.. _location_arithmetics:

Location arithmetic for algebra mode
======================================

Position a shape relative to the XY plane
---------------------------------------------

For the following use the helper function:

.. code-block:: build123d

    def location_symbol(location: Location, scale: float = 1) -> Compound:
        return Compound.make_triad(axes_scale=scale).locate(location)

    def plane_symbol(plane: Plane, scale: float = 1) -> Compound:
        triad = Compound.make_triad(axes_scale=scale)
        circle = Circle(scale * .8).edge()
        return (triad + circle).locate(plane.location)

1. **Positioning at a location**

    .. code-block:: build123d

    loc = Location((0.1, 0.2, 0.3), (10, 20, 30))

    face = loc * Rectangle(1, 2)

    show_object(face, name="face")
    show_object(location_symbol(loc), name="location")

.. image:: assets/location-example-01.png

2) **Positioning on a plane**

    .. code-block:: build123d

    plane = Plane.XZ

    face = plane * Rectangle(1, 2)

    show_object(face, name="face")
    show_object(plane_symbol(plane), name="plane")

.. image:: assets/location-example-07.png

Note: The ``x``-axis and the ``y``-axis of the plane are on the ``x``-axis and the ``z``-axis of the world coordinate system (red and blue axis).

Relative positioning to a plane
------------------------------------

1. **Position an object on a plane relative to the plane**

    .. code-block:: build123d

    loc = Location((0.1, 0.2, 0.3), (10, 20, 30))

    face = loc * Rectangle(1,2)

    box = Plane(loc) * Pos(0.2, 0.4, 0.1) * Box(0.2, 0.2, 0.2)
    # box = Plane(face.location) * Pos(0.2, 0.4, 0.1) * Box(0.2, 0.2, 0.2)
    # box = loc * Pos(0.2, 0.4, 0.1) * Box(0.2, 0.2, 0.2)

    show_object(face, name="face")
    show_object(location_symbol(loc), name="location")
    show_object(box, name="box")

.. image:: assets/location-example-02.png

The ``X``, ``Y``, ``Z`` components of ``Pos(0.2, 0.4, 0.1)`` are relative to the ``x``-axis, ``y``-axis or
``z``-axis of the underlying location ``loc``.

Note: ``Plane(loc) *``, ``Plane(face.location) *`` and ``loc *`` are equivalent in this example.

2. **Rotate an object on a plane relative to the plane**

    .. code-block:: build123d

    loc = Location((0.1, 0.2, 0.3), (10, 20, 30))

    face = loc * Rectangle(1,2)

    box = Plane(loc) * Rot(Z=80) * Box(0.2, 0.2, 0.2)

    show_object(face, name="face")
    show_object(location_symbol(loc), name="location")
    show_object(box, name="box")

.. image:: assets/location-example-03.png

The box is rotated via ``Rot(Z=80)`` around the ``z``-axis of the underlying location
(and not of the z-axis of the world).

More general:

    .. code-block:: build123d

    loc = Location((0.1, 0.2, 0.3), (10, 20, 30))

    face = loc * Rectangle(1,2)

    box = loc * Rot(20, 40, 80) * Box(0.2, 0.2, 0.2)

    show_object(face, name="face")
    show_object(location_symbol(loc), name="location")
    show_object(box, name="box")

.. image:: assets/location-example-04.png

The box is rotated via ``Rot(20, 40, 80)`` around all three axes relative to the plane.

3. **Rotate and position an object relative to a location**

    .. code-block:: build123d

    loc = Location((0.1, 0.2, 0.3), (10, 20, 30))

    face = loc * Rectangle(1,2)

    box = loc * Rot(20, 40, 80) * Pos(0.2, 0.4, 0.1) * Box(0.2, 0.2, 0.2)

    show_object(face, name="face")
    show_object(location_symbol(loc), name="location")
    show_object(box, name="box")
    show_object(location_symbol(loc * Rot(20, 40, 80), 0.5), options={"color":(0, 255, 255)}, name="local_location")

.. image:: assets/location-example-05.png

The box is positioned via ``Pos(0.2, 0.4, 0.1)`` relative to the location ``loc * Rot(20, 40, 80)``

4. **Position and rotate an object relative to a location**

    .. code-block:: build123d

    loc = Location((0.1, 0.2, 0.3), (10, 20, 30))

    face = loc * Rectangle(1,2)

    box = loc * Pos(0.2, 0.4, 0.1) * Rot(20, 40, 80) * Box(0.2, 0.2, 0.2)

    show_object(face, name="face")
    show_object(location_symbol(loc), name="location")
    show_object(box, name="box")
    show_object(location_symbol(loc * Pos(0.2, 0.4, 0.1), 0.5), options={"color":(0, 255, 255)}, name="local_location")

.. image:: assets/location-example-06.png

Note: This is the same as ``box = loc * Location((0.2, 0.4, 0.1), (20, 40, 80)) * Box(0.2, 0.2, 0.2)``