godot-docs/tutorials/math/matrices_and_transforms.rst

.. _doc_matrices_and_transforms:

Matrices and transforms
=======================

Introduction
------------

Before reading this tutorial, it is advised to read the previous one
about :ref:`doc_vector_math` as this one is a direct continuation.

This tutorial will be about *transformations* and will cover a little
about matrices (but not in-depth).

Transformations are most of the time applied as translation, rotation
and scale so they will be considered as priority here.

Oriented coordinate system (OCS)
--------------------------------

Imagine we have a spaceship somewhere in space. In Godot this is easy,
just move the ship somewhere and rotate it:

.. image:: img/tutomat1.png

Ok, so in 2D this looks simple, a position and an angle for a rotation.
But remember, we are grown ups here and don't use angles (plus, angles
are not really even that useful when working in 3D).

We should realize that at some point, someone *designed* this
spaceship. Be it for 2D in a drawing such as Paint.net, Gimp,
Photoshop, etc. or in 3D through a 3D DCC tool such as Blender, Max,
Maya, etc.

When it was designed, it was not rotated. It was designed in its own
*coordinate system*.

.. image:: img/tutomat2.png

This means that the tip of the ship has a coordinate, the fin has
another, etc. Be it in pixels (2D) or vertices (3D).

So, let's recall again that the ship was somewhere in space:

.. image:: img/tutomat3.png

How did it get there? What moved it and rotated it from the place it was
designed to its current position? The answer is... a **transform**, the
ship was *transformed* from their original position to the new one. This
allows the ship to be displayed where it is.

But transform is too generic of a term to describe this process. To solve this
puzzle, we will superimpose the ship's original design position at their
current position:

.. image:: img/tutomat4.png

So, we can see that the "design space" has been transformed too. How can
we best represent this transformation? Let's use 3 vectors for this (in
2D), a unit vector pointing towards X positive, a unit vector pointing
towards Y positive and a translation.

.. image:: img/tutomat5.png

Let's call the 3 vectors "X", "Y" and "Origin", and let's also
superimpose them over the ship so it makes more sense:

.. image:: img/tutomat6.png

Ok, this is nicer, but it still does not make sense. What do X,Y and
Origin have to do with how the ship got there?

Well, let's take the point from top tip of the ship as reference:

.. image:: img/tutomat7.png

And let's apply the following operation to it (and to all the points in
the ship too, but we'll track the top tip as our reference point):

::

    var new_pos = pos - origin

Doing this to the selected point will move it back to the center:

.. image:: img/tutomat8.png

This was expected, but then let's do something more interesting. Use the
dot product of X and the point, and add it to the dot product of Y and
the point:

::

    var final_pos = Vector2(x.dot(new_pos), y.dot(new_pos))

Then what we have is.. wait a minute, it's the ship in its design
position!

.. image:: img/tutomat9.png

How did this black magic happen? The ship was lost in space, and now
it's back home!

It might seem strange, but it does have plenty of logic. Remember, as
we have seen in the :ref:`doc_vector_math`, what
happened is that the distance to X axis, and the distance to Y axis
were computed. Calculating distance in a direction or plane was one of
the uses for the dot product. This was enough to obtain back the
design coordinates for every point in the ship.

So, what we have been working with so far (with X, Y and Origin) is an
*Oriented Coordinate System*. X an Y are the **Basis**, and *Origin*
is the offset.

Basis
-----

We know what the Origin is. It's where the 0,0 (origin) of the design
coordinate system ended up after being transformed to a new position.
This is why it's called *Origin*, But in practice, it's just an offset
to the new position.

The Basis is more interesting. The basis is the direction of X and Y in the OCS
from the new, transformed location. It tells what has changed, in either 2D or
3D. The Origin (offset) and Basis (direction) communicate "Hey, the original X
and Y axes of your design are *right here*, pointing towards *these
directions*."

So, let's change the representation of the basis. Instead of 2 vectors,
let's use a *matrix*.

.. image:: img/tutomat10.png

The vectors are up there in the matrix, horizontally. The next problem
now is that.. what is this matrix thing? Well, we'll assume you've never
heard of a matrix.

Transforms in Godot
-------------------

This tutorial will not explain matrix math (and their operations) in
depth, only its practical use. There is plenty of material for that,
which should be a lot simpler to understand after completing this
tutorial. We'll just explain how to use transforms.

Transform2D
-----------

:ref:`class_Transform2D` is a 3x2 matrix. It has 3 Vector2 elements and
it's used for 2D. The "X" axis is the element 0, "Y" axis is the element 1 and
"Origin" is element 2. It's not divided in basis/origin for convenience, due to
its simplicity.

::

    var m = Transform2D()
    var x = m[0] # 'X'
    var y = m[1] # 'Y'
    var o = m[2] # 'Origin'
    

Most operations will be explained with this datatype (Transform2D), but the
same logic applies to 3D.

Identity
--------

By default, Transform2D is created as an "identity" matrix. This means:

-  'X' Points right: Vector2(1,0)
-  'Y' Points up (or down in pixels): Vector2(0,1)
-  'Origin' is the origin Vector2(0,0)

.. image:: img/tutomat11.png

It's easy to guess that an *identity* matrix is just a matrix that
aligns the transform to its parent coordinate system. It's an *OCS*
that hasn't been translated, rotated or scaled. All transform types in
Godot are created with *identity*.

Operations
----------

Rotation
--------

Rotating Transform2D is done by using the "rotated" function:

::

    var m = Transform2D()
    m = m.rotated(PI/2) # rotate 90°

.. image:: img/tutomat12.png

Translation
-----------

There are two ways to translate a Transform2D, the first one is just moving
the origin:

::

    # Move 2 units to the right
    var m = Transform2D()
    m = m.rotated(PI/2) # rotate 90°
    m[2]+=Vector2(2,0)

.. image:: img/tutomat13.png

This will always work in global coordinates.

If instead, translation is desired in *local* coordinates of the
matrix (towards where the *basis* is oriented), there is the
:ref:`Transform2D.translated() <class_Transform2D_translated>`
method:

::

    # Move 2 units towards where the basis is oriented
    var m = Transform2D()
    m = m.rotated(PI/2) # rotate 90°
    m = m.translated( Vector2(2,0) )

.. image:: img/tutomat14.png

You could also transform the global coordinates to local coordinates manually:

::

    var local_pos = this_transform.xform_inv(point)

But even better, there are helper functions for this as you can read in the next sections.

Local to global coordinates and vice versa
------------------------------------------

There are helper methods for converting between local and global coordinates.

There are :ref:`Node2D.to_local() <class_Node2D_to_local>` and :ref:`Node2D.to_global() <class_Node2D_to_global>` for 2D
as well as :ref:`Spatial.to_local() <class_Spatial_to_local>` and :ref:`Spatial.to_global() <class_Spatial_to_global>` for 3D.

Scale
-----

A matrix can be scaled too. Scaling will multiply the basis vectors by a
vector (X vector by x component of the scale, Y vector by y component of
the scale). It will leave the origin alone:

::

    # Make the basis twice its size.
    var m = Transform2D()
    m = m.scaled( Vector2(2,2) )

.. image:: img/tutomat15.png

These kind of operations in matrices are accumulative. It means every
one starts relative to the previous one. For those who have been living
on this planet long enough, a good reference of how transform works is
this:

.. image:: img/tutomat16.png

A matrix is used similarly to a turtle. The turtle most likely had a
matrix inside (and you are likely learning this many years *after*
discovering Santa is not real).

Transform
---------

Transform is the act of switching between coordinate systems. To convert
a position (either 2D or 3D) from "designer" coordinate system to the
OCS, the "xform" method is used.

::

    var new_pos = m.xform(pos)

And only for basis (no translation):

::

    var new_pos = m.basis_xform(pos)

Post - multiplying is also valid:

::

    var new_pos = m * pos

Inverse transform
-----------------

To do the opposite operation (what we did up there with the rocket), the
"xform_inv" method is used:

::

    var new_pos = m.xform_inv(pos)

Only for Basis:

::

    var new_pos = m.basis_xform_inv(pos)

Or pre-multiplication:

::

    var new_pos = pos * m

Orthonormal matrices
--------------------

However, if the matrix has been scaled (vectors are not unit length),
or the basis vectors are not orthogonal (90°), the inverse transform
will not work.

In other words, inverse transform is only valid in *orthonormal*
matrices. For this, these cases an affine inverse must be computed.

The transform, or inverse transform of an identity matrix will return
the position unchanged:

::

    # Does nothing, pos is unchanged
    pos = Transform2D().xform(pos)

Affine inverse
--------------

The affine inverse is a matrix that does the inverse operation of
another matrix, no matter if the matrix has scale or the axis vectors
are not orthogonal. The affine inverse is calculated with the
affine_inverse() method:

::

    var mi = m.affine_inverse()
    var pos = m.xform(pos)
    pos = mi.xform(pos)
    # pos is unchanged

If the matrix is orthonormal, then:

::

    # if m is orthonormal, then
    pos = mi.xform(pos)
    # is the same is
    pos = m.xform_inv(pos)

Matrix multiplication
---------------------

Matrices can be multiplied. Multiplication of two matrices "chains"
(concatenates) their transforms.

However, as per convention, multiplication takes place in reverse
order.

Example:

::

    var m = more_transforms * some_transforms

To make it a little clearer, this:

::

    pos = transform1.xform(pos)
    pos = transform2.xform(pos)

Is the same as:

::

    # note the inverse order
    pos = (transform2 * transform1).xform(pos)

However, this is not the same:

::

    # yields a different results
    pos = (transform1 * transform2).xform(pos)

Because in matrix math, A * B is not the same as B * A.

Multiplication by inverse
-------------------------

Multiplying a matrix by its inverse, results in identity:

::

    # No matter what A is, B will be identity
    B = A.affine_inverse() * A

Multiplication by identity
--------------------------

Multiplying a matrix by identity, will result in the unchanged matrix:

::

    # B will be equal to A
    B = A * Transform2D()

Matrix tips
-----------

When using a transform hierarchy, remember that matrix multiplication is
reversed! To obtain the global transform for a hierarchy, do:

::

    var global_xform = parent_matrix * child_matrix

For 3 levels:

::

    var global_xform = gradparent_matrix * parent_matrix * child_matrix

To make a matrix relative to the parent, use the affine inverse (or
regular inverse for orthonormal matrices).

::

    # transform B from a global matrix to one local to A
    var B_local_to_A = A.affine_inverse() * B

Revert it just like the example above:

::

    # transform back local B to global B
    var B = A * B_local_to_A

OK, hopefully this should be enough! Let's complete the tutorial by
moving to 3D matrices.

Matrices & transforms in 3D
---------------------------

As mentioned before, for 3D, we deal with 3 :ref:`Vector3 <class_Vector3>`
vectors for the rotation matrix, and an extra one for the origin.

Basis
-----

Godot has a special type for a 3x3 matrix, named :ref:`Basis <class_basis>`.
It can be used to represent a 3D rotation and scale. Sub vectors can be
accessed as:

::

    var m = Basis()
    var x = m[0] # Vector3
    var y = m[1] # Vector3
    var z = m[2] # Vector3

Or, alternatively as:

::

    var m = Basis()
    var x = m.x # Vector3
    var y = m.y # Vector3
    var z = m.z # Vector3

Basis is also initialized to Identity by default:

.. image:: img/tutomat17.png

Rotation in 3D
--------------

Rotation in 3D is more complex than in 2D (translation and scale are the
same), because rotation is an implicit 2D operation. To rotate in 3D, an
*axis*, must be picked. Rotation, then, happens around this axis.

The axis for the rotation must be a *normal vector*. As in, a vector
that can point to any direction, but length must be one (1.0).

::

    #rotate in Y axis
    var m3 = Basis()
    m3 = m3.rotated( Vector3(0,1,0), PI/2 )

Transform
---------

To add the final component to the mix, Godot provides the
:ref:`Transform <class_Transform>` type. Transform has two members:

-  *basis* (of type :ref:`Basis <class_Basis>`)
-  *origin* (of type :ref:`Vector3 <class_Vector3>`)

Any 3D transform can be represented with Transform, and the separation
of basis and origin makes it easier to work translation and rotation
separately.

An example:

::

    var t = Transform()
    pos = t.xform(pos) # transform 3D position
    pos = t.basis.xform(pos) # (only rotate)
    pos = t.origin + pos # (only translate)