1. Current Transformation

Definition: The current transformation (often called CTM) is the matrix that represents all the transformations currently applied to objects in a graphics scene.

• It is used to transform object coordinates into world or screen coordinates.
• Every vertex of an object is multiplied by the CTM before rendering.

Key Points:

• The CTM can represent translation, rotation, scaling, shear, or any combination.
• When you apply a transformation (e.g., rotate, translate), it modifies the CTM, not the object coordinates directly.

Example:

• Suppose the CTM is initially the identity matrix.
• Applying glTranslatef(2, 0, 0) updates the CTM.
• All subsequent drawing commands will have their vertices translated by (2, 0, 0) automatically.

2. Matrix Stack

Definition: A matrix stack is a stack data structure used to save and restore transformation matrices.

• Common in OpenGL to manage hierarchical transformations.

• Two main operations:

  1. Push – Save the current CTM onto the stack.
  2. Pop – Restore the last saved CTM from the stack.

Purpose:

• Allows local transformations for individual objects without affecting others.
• Essential for hierarchical modeling (e.g., a robot arm: shoulder → elbow → hand).

3. How Matrix Stack Works

Workflow Example:

1. Start with identity CTM.
2. Draw a robot body: apply translation and scaling.
3. Push CTM before transforming the arm.
4. Apply rotation for the arm and draw it.
5. Pop CTM to revert to the body’s transformation.
6. Draw another object using the original body transformation.

Diagram Conceptually:

CTM (identity)
   |
   |-- translate body
   |-- scale body
   |-- push CTM
        |-- rotate arm
        |-- draw arm
   |-- pop CTM
   |-- draw other body parts

4. OpenGL Functions for Matrix Stack

5. Benefits of Using Matrix Stack

• Simplifies complex hierarchical transformations.
• Avoids manual calculation of inverse transformations.
• Provides modularity: local transformations don’t interfere with global ones.

6. Summary Table