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Basis¶
A 3×3 matrix for representing 3D rotation and scale.
Description¶
The Basis built-in Variant type is a 3×3 matrix used to represent 3D rotation, scale, and shear. It is frequently used within a Transform3D.
A Basis is composed by 3 axis vectors, each representing a column of the matrix: x, y, and z. The length of each axis (Vector3.length) influences the basis's scale, while the direction of all axes influence the rotation. Usually, these axes are perpendicular to one another. However, when you rotate any axis individually, the basis becomes sheared. Applying a sheared basis to a 3D model will make the model appear distorted.
A Basis is orthogonal if its axes are perpendicular to each other. A basis is normalized if the length of every axis is 1. A basis is uniform if all axes share the same length (see get_scale). A basis is orthonormal if it is both orthogonal and normalized, which allows it to only represent rotations. A basis is conformal if it is both orthogonal and uniform, which ensures it is not distorted.
For a general introduction, see the Matrices and transforms tutorial.
Note: Redot uses a right-handed coordinate system, which is a common standard. For directions, the convention for built-in types like Camera3D is for -Z to point forward (+X is right, +Y is up, and +Z is back). Other objects may use different direction conventions. For more information, see the 3D asset direction conventions tutorial.
Note: The basis matrices are exposed as column-major order, which is the same as OpenGL. However, they are stored internally in row-major order, which is the same as DirectX.
Note
There are notable differences when using this API with C#. See C# API differences to GDScript for more information.
Tutorials¶
Properties¶
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Constructors¶
Basis() |
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Basis(from: Quaternion) |
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Methods¶
determinant() const |
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from_euler(euler: Vector3, order: int = 2) static |
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from_scale(scale: Vector3) static |
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get_rotation_quaternion() const |
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get_scale() const |
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inverse() const |
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is_conformal() const |
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is_equal_approx(b: Basis) const |
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is_finite() const |
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looking_at(target: Vector3, up: Vector3 = Vector3(0, 1, 0), use_model_front: bool = false) static |
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orthonormalized() const |
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transposed() const |
Operators¶
operator !=(right: Basis) |
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operator *(right: Basis) |
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operator *(right: Vector3) |
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operator *(right: float) |
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operator *(right: int) |
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operator /(right: float) |
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operator /(right: int) |
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operator ==(right: Basis) |
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operator [](index: int) |
Constants¶
IDENTITY = Basis(1, 0, 0, 0, 1, 0, 0, 0, 1) 🔗
The identity basis. This is a basis with no rotation, no shear, and its scale being 1. This means that:
The x points right (Vector3.RIGHT);
The y points up (Vector3.UP);
The z points back (Vector3.BACK).
var basis := Basis.IDENTITY
print("| X | Y | Z")
print("| %s | %s | %s" % [basis.x.x, basis.y.x, basis.z.x])
print("| %s | %s | %s" % [basis.x.y, basis.y.y, basis.z.y])
print("| %s | %s | %s" % [basis.x.z, basis.y.z, basis.z.z])
# Prints:
# | X | Y | Z
# | 1 | 0 | 0
# | 0 | 1 | 0
# | 0 | 0 | 1
This is identical to creating 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) 🔗
When any basis is multiplied by FLIP_X, it negates all components of the x axis (the X column).
When FLIP_X is multiplied by any basis, it negates the Vector3.x component of all axes (the X row).
FLIP_Y = Basis(1, 0, 0, 0, -1, 0, 0, 0, 1) 🔗
When any basis is multiplied by FLIP_Y, it negates all components of the y axis (the Y column).
When FLIP_Y is multiplied by any basis, it negates the Vector3.y component of all axes (the Y row).
FLIP_Z = Basis(1, 0, 0, 0, 1, 0, 0, 0, -1) 🔗
When any basis is multiplied by FLIP_Z, it negates all components of the z axis (the Z column).
When FLIP_Z is multiplied by any basis, it negates the Vector3.z component of all axes (the Z row).
Property Descriptions¶
Vector3 x = Vector3(1, 0, 0) 🔗
The basis's X axis, and the column 0 of the matrix.
On the identity basis, this vector points right (Vector3.RIGHT).
Vector3 y = Vector3(0, 1, 0) 🔗
The basis's Y axis, and the column 1 of the matrix.
On the identity basis, this vector points up (Vector3.UP).
Vector3 z = Vector3(0, 0, 1) 🔗
The basis's Z axis, and the column 2 of the matrix.
On the identity basis, this vector points back (Vector3.BACK).
Constructor Descriptions¶
Constructs a Basis identical to the IDENTITY.
Constructs a Basis as a copy of the given Basis.
Basis Basis(axis: Vector3, angle: float)
Constructs a Basis that only represents rotation, rotated around the axis by the given angle, in radians. The axis must be a normalized vector.
Note: This is the same as using rotated on the IDENTITY basis. With more than one angle consider using from_euler, instead.
Basis Basis(from: Quaternion)
Constructs a Basis that only represents rotation from the given Quaternion.
Note: Quaternions only store rotation, not scale. Because of this, conversions from Basis to Quaternion cannot always be reversed.
Basis Basis(x_axis: Vector3, y_axis: Vector3, z_axis: Vector3)
Constructs a Basis from 3 axis vectors. These are the columns of the basis matrix.
Method Descriptions¶
Returns the determinant of this basis's matrix. For advanced math, this number can be used to determine a few attributes:
If the determinant is exactly
0, the basis is not invertible (see inverse).If the determinant is a negative number, the basis represents a negative scale.
Note: If the basis's scale is the same for every axis, its determinant is always that scale by the power of 2.
Basis from_euler(euler: Vector3, order: int = 2) static 🔗
Constructs a new Basis that only represents rotation from the given Vector3 of Euler angles, in radians.
The Vector3.x should contain the angle around the x axis (pitch).
The Vector3.y should contain the angle around the y axis (yaw).
The Vector3.z should contain the angle around the z axis (roll).
# Creates a Basis whose z axis points down.
var my_basis = Basis.from_euler(Vector3(TAU / 4, 0, 0))
print(my_basis.z) # Prints (0, -1, 0).
// Creates a Basis whose z axis points down.
var myBasis = Basis.FromEuler(new Vector3(Mathf.Tau / 4.0f, 0.0f, 0.0f));
GD.Print(myBasis.Z); // Prints (0, -1, 0).
The order of each consecutive rotation can be changed with order (see EulerOrder constants). By default, the YXZ convention is used (@GlobalScope.EULER_ORDER_YXZ): the basis rotates first around the Y axis (yaw), then X (pitch), and lastly Z (roll). When using the opposite method get_euler, this order is reversed.
Basis from_scale(scale: Vector3) static 🔗
Constructs a new Basis that only represents scale, with no rotation or shear, from the given scale vector.
var my_basis = Basis.from_scale(Vector3(2, 4, 8))
print(my_basis.x) # Prints (2, 0, 0).
print(my_basis.y) # Prints (0, 4, 0).
print(my_basis.z) # Prints (0, 0, 8).
var myBasis = Basis.FromScale(new Vector3(2.0f, 4.0f, 8.0f));
GD.Print(myBasis.X); // Prints (2, 0, 0).
GD.Print(myBasis.Y); // Prints (0, 4, 0).
GD.Print(myBasis.Z); // Prints (0, 0, 8).
Note: In linear algebra, the matrix of this basis is also known as a diagonal matrix.
Vector3 get_euler(order: int = 2) const 🔗
Returns this basis's rotation as a Vector3 of Euler angles, in radians.
The order of each consecutive rotation can be changed with order (see EulerOrder constants). By default, the YXZ convention is used (@GlobalScope.EULER_ORDER_YXZ): Z (roll) is calculated first, then X (pitch), and lastly Y (yaw). When using the opposite method from_euler, this order is reversed.
Note: Euler angles are much more intuitive but are not suitable for 3D math. Because of this, consider using the get_rotation_quaternion method instead, which returns a Quaternion.
Note: In the Inspector dock, a basis's rotation is often displayed in Euler angles (in degrees), as is the case with the Node3D.rotation property.
Quaternion get_rotation_quaternion() const 🔗
Returns this basis's rotation as a Quaternion.
Note: Quatenions are much more suitable for 3D math but are less intuitive. For user interfaces, consider using the get_euler method, which returns Euler angles.
Returns the length of each axis of this basis, as a Vector3. If the basis is not sheared, this is the scaling factor. It is not affected by rotation.
var my_basis = Basis(
Vector3(2, 0, 0),
Vector3(0, 4, 0),
Vector3(0, 0, 8)
)
# Rotating the Basis in any way preserves its scale.
my_basis = my_basis.rotated(Vector3.UP, TAU / 2)
my_basis = my_basis.rotated(Vector3.RIGHT, TAU / 4)
print(my_basis.get_scale()) # Prints (2, 4, 8).
var myBasis = new Basis(
Vector3(2.0f, 0.0f, 0.0f),
Vector3(0.0f, 4.0f, 0.0f),
Vector3(0.0f, 0.0f, 8.0f)
);
// Rotating the Basis in any way preserves its scale.
myBasis = myBasis.Rotated(Vector3.Up, Mathf.Tau / 2.0f);
myBasis = myBasis.Rotated(Vector3.Right, Mathf.Tau / 4.0f);
GD.Print(myBasis.Scale); // Prints (2, 4, 8).
Note: If the value returned by determinant is negative, the scale is also negative.
Returns the inverse of this basis's matrix.
Returns true if this basis is conformal. A conformal basis is both orthogonal (the axes are perpendicular to each other) and uniform (the axes share the same length). This method can be especially useful during physics calculations.
bool is_equal_approx(b: Basis) const 🔗
Returns true if this basis and b are approximately equal, by calling @GlobalScope.is_equal_approx on all vector components.
Returns true if this basis is finite, by calling @GlobalScope.is_finite on all vector components.
Basis looking_at(target: Vector3, up: Vector3 = Vector3(0, 1, 0), use_model_front: bool = false) static 🔗
Creates a new Basis with a rotation such that the forward axis (-Z) points towards the target position.
By default, the -Z axis (camera forward) is treated as forward (implies +X is right). 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.
The up axis (+Y) points as close to the up vector as possible while staying perpendicular to the forward axis. The returned basis is orthonormalized (see orthonormalized). The target and up vectors cannot be Vector3.ZERO, and cannot be parallel to each other.
Basis orthonormalized() const 🔗
Returns the orthonormalized version of this basis. An orthonormal basis is both orthogonal (the axes are perpendicular to each other) and normalized (the axes have a length of 1), which also means it can only represent rotation.
It is often useful to call this method to avoid rounding errors on a rotating basis:
# Rotate this Node3D every frame.
func _process(delta):
basis = basis.rotated(Vector3.UP, TAU * delta)
basis = basis.rotated(Vector3.RIGHT, TAU * delta)
basis = basis.orthonormalized()
// Rotate this Node3D every frame.
public override void _Process(double delta)
{
Basis = Basis.Rotated(Vector3.Up, Mathf.Tau * (float)delta)
.Rotated(Vector3.Right, Mathf.Tau * (float)delta)
.Orthonormalized();
}
Basis rotated(axis: Vector3, angle: float) const 🔗
Returns this basis rotated around the given axis by angle (in radians). The axis must be a normalized vector (see Vector3.normalized).
Positive values rotate this basis clockwise around the axis, while negative values rotate it counterclockwise.
var my_basis = Basis.IDENTITY
var angle = TAU / 2
my_basis = my_basis.rotated(Vector3.UP, angle) # Rotate around the up axis (yaw).
my_basis = my_basis.rotated(Vector3.RIGHT, angle) # Rotate around the right axis (pitch).
my_basis = my_basis.rotated(Vector3.BACK, angle) # Rotate around the back axis (roll).
var myBasis = Basis.Identity;
var angle = Mathf.Tau / 2.0f;
myBasis = myBasis.Rotated(Vector3.Up, angle); // Rotate around the up axis (yaw).
myBasis = myBasis.Rotated(Vector3.Right, angle); // Rotate around the right axis (pitch).
myBasis = myBasis.Rotated(Vector3.Back, angle); // Rotate around the back axis (roll).
Basis scaled(scale: Vector3) const 🔗
Returns this basis with each axis's components scaled by the given scale's components.
The basis matrix's rows are multiplied by scale's components. This operation is a global scale (relative to the parent).
var my_basis = Basis(
Vector3(1, 1, 1),
Vector3(2, 2, 2),
Vector3(3, 3, 3)
)
my_basis = my_basis.scaled(Vector3(0, 2, -2))
print(my_basis.x) # Prints (0, 2, -2).
print(my_basis.y) # Prints (0, 4, -4).
print(my_basis.z) # Prints (0, 6, -6).
var myBasis = new Basis(
new Vector3(1.0f, 1.0f, 1.0f),
new Vector3(2.0f, 2.0f, 2.0f),
new Vector3(3.0f, 3.0f, 3.0f)
);
myBasis = myBasis.Scaled(new Vector3(0.0f, 2.0f, -2.0f));
GD.Print(myBasis.X); // Prints (0, 2, -2).
GD.Print(myBasis.Y); // Prints (0, 4, -4).
GD.Print(myBasis.Z); // Prints (0, 6, -6).
Basis slerp(to: Basis, weight: float) const 🔗
Performs a spherical-linear interpolation with the to basis, given a weight. Both this basis and to should represent a rotation.
Example: Smoothly rotate a Node3D to the target basis over time, with a Tween.
var start_basis = Basis.IDENTITY
var target_basis = Basis.IDENTITY.rotated(Vector3.UP, TAU / 2)
func _ready():
create_tween().tween_method(interpolate, 0.0, 1.0, 5.0).set_trans(Tween.TRANS_EXPO)
func interpolate(weight):
basis = start_basis.slerp(target_basis, weight)
float tdotx(with: Vector3) const 🔗
Returns the transposed dot product between with and the x axis (see transposed).
This is equivalent to basis.x.dot(vector).
float tdoty(with: Vector3) const 🔗
Returns the transposed dot product between with and the y axis (see transposed).
This is equivalent to basis.y.dot(vector).
float tdotz(with: Vector3) const 🔗
Returns the transposed dot product between with and the z axis (see transposed).
This is equivalent to basis.z.dot(vector).
Returns the transposed version of this basis. This turns the basis matrix's columns into rows, and its rows into columns.
var my_basis = Basis(
Vector3(1, 2, 3),
Vector3(4, 5, 6),
Vector3(7, 8, 9)
)
my_basis = my_basis.transposed()
print(my_basis.x) # Prints (1, 4, 7).
print(my_basis.y) # Prints (2, 5, 8).
print(my_basis.z) # Prints (3, 6, 9).
var myBasis = new Basis(
new Vector3(1.0f, 2.0f, 3.0f),
new Vector3(4.0f, 5.0f, 6.0f),
new Vector3(7.0f, 8.0f, 9.0f)
);
myBasis = myBasis.Transposed();
GD.Print(myBasis.X); // Prints (1, 4, 7).
GD.Print(myBasis.Y); // Prints (2, 5, 8).
GD.Print(myBasis.Z); // Prints (3, 6, 9).
Operator Descriptions¶
bool operator !=(right: Basis) 🔗
Returns true if the components of both Basis matrices are not equal.
Note: Due to floating-point precision errors, consider using is_equal_approx instead, which is more reliable.
Basis operator *(right: Basis) 🔗
Transforms (multiplies) the right basis by this basis.
This is the operation performed between parent and child Node3Ds.
Vector3 operator *(right: Vector3) 🔗
Transforms (multiplies) the right vector by this basis, returning a Vector3.
# Basis that swaps the X/Z axes and doubles the scale.
var my_basis = Basis(Vector3(0, 2, 0), Vector3(2, 0, 0), Vector3(0, 0, 2))
print(my_basis * Vector3(1, 2, 3)) # Prints (4, 2, 6)
// Basis that swaps the X/Z axes and doubles the scale.
var myBasis = new Basis(new Vector3(0, 2, 0), new Vector3(2, 0, 0), new Vector3(0, 0, 2));
GD.Print(myBasis * new Vector3(1, 2, 3)); // Prints (4, 2, 6)
Basis operator *(right: float) 🔗
Multiplies all components of the Basis by the given float. This affects the basis's scale uniformly, resizing all 3 axes by the right value.
Basis operator *(right: int) 🔗
Multiplies all components of the Basis by the given int. This affects the basis's scale uniformly, resizing all 3 axes by the right value.
Basis operator /(right: float) 🔗
Divides all components of the Basis by the given float. This affects the basis's scale uniformly, resizing all 3 axes by the right value.
Basis operator /(right: int) 🔗
Divides all components of the Basis by the given int. This affects the basis's scale uniformly, resizing all 3 axes by the right value.
bool operator ==(right: Basis) 🔗
Returns true if the components of both Basis matrices are exactly equal.
Note: Due to floating-point precision errors, consider using is_equal_approx instead, which is more reliable.
Vector3 operator [](index: int) 🔗
Accesses each axis (column) of this basis by their index. Index 0 is the same as x, index 1 is the same as y, and index 2 is the same as z.
Note: In C++, this operator accesses the rows of the basis matrix, not the columns. For the same behavior as scripting languages, use the set_column and get_column methods.