geometrical transformation in computer graphics

geometrical transformation

Geometrical transformation in computer graphics refers to the manipulation of objects' positions, sizes, orientations, or shapes within a digital environment. These transformations are crucial for achieving desired visual effects, animations, or representations. Here are some common types of geometrical transformations:

1. Translation: Translation involves shifting an object from one position to another along specified directions by certain distances. In 2D graphics, this typically means moving objects horizontally and vertically, while in 3D graphics, it extends to movements along the x, y, and z axes.

2. Rotation: Rotation alters the orientation of an object by turning it around a fixed point, axis, or origin. In 2D graphics, objects rotate clockwise or counterclockwise by specified angles, while in 3D graphics, rotations occur in three-dimensional space around the x, y, and z axes.

3. Scaling: Scaling adjusts the size of an object by enlarging or reducing its dimensions along each axis. It involves multiplying or dividing the coordinates of the object's vertices by scaling factors along the x, y, and z axes. Scaling can result in uniform scaling (maintaining aspect ratio) or non-uniform scaling (changing aspect ratio).

4. Shearing: Shearing distorts the shape of an object by skewing it along one or more axes. It involves shifting the coordinates of the object's vertices in a linear manner along specific directions, causing the object to deform.

5. Reflection: Reflection mirrors an object across a specified line or plane, known as the mirror line or reflection axis. This transformation creates a mirrored image of the object on the opposite side of the mirror line.

6. Projection: Projection converts 3D objects into 2D representations for display on a 2D surface, such as a computer screen. Common projection methods include perspective projection, orthogonal projection, and oblique projection, each producing different visual effects.

Geometrical transformations are fundamental to computer graphics, used extensively in graphics software, video games, virtual reality applications, architectural visualization, and various other fields where digital representations of objects are required.

Homogeneous Coordinates

Homogeneous coordinates and matrix representation play crucial roles in 2D transformations in computer graphics. Let's delve into each concept:

Homogeneous coordinates provide a mathematical framework for representing points in 2D space, including points at infinity, in a consistent manner. In homogeneous coordinates, a 2D point (x, y) is represented as ((x_h , y_h , w_h )), where (x_h = x.w_h), (y_h = y.w_h ), and (w_h ) is a non-zero scaling factor. This representation allows for the representation of translations as well as perspective transformations, which are challenging to represent using Cartesian coordinates alone. Homogeneous coordinates also facilitate the use of matrices for representing transformations.

Matrix Representation of 2D Transformations

Matrices provide a concise and efficient way to represent 2D transformations such as translation, rotation, scaling, shearing, and reflection. Each transformation is represented by a 3x3 transformation matrix. The typical form of a 2D transformation matrix is:

a	b	tx
c	d	ty
0	0	1

Where:
- (a) and (d) represent scaling factors for the x and y axes, respectively.
- (b) and (c) represent shearing factors.
- (tx) and (ty) represent translation amounts along the x and y axes, respectively.

For example:
- Translation by (dx) units in the x-direction and (dy) units in the y-direction is represented as:

1	0	dx
0	1	dy
0	0	1

- Rotation by angle θ around the origin is represented as:

cosθ	-sinθ	0
sinθ	cosθ	0
0	0	1

- Scaling by factors (sx) and (sy) along the x and y axes, respectively, is represented as:

sx	0	0
0	sy	0
0	0	1

To apply a transformation to a point represented in homogeneous coordinates, you multiply the transformation matrix by the point's homogeneous coordinate vector.
Using matrices for 2D transformations offers advantages such as efficient computation, easy combination of multiple transformations, and compatibility with hardware graphics pipelines.