Model Transform Example

Problem (2025 Paper 3 Question 4(a)(i))

Transform a cylinder (radius 1, height 2, centred at origin, aligned along y-axis) to have:

• Radius 2
• Ends at $(1, 2, 3)$ and $(2, 4, 5)$

Solution

Order: Scale → Rotate → Translate

Compute Target Properties

• Axis vector: $\mathbf{v} = (2-1, 4-2, 5-3) = (1, 2, 2)$
• Length: $|\mathbf{v}| = \sqrt{1^2 + 2^2 + 2^2} = 3$
• Midpoint: $\mathbf{m} = \left(\frac{1+2}{2}, \frac{2+4}{2}, \frac{3+5}{2}\right) = (1.5, 3, 4)$

1. Scale

Original: radius 1, height 2. Target: radius 2, height 3.

• $s_x = 2$, $s_z = 2$ (double radius)
• $s_y = 1.5$ (height 3 from 2: $3/2 = 1.5$)

$\mathbf{S} = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 1.5 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

2. Rotate

Align y-axis $(0,1,0)$ with target axis $(1, 2, 2)$.

Step A: Zero z-component. Rotate about x-axis by $\theta = 45°$.

Step B: Zero x-component. Rotate about z-axis by $\phi = \arcsin(1/3) \approx 19.47°$.

The rotation aligning the axis: $\mathbf{R}^{-1} = \mathbf{R}_x(-45°) \cdot \mathbf{R}_z(-\arcsin(1/3))$

3. Translate

Move origin to midpoint $(1.5, 3, 4)$: $\mathbf{T} = \begin{bmatrix} 1 & 0 & 0 & 1.5 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Overall Model Matrix

$\mathbf{M} = \mathbf{T} \cdot \mathbf{R}_x(-45°) \cdot \mathbf{R}_z(-\arcsin(1/3)) \cdot \mathbf{S}$

Summary

• Compute geometric properties first (axis, length, midpoint)
• Apply SRT order: Scale, Rotate, Translate
• Rotate by inverse of the rotation that aligns axis to canonical

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Past Paper Questions

2025 Paper 3 Question 4(a)(i): Transform a cylinder to have specified radius and endpoints. [6 marks]