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# Basis¶

3×3 matrix datatype.

## Description¶

3×3 matrix used for 3D rotation and scale. Almost always used as an orthogonal basis for a Transform3D.

Contains 3 vector fields X, Y and Z as its columns, which are typically interpreted as the local basis vectors of a transformation. For such use, it is composed of a scaling and a rotation matrix, in that order (M = R.S).

Can also be accessed as array of 3D vectors. These vectors are normally orthogonal to each other, but are not necessarily normalized (due to scaling).

## Properties¶

 Vector3 x `Vector3(1, 0, 0)` Vector3 y `Vector3(0, 1, 0)` Vector3 z `Vector3(0, 0, 1)`

## Constructors¶

 Basis Basis ( ) Basis Basis ( Basis from ) Basis Basis ( Vector3 axis, float angle ) Basis Basis ( Quaternion from ) Basis Basis ( Vector3 x_axis, Vector3 y_axis, Vector3 z_axis )

## Methods¶

 float determinant ( ) const Basis from_euler ( Vector3 euler, int order=2 ) static Basis from_scale ( Vector3 scale ) static Vector3 get_euler ( int order=2 ) const Quaternion get_rotation_quaternion ( ) const Vector3 get_scale ( ) const Basis inverse ( ) const bool is_equal_approx ( Basis b ) const bool is_finite ( ) const Basis looking_at ( Vector3 target, Vector3 up=Vector3(0, 1, 0), bool use_model_front=false ) static Basis orthonormalized ( ) const Basis rotated ( Vector3 axis, float angle ) const Basis scaled ( Vector3 scale ) const Basis slerp ( Basis to, float weight ) const float tdotx ( Vector3 with ) const float tdoty ( Vector3 with ) const float tdotz ( Vector3 with ) const Basis transposed ( ) const

## Operators¶

 bool operator != ( Basis right ) Basis operator * ( Basis right ) Vector3 operator * ( Vector3 right ) Basis operator * ( float right ) Basis operator * ( int right ) bool operator == ( Basis right ) Vector3 operator [] ( int index )

## Constants¶

IDENTITY = `Basis(1, 0, 0, 0, 1, 0, 0, 0, 1)`

The identity basis, with no rotation or scaling applied.

This is identical to calling `Basis()` without any parameters. This constant can be used to make your code clearer, and for consistency with C#.

FLIP_X = `Basis(-1, 0, 0, 0, 1, 0, 0, 0, 1)`

The basis that will flip something along the X axis when used in a transformation.

FLIP_Y = `Basis(1, 0, 0, 0, -1, 0, 0, 0, 1)`

The basis that will flip something along the Y axis when used in a transformation.

FLIP_Z = `Basis(1, 0, 0, 0, 1, 0, 0, 0, -1)`

The basis that will flip something along the Z axis when used in a transformation.

## Property Descriptions¶

Vector3 x = `Vector3(1, 0, 0)`

The basis matrix's X vector (column 0). Equivalent to array index `0`.

Vector3 y = `Vector3(0, 1, 0)`

The basis matrix's Y vector (column 1). Equivalent to array index `1`.

Vector3 z = `Vector3(0, 0, 1)`

The basis matrix's Z vector (column 2). Equivalent to array index `2`.

## Constructor Descriptions¶

Basis Basis ( )

Constructs a default-initialized Basis set to IDENTITY.

Basis Basis ( Basis from )

Constructs a Basis as a copy of the given Basis.

Basis Basis ( Vector3 axis, float angle )

Constructs a pure rotation basis matrix, rotated around the given `axis` by `angle` (in radians). The axis must be a normalized vector.

Basis Basis ( Quaternion from )

Constructs a pure rotation basis matrix from the given quaternion.

Basis Basis ( Vector3 x_axis, Vector3 y_axis, Vector3 z_axis )

Constructs a basis matrix from 3 axis vectors (matrix columns).

## Method Descriptions¶

float determinant ( ) const

Returns the determinant of the basis matrix. If the basis is uniformly scaled, its determinant is the square of the scale.

A negative determinant means the basis has a negative scale. A zero determinant means the basis isn't invertible, and is usually considered invalid.

Basis from_euler ( Vector3 euler, int order=2 ) static

Constructs a pure rotation Basis matrix from Euler angles in the specified Euler rotation order. By default, use YXZ order (most common). See the EulerOrder enum for possible values.

Basis from_scale ( Vector3 scale ) static

Constructs a pure scale basis matrix with no rotation or shearing. The scale values are set as the diagonal of the matrix, and the other parts of the matrix are zero.

Vector3 get_euler ( int order=2 ) const

Returns the basis's rotation in the form of Euler angles. The Euler order depends on the `order` parameter, by default it uses the YXZ convention: when decomposing, first Z, then X, and Y last. The returned vector contains the rotation angles in the format (X angle, Y angle, Z angle).

Consider using the get_rotation_quaternion method instead, which returns a Quaternion quaternion instead of Euler angles.

Quaternion get_rotation_quaternion ( ) const

Returns the basis's rotation in the form of a quaternion. See get_euler if you need Euler angles, but keep in mind quaternions should generally be preferred to Euler angles.

Vector3 get_scale ( ) const

Assuming that the matrix is the combination of a rotation and scaling, return the absolute value of scaling factors along each axis.

Basis inverse ( ) const

Returns the inverse of the matrix.

bool is_equal_approx ( Basis b ) const

Returns `true` if this basis and `b` are approximately equal, by calling `is_equal_approx` on each component.

bool is_finite ( ) const

Returns `true` if this basis is finite, by calling @GlobalScope.is_finite on each component.

Basis looking_at ( Vector3 target, Vector3 up=Vector3(0, 1, 0), bool use_model_front=false ) static

Creates a Basis with a rotation such that the forward axis (-Z) points towards the `target` position.

The up axis (+Y) points as close to the `up` vector as possible while staying perpendicular to the forward axis. The resulting Basis is orthonormalized. The `target` and `up` vectors cannot be zero, and cannot be parallel to each other.

If `use_model_front` is `true`, the +Z axis (asset front) is treated as forward (implies +X is left) and points toward the `target` position. By default, the -Z axis (camera forward) is treated as forward (implies +X is right).

Basis orthonormalized ( ) const

Returns the orthonormalized version of the matrix (useful to call from time to time to avoid rounding error for orthogonal matrices). This performs a Gram-Schmidt orthonormalization on the basis of the matrix.

Basis rotated ( Vector3 axis, float angle ) const

Introduce an additional rotation around the given axis by `angle` (in radians). The axis must be a normalized vector.

Basis scaled ( Vector3 scale ) const

Introduce an additional scaling specified by the given 3D scaling factor.

Basis slerp ( Basis to, float weight ) const

Assuming that the matrix is a proper rotation matrix, slerp performs a spherical-linear interpolation with another rotation matrix.

float tdotx ( Vector3 with ) const

Transposed dot product with the X axis of the matrix.

float tdoty ( Vector3 with ) const

Transposed dot product with the Y axis of the matrix.

float tdotz ( Vector3 with ) const

Transposed dot product with the Z axis of the matrix.

Basis transposed ( ) const

Returns the transposed version of the matrix.

## Operator Descriptions¶

bool operator != ( Basis right )

Returns `true` if the Basis matrices are not equal.

Note: Due to floating-point precision errors, consider using is_equal_approx instead, which is more reliable.

Basis operator * ( Basis right )

Composes these two basis matrices by multiplying them together. This has the effect of transforming the second basis (the child) by the first basis (the parent).

Vector3 operator * ( Vector3 right )

Transforms (multiplies) the Vector3 by the given Basis matrix.

Basis operator * ( float right )

This operator multiplies all components of the Basis, which scales it uniformly.

Basis operator * ( int right )

This operator multiplies all components of the Basis, which scales it uniformly.

bool operator == ( Basis right )

Returns `true` if the Basis matrices are exactly equal.

Note: Due to floating-point precision errors, consider using is_equal_approx instead, which is more reliable.

Vector3 operator [] ( int index )

Access basis components using their index. `b[0]` is equivalent to `b.x`, `b[1]` is equivalent to `b.y`, and `b[2]` is equivalent to `b.z`.