Coordinate
Systems
Dr. Midori Kitagawa
Coordinate Systems
 Coordinates
are an ordered set of values
which specify a location relative to some
origin.
 The order of coordinates is important.
 There are a variety of coordinate systems.
2D and 3D coordinate systems
A
2D coordinate system is used to specify
all locations in 2D space.
 A 3D coordinate system is used to specify
all locations in 3D space.
Cartesian coordinate systems
 Also
called a rectangular coordinate
system.
 2D and 3D Cartesian coordinate systems
are commonly used.
2D Cartesian coordinate
system
 In
the 2D Cartesian coordinate system,
each location is specified by an ordered
set of two distances, an x-coordinate and
a y-coordinate, represented as (x, y).
2D Cartesian coordinate
system
 Coordinates
(x, y) are a ordered set
because their order matters.
 The x-coordinate of a point location
comes first; the y-coordinate comes next.
 For instance, (4, 2) and (2, 4) specify two
different locations.
3D Cartesian coordinate
system
 In
the 3D Cartesian coordinate system,
each location is specified by an ordered
set of three coordinates, an x-coordinate,
a y-coordinate, and a z-coordinate, (x, y,
z).
Other 3D coordinate systems
 In
addition to the 3D Cartesian
coordinate system, there are other types
of 3D coordinate systems, such as
spherical coordinate system and
cylindrical coordinate system.
Spherical coordinate system
 Spherical
coordinate system is so called
the azimuthal coordinate system.
 In the spherical coordinate system, each
location is specified by two angles (θ and
φ) and a distance (r).
Spherical coordinate system
 The
spherical coordinate system is often
used to aim a camera at an object in the
world space by specifying the camera's
angle around the horizon φ, its angle
above the horizon θ, and its distance from
the object r.
Cylindrical coordinate system
 In
the cylindrical coordinate system each
location is specified by two distances (ρ
and z) and one angle (φ).
Right-handed and left-handed 3D
Cartesian coordinate systems
 There
are two types of 3D Cartesian
coordinate systems: Right-handed and
Left-handed.
How to tell if it’s a Right-handed or
left-handed coordinate system
 Align
your right-hand thumb with the x-
axis.
 If you can curl your right-hand fingers from
the y-axis to the z-axis to make a fist, that’s
a right-handed system.
Positive rotation in a righthanded coordinate system
 In
the right-handed coordinate system the
direction in which your hand closes to
make a fist is the direction of a positive
rotation around any axis represented by
the extended right-hand thumb.
Positive rotation in a lefthanded coordinate system
 In
the left-handed coordinate system the
direction in which your hand closes to
make a fist is the direction of a positive
rotation around any axis represented by
the extended left-hand thumb.
Right-handed coordinate
system in 3D applications



Many 3D animation applications, such as
Maya, Houdini, and Motion Builder, use righthanded 3D Cartesian coordinate systems.
The y-axis is the vertical axis in many 3D
applications but in some software the z-axis is
the vertical axis.
Be aware of the orientation of the major axes
when you transfer an object or a scene from
one software to another.
Object space and world
space
 Each
object is defined in its object space
(also called object coordinate system or
local coordinate system.)
 A scene containing a single object or
multiple objects is defined in the world
space (also called world coordinate
system or global coordinate system.)
Object space and world
space



Object space and world space work
together.
For instance, if the geometric center of an
object is at the origin of its object space and
if the object is placed at the origin of the
world space, it will be centered at origin of
the world space.
However, if the geometric center of an object
is not centered at the origin of its object
space, it will be off-centered in the world
space even if it’s placed at the origin of the
world space.