Mobile 3D Graphics API (M3G)

 
 
Mobile 3D Graphics API (M3G) is an optional package which can be used to existing J2ME MIDP 1.0 or MIDP 2.0. profiles. The specification was defined under JSR-184. The target platform of this optional API is J2ME/CLDC 1.1.

More information about M3G can be found at: http://developers.sun.com/techtopics/mobility/apis/articles/3dgraphics/

Version 2.2 of the J2ME Wireless Toolkit provides support for mobile 3D graphics by implementing JSR 184.
The latest M3G API version can be downloaded from: http://www.forum.nokia.com/main/resources/technologies/java/

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Matrix arithmetic and coordinate system.




Information
Coordinate systems

Coordinate systems are used to describe the locations of points in space.
In m3g the orthographic coordinate system is used. In an orthographic coordinate system, the X, Y and Z axis are perpendicular to each other. The right-hand rule is used, with +Y (index finger) being up, +X (thumb) horizontal to the right, and +Z (middle) directed toward the viewer.

Right-hand coordinate system.

All angles or rotational representations are in degrees. A positive rotation is counter clockwise.

Right-hand coordinate system with rotation.

In m3g, three coordinate systems plays an important role:
  • World coordinate system.

    Is the base on which all other coordinates are being defined.

  • Camera coordinate system.

    This coordinate system is aligned with the eye or camera. The +Y axis is up, and the -Z axis points out off the camera. In order to render the world, everything in the scene must be transformed to camera coordinates.

  • Local (or object) coordinate system.

    Each object in the scene is defined in its own local coordinate system. By using a local coordinate system you can create many copies of the object, in different positions and orientations.
    Coordinate systems.
Matrix arithmetic

A 3x3 matrix can be used to represent a scaling [S] (resize an object) or rotation [R] (rotate an object about the origin by an angle Θ).

A 3x1 matrix can be used to represent a translation [T] (moves an object).

A 4x4 matrix can be used to combine a scaling or rotation with a translation.

Matrix.

The meaning of values a00...a23 are explained in the figures below:

Vectors

Rotation matrix

Translation matrix



A 4x4 matrix can store a single transformation (= rotation, scaling or translation) or a sequence of transformations. The latter is called a composite transformation. The matrix of a composite transformation is obtained by multiplying the matrices of the individual transformations.

The order of multiplying the matrices matters. Composite transformations are built from left to right. The product [S][R][T] (in that order) is the matrix of the composite transformation that first scales, then rotates, then translates. The matrix produced by the product [S][R][T] is different from the matrix produced by the product [T][R][S].

Rotations around the XYZ axes are necessary in order to describe any orientation of a body in space. Rotations about each axis are often used to transform between different coordinate systems. By using multiple coordinate systems, graphics algorithms are made easier to understand and to implement. In the figure below the two coordinate systems represent point P (in 2D space) in different ways.

A point in two different coordinate systems

The three rotation matrices are given below:

Rotation tx around the X-axis:

left bracket
x'
y'
z'
right bracket  =  left bracket
1 0 0
0 cos(tx) sin(tx)
0 -sin(tx) cos(tx)
right bracket left bracket
x
y
z
right bracket


Rotation ty around the Y-axis:

left bracket
x'
y'
z'
right bracket  =  left bracket
cos(ty) 0 -sin(ty)
0 1 0
sin(ty) 0 cos(ty)
right bracket left bracket
x
y
z
right bracket


Rotation tz around the Z-axis:

left bracket
x'
y'
z'
right bracket  =  left bracket
cos(tz) sin(tz) 0
-sin(tz) cos(tz) 0
0 0 1
right bracket left bracket
x
y
z
right bracket


In games development, the particular order of rotations is to rotate about the Y-axis first, then the X-axis, then the Z-axis:

left bracket
x'
y'
z'
right bracket = Rz(t) Rx(t) Ry(t) left bracket
x
y
z
right bracket


Rz(t) Rx(t) Ry(t) combined in a single matrix is:

left bracket
cos(tz) cos(ty) + sin(tz) sin(tx) sin(ty)    sin(tz) cos(tx)    -cos(tz) sin(ty) + sin(tz) sin(tx) cos(ty)
-sin(tz) cos(ty) + cos(tz) sin(tx) sin(ty)    cos (tz) cos(tx)    sin(tz) sin(ty) + cos(tz) sin(tx) cos(ty)
cos(tx) sin(ty)    -sin(tx)    cos(tx) cos(ty)
right bracket


Rotation and translation about or in a particular axis are also indicated by a name:
  • Pitch :Rotation about X-axis
  • Yaw :Rotation about Y-axis
  • Roll :Rotation about Z-axis
  • Strafe :Translation in X-axis
  • Fly :Translation in Y-axis
  • Move :Translation in Z-axis