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Mathematics · 5. Coordinates, vectors and transformations

Describing geometry with coordinates, transformation matrices and Bézier curves

Connect visual geometry to coordinate rules, vectors, 2D transformation matrices (rotation, reflection, scaling, shear) and parametric Bézier curves in computer graphics.

40 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

What information uniquely specifies a transformation and constructs smooth parametric curves?

Curriculum coverage

Vietnam Mathematics 6–10; Cambridge IGCSE / AS-Level transformations, matrices and parametric curves.

Model scope and limitations

Standard matrix rules usually assume the origin as centre; another centre requires translating to and from the origin.

Learning objectives

  • Fully describe translations, rotations, reflections and enlargements with invariants.
  • Represent 2D transformations via 2×2 matrices: rotation R(θ), reflection M(θ), scaling S and shear.
  • Compute composite transformation matrix products and identify orientation reversal via det(M).
  • Understand parametric quadratic and cubic Bézier curves via Bernstein polynomials and de Casteljau algorithm.

Prerequisite knowledge

  • Coordinate plane and negative numbers.
  • Matrix multiplication and 2D vectors.

Core knowledge

Isometry & Invariants

Translations, rotations and reflections preserve lengths and angles. Direct isometries (translations, rotations) preserve orientation (det = +1); opposite isometries (reflections) reverse orientation (det = −1).

Enlargement & Similarity

An enlargement centred at C with factor k maps P to P′ with vector CP′ = k·CP. Area scales by k². The matrix k·I has determinant k².

2D Transformation Matrices & Composition

Every linear map about the origin is a 2×2 matrix. Rotation by θ: R(θ) = [[cos θ, −sin θ], [sin θ, cos θ]] (det = 1). Reflection across line at angle θ: M(θ) = [[cos 2θ, sin 2θ], [sin 2θ, −cos 2θ]] (det = −1). Scaling: S = diag(s_x, s_y). Shear: [[1, k], [0, 1]]. Composition of A followed by B is matrix product B · A (non-commutative: B·A ≠ A·B).

Parametric Bézier curves & de Casteljau algorithm

Cubic Bézier curve defined by 4 control points P₀, P₁, P₂, P₃: B(t) = (1−t)³P₀ + 3(1−t)²t P₁ + 3(1−t)t² P₂ + t³P₃ (t ∈ [0, 1]). Endpoints B(0) = P₀, B(1) = P₃. Tangents B′(0) = 3(P₁−P₀), B′(1) = 3(P₃−P₂). Lies within the convex hull. De Casteljau’s recursive linear interpolation is the foundation of TrueType fonts and CAD graphics.

Worked example

Example 1 (Rotation matrix): 90° anticlockwise rotation R(90°) = [[0, −1], [1, 0]] maps (3, −2) to (2, 3). Example 2 (Cubic Bézier): Points P₀(0,0), P₁(0,2), P₂(2,2), P₃(2,0) at t = 0.5 yield B(0.5) = (1, 1.5).

Misconceptions to avoid

  • A rotation needs centre, angle and direction; rotating about I ≠ O requires translating I to origin, applying rotation matrix, and translating back.
  • Matrix multiplication is non-commutative (B·A ≠ A·B); applying A then B requires multiplying B on the left: Y = (B·A)·X.
  • Control points P₁ and P₂ in a cubic Bézier curve typically do NOT lie on the curve itself, but act as directional tangent anchors.

Virtual experiment procedure: 2D transformations

  1. Predict one key point image before running the simulation.
  2. Apply the rule to every vertex and reconnect in order.
  3. Check lengths, angles, orientation and centre distances as appropriate.

Safety and cautions

  • Do not infer from a distorted display; verify with grid and numerical coordinates.

Evidence to collect

  • A preimage–image coordinate table, transformation rule and checked invariants.
Open the virtual experiment
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<iframe src="https://visuallab.win/math-lab/?module=geom_transformations_2d" width="100%" height="560" frameborder="0" allow="fullscreen" loading="lazy"></iframe>

Diagnostic check

1. Reflection in the y-axis maps (x,y) to?

  1. (−x,y)
  2. (x,−y)
  3. (y,x)
Show answer and explanation

Correct answer: (−x,y)

The x-coordinate changes sign; y stays.

2. Which transformation preserves orientation?

  1. Translation
  2. Reflection
  3. None
Show answer and explanation

Correct answer: Translation

Translation does not reverse vertex order.

3. Which matrix represents a 90° anticlockwise rotation about the origin?

  1. [[0, −1], [1, 0]]
  2. [[0, 1], [−1, 0]]
  3. [[1, 0], [0, 1]]
Show answer and explanation

Correct answer: [[0, −1], [1, 0]]

Substituting θ = 90° into [[cos θ, −sin θ], [sin θ, cos θ]] yields [[0, −1], [1, 0]].

4. What algebraic feature proves a 2×2 isometry matrix is orientation-reversing?

  1. Determinant det(M) = −1
  2. Determinant det(M) = +1
  3. Determinant det(M) = 0
Show answer and explanation

Correct answer: Determinant det(M) = −1

Isometries satisfy |det(M)| = 1. When det(M) = −1, vertex orientation is flipped (like a reflection).

Application practice

1. Translation by (−2,3) maps (5,1) to?

  1. (3,4)
  2. (7,−2)
  3. (−10,3)
Show answer and explanation

Correct answer: (3,4)

Add coordinates: (5−2, 1+3) = (3,4).

2. A 180° rotation about origin maps (4,−1) to?

  1. (−4,1)
  2. (1,4)
  3. (4,1)
Show answer and explanation

Correct answer: (−4,1)

(x,y) → (−x,−y).

3. What does enlargement factor −2 mean?

  1. Double size on opposite side of centre
  2. Half size on same side
  3. Reflection only
Show answer and explanation

Correct answer: Double size on opposite side of centre

Magnitude 2 enlarges; the negative sign puts the image across the centre.

4. For a cubic Bézier curve with control points P₀, P₁, P₂, P₃, what is the tangent vector at the start point P₀ (t = 0)?

  1. 3 · (vec(P₁) − vec(P₀))
  2. vec(P₁) − vec(P₀)
  3. vec(P₃) − vec(P₀)
Show answer and explanation

Correct answer: 3 · (vec(P₁) − vec(P₀))

The derivative B′(0) = 3(P₁ − P₀), so the start tangent is directed along P₀P₁ with magnitude scaled by 3.

5. Using rotation matrix R(90°) = [[0, −1], [1, 0]], what is the image of point A(4, −3) under 90° anticlockwise rotation?

  1. (3, 4)
  2. (−3, −4)
  3. (4, 3)
Show answer and explanation

Correct answer: (3, 4)

Matrix product: x′ = 3, y′ = 4. Image coordinates are (3, 4).

Academic references

  1. Cambridge IGCSE Mathematics 0580 — official syllabus and resources
  2. Cambridge AS & A Level Mathematics 9709 — official syllabus
  3. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

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WebLab STEM
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