the concatenation is when we also first rotate and then move. The matrices shown above are valid for the second case, while those for column vectors are transposed. The rule applies, which for multiplication with vectors means that you can switch the multiplication order by transposing the matrix. The interesting thing about this matrix chaining is that a new coordinate system is defined by each such transformation. This can be extended as desired. For example, the propeller of the aircraft may be a separate model, which is then placed by translation to the aircraft nose. This translation only needs to describe the shift from the model coordinate system to the propeller coordinate system. In order to draw the entire aircraft, the transformation matrix for the aircraft is first determined, the points are transformed, and then the propeller model matrix is multiplied to the matrix of the aircraft, and then the propeller points are transformed. The matrix calculated in this way is also called the world matrix. It must be determined for each object in the world before rendering. The application can introduce changes here, for example change the position of the aircraft according to the speed after each frame. Camera Transformation In addition to the objects, the scene also defines a virtual camera or viewer that indicates the position and direction of view relative to which the scene is rendered. The scene is transformed so that the camera is at the origin looking along the Z axis. The resulting coordinate system is called the camera coordinate system and the transformation is called camera transformation or View Transformation. The view matrix is usually determined from camera position, target point (where the camera looks) and an "up vector" ("up" from the viewer's viewpoint). First three auxiliary vectors are required: With normal(v) = normalization of the vector v; cross(v1, v2) = cross product of v1 and v2. Finally, the matrix: with dot(v1, v2) = dot product of v1 and v2. Projection The 3D projection step transforms the view volume into a cube with the corner point coordinates (-1, -1, 0) and (1, 1, 1); Occasionally other target volumes are also used. This step is called projection, even though it transforms a volume into another volume, since the resulting Z coordinates are not stored in the image, but are only used in Z-buffering in the later rastering step. In a perspective illustration, a central