Location arithmetic for algebra mode

Position a shape relative to the XY plane

For the following use the helper function:

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

   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")
   

2. Positioning on a plane

   plane = Plane.XZ
   
   face = plane * Rectangle(1, 2)
   
   show_object(face, name="face")
   show_object(plane_symbol(plane), name="plane")
   

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

   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")
   

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

   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")
   

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:

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")

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

   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")
   

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

   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")
   

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)