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Coordinate Systems Dr. Midori Kitagawa

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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.

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**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.

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**Cartesian coordinate systems**

Also called a rectangular coordinate system. 2D and 3D Cartesian coordinate systems are commonly used.

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**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).

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**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.

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**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).

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**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.

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**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).

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**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.

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**Cylindrical coordinate system**

In the cylindrical coordinate system each location is specified by two distances (ρ and z) and one angle (φ).

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**Right-handed and left-handed 3D Cartesian coordinate systems**

There are two types of 3D Cartesian coordinate systems: Right-handed and Left-handed.

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**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.

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**Positive rotation in a right-handed 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.

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**Positive rotation in a left-handed 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.

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**Right-handed coordinate system in 3D applications**

Many 3D animation applications, such as Maya, Houdini, and Motion Builder, use right-handed 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.

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**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.)

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**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.

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