---
title: "Scaling about arbitrary fixed point"
subject: "Computer Graphics"
chapter: 3
key: "semester_5/computer_graphics:scaling-about-fixed-point"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/scaling-about-fixed-point
site: BCT Engineering
---

# Scaling about arbitrary fixed point

Computer Graphics · Chapter 3: Two –Dimensional Transformations

- Asked in: 2069 Chaitra, 6 marks
- Video: [Fixed point Scaling: Explanation With example](https://www.youtube.com/watch?v=oOL-_J585wc) — LectureLoops By Rahul Shukla
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/scaling-about-fixed-point
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3

## Question

Derive the 2D transformation matrix for scaling with respect to an arbitrary fixed point.

## Short answer

Translate fixed point to origin, scale, then translate back: M = [sx 0 xf(1-sx); 0 sy yf(1-sy); 0 0 1]. Transformed point: x' = x·sx + xf(1-sx), y' = y·sy + yf(1-sy).

## Answer

Scaling about the origin moves the object away from or toward the origin. To scale an object about a fixed point (xf, yf) so that this point stays unchanged, the object is moved so that the fixed point is at the origin, scaled, and then moved back.

#### Steps
1. Translate the fixed point to the origin: T(−xf, −yf)
2. Scale about the origin: S(sx, sy)
3. Translate back: T(xf, yf)

```text
M = T(xf, yf) · S(sx, sy) · T(−xf, −yf)

    | 1 0 xf |   | sx 0 0 |   | 1 0 −xf |
  = | 0 1 yf | · | 0 sy 0 | · | 0 1 −yf |
    | 0 0  1 |   | 0  0 1 |   | 0 0   1 |
```

Multiply the right two matrices:

```text
| sx 0 −sx·xf |
| 0 sy −sy·yf |
| 0  0    1   |
```

Then the left matrix:

```text
        | sx  0   xf(1 − sx) |
   M =  | 0   sy  yf(1 − sy) |
        | 0   0        1     |
```

So x' = x·sx + xf(1 − sx) and y' = y·sy + yf(1 − sy).

**Check:** for (x, y) = (xf, yf): x' = xf·sx + xf − xf·sx = xf, so the fixed point stays unchanged. Example: scaling by 2 about (4,2) gives x' = 2x − 4, y' = 2y − 2.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is scaling about (xf, yf) done?
   A. T(xf,yf)·S·T(-xf,-yf)
   B. S·T(xf,yf)
   C. T(-xf,-yf)·S·T(xf,yf)
   D. R·S·R
   Answer: A. Move the fixed point to the origin, scale, then move back. (Key point: Steps)

2. What is the translation entry in the top row?
   A. xf·sx
   B. xf(1 - sx)
   C. -xf·sx
   D. xf(1 + sx)
   Answer: B. Multiplying the three matrices gives xf(1 - sx). (Key point: Matrix)

3. What confirms the matrix?
   A. The origin is fixed
   B. Determinant equals 1
   C. The fixed point maps to itself
   D. All points move equally
   Answer: C. x' = xf·sx + xf - xf·sx = xf. (Key point: Check)

4. Scaling by 2 about (4,2) gives?
   A. x' = 2x + 4, y' = 2y + 2
   B. x' = 2x, y' = 2y
   C. x' = x/2 + 2, y' = y/2 + 1
   D. x' = 2x - 4, y' = 2y - 2
   Answer: D. xf(1-2) = -4 and yf(1-2) = -2. (Key point: Example)

## Related in Computer Graphics

- [3D scaling about arbitrary point](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ascale-3d-arbitrary-point) — Computer Graphics, chapter 4
- [Derive perspective projection matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aperspective-projection-matrix-derive) — Computer Graphics, chapter 4, asked 3×
- [Rotation about a pivot point](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Arotation-about-pivot-point-derive) — Computer Graphics, chapter 3, asked 2×
- [Derive parallel projection matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aparallel-projection-matrix-derive) — Computer Graphics, chapter 4, asked 2×
- [Derive oblique projection matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aoblique-projection-matrix-derive) — Computer Graphics, chapter 4, asked 3×
- [2D reflection matrices and line y+x=0](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Areflection-matrices-axes-y-plus-x) — Computer Graphics, chapter 3

## Related in other subjects

- [Bar stiffness matrix and 2D truss element](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Abar-stiffness-truss-rotation) — Computational Techniques in Civil Engineering, chapter 4
- [4-point DFT by linear transformation](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adft-linear-transformation) — Signals and Systems, chapter 3
- [Three moment equation: derive and apply to fixed beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Athree-moment-derive-fixed-beam-udl) — Theory of Structures II, chapter 2
- [Fixed point: sin x + 3x - 2](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Afixed-point-sinx-3x-2) — Numerical Method, chapter 2, asked 2×