2D Transformations

CSC-214 · Semester III · Computer Graphics

intermediate25 minAvailable

2D Transformations

Compose translate · scale · shear · rotate on a triangle.

translate Xtranslate Yscale Xscale Yshear Xshear Y

rotation 0°

Transformed vertices (grid coords)

P0: (13, 7)

P1: (18, 7)

P2: (13, 12)

Order of application

Scale and shear around the origin first, then rotate, then translate. In matrix form a column vector is multiplied as T · R · Sh · S · P — rightmost applied first. Swap the matrix order and you get a completely different shape.

Lab brief

Objective

Apply translation, rotation, scaling and composite transformations to a 2D shape and see the matrices act on every vertex.

Theory & concepts

In homogeneous coordinates, translation, rotation and scaling are all 3x3 matrices. Composing transformations is multiplying matrices, with the rightmost matrix applied first. The visualization shows the shape and its transformed copy, with the composite matrix and per-vertex results updated live.

Rendering diagram…

2D transformation pipeline in homogeneous coordinates

Transform	3x3 matrix
Translate (tx, ty)	[1 0 tx; 0 1 ty; 0 0 1]
Rotate theta	[cos -sin 0; sin cos 0; 0 0 1]
Scale (sx, sy)	[sx 0 0; 0 sy 0; 0 0 1]

All three matrices act on the homogeneous point (x, y, 1). To rotate about an arbitrary pivot p, compose T(p) * R * T(-p) - the rightmost matrix is applied to the point first.

Prerequisites

Lesson: Homogeneous coordinates and composite transformations
Lesson: 2D rotation

Instructions

1Pick a shape from the shape library (triangle, square, house).
2Apply a single translation and drag the vector; watch every vertex move equally.
3Apply a rotation about the origin, then about a pivot point you choose.
4Compose scale then rotate and rotate then scale; observe the different results.
5Read the combined 3x3 matrix and verify it equals the ordered product of the individual matrices.
6Reset and verify the inverse transform returns the shape to its original position.

Tasks

Rotate about a pivot
Build T * R * T(-pivot) and confirm the shape spins in place.
Order sensitivity
Apply R then S and S then R to the same shape and explain the difference.
Verify the matrix
Compute one transformed vertex by hand and compare with the visualization.

Self-check

The transformed shape matches the expected matrix result.
The student composes pivot rotation correctly as T * R * T(-pivot).
The student explains why matrix order changes the outcome.

Environment

shape librarytransformation controlscomposite matrix displaypivot point pickerundo/inverse