Transformations in Graphics

Geometric Transformations

• Introduction of Transformations in Graphics

  • Transformations allow for a wide range of creative possibilities in graphic design and animation.

• Definition of Transformations in Graphics

  • Geometric Transformations are fundamental operations or techniques used in multimedia graphics to change or manipulate the position, size, or orientation of objects in 2D or 3D graphic space for animation, modeling, and visual effects.
  • Transformations is a technique used to modify the appearance and position of graphical objects, such as images, shapes, and text. 
  • In Graphics, transformation refers to the process of changing the position, size, or orientation of an object or image on a two-dimensional plane.
  • In multimedia graphics, Translation, Rotation, Scaling, Reflection, and Shearing are called Geometric Transformations.

• Features of Transformations in Graphics

• • Each transformation activity is a single entity. It can be denoted by a unique name or symbol.
  • It is also possible to combine two or more different transformation techniques to create complex effects and animations in graphics. For example, a rotating and scaling transformation can be used to create a spinning, growing, or shrinking animation.
  • Transformations can also be used to create a sense of depth and perspective in 3D graphics by applying different scaling and positioning values to objects based on their distance from the viewer.

• Types of Transformations in Graphics

  • Geometric Transformations in Multimedia Graphics are as follows –

2. Rotation

• Rotation is a transformation that turns an object about a fixed point called the pivot point.
• It is usually rotation that happens around the origin (0,0).
• Mathematically, 

• Types of Rotation
  • Clockwise rotation
  • Anticlockwise rotation
• For example, rotating a point 90° around the origin changes its orientation.

3. Scaling

• Scaling is a transformation that changes the size of an object.
• It can enlarge or shrink the object.
• Mathematically,

• Types of Scaling
  • Uniform Scaling (same factor in both directions)
  • Non-uniform Scaling (different factors)
• For Example,

If S_x = 2 and S_y = 2, the object becomes twice as large.

4. Reflection

• Reflection is a transformation that creates a mirror image of an object with respect to a line or axis.
• Types of Reflection
  • Reflection about the X-axis.
  • Reflection about the Y-axis.
  • Reflection about Origin.
  • Reflection about the line y = x.
• For example

5. Shearing

• Shearing is a transformation that slants or skews the shape of an object.
• It changes the shape but keeps the area the same.
• Types of Shearing
  • X-shear
  • Y-shear
• Mathematically,

• Transformation – Inverse Transformation
  • Translation (Tx, Ty)  –  Translation (-Tx, -Ty)
  • Scaling (Sx, Sy)  –  Scaling (1/Sx, 1/Sy)
  • Rotation θ – Rotation −θ
  • Reflection – Reflection again on the same axis
• Uses of Inverse Transformation
  • In undoing transformations
  • In animation correction
  • In the coordinate system changes

• A composite transformation is a combination of two or more transformations applied to an object in sequence.
• Instead of applying transformations one by one, they can be combined into a single transformation matrix.
• Mathematically, an object may undergo Translation, Rotation, and Scaling. These together form a composite transformation.

• The order of transformation Scale → Rotate → Translate is different from Translate → Rotate → Scale.
• Example of Composite Transformation –

• Advantages of Composite Transformation
  • This transformation reduces the number of calculations.
  • This transformation provides faster graphics processing.
  • This transformation makes object manipulation easier.
  • This transformation is used in animation and modeling.

Polygon Representation

• Polygon representation is a fundamental technique in multimedia graphics used to model objects by dividing their surfaces into polygons, making it easier to render complex shapes efficiently in computer graphics systems.
• Polygon representation is a method used in computer and multimedia graphics to represent 3D objects or shapes using polygons.
• A polygon is a closed figure formed by connecting three or more straight line segments.
• Most 3D objects in graphics are represented by many small polygons, usually triangles or quadrilaterals.
• Polygon representation is a technique used to model and display objects by dividing their surfaces into polygons. For example, a cube can be represented by 6 square polygons. A complex object like a car can be represented by thousands of polygons.
• Components of Polygon Representation – Following components
  • Vertices
    • It is points in 3D space.
    • Each vertex has coordinates (x, y, z).
  • Edges
    • This is the line connecting two vertices.
  • Faces
    • This is the closed surface formed by edges.
• Example

• Methods of Polygon Representation

• • This method stores the coordinates of all vertices.
  • Example –

• • This method stores pairs of vertices that form edges.
  • For example –

• • This method stores polygons using edges or vertices.
  • For example –

• Types of Polygons
  • Convex Polygon
    • All interior angles are less than 180°.
    • Any line between two points lies inside the polygon.
  • Concave Polygon
    • This polygon has at least one interior angle that is greater than 180°.
• Advantages of Polygon Representation
  • It is simple and efficient.
  • It is easy to store in computer memory.
  • It is suitable for rendering and animation.
  • It is widely used in 3D modeling.
• Applications of Polygon Representation
  • The polygon representation concept is used in –
    • 3D animation work.
    • Computer games.
    • Virtual reality work.
    • CAD systems
    • Multimedia graphics
• Polygon Representation Tools
  • Examples include games like Minecraft and 3D modeling tools such as Blender.