:	Return the determinant of a matrix.
Return type:	float