---
title: "Shear cube w.r.t. ZX plane, constant 3"
subject: "Computer Graphics"
chapter: 4
key: "semester_5/computer_graphics:cube-shear-zx-constant-3"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/cube-shear-zx-constant-3
site: BCT Engineering
---

# Shear cube w.r.t. ZX plane, constant 3

Computer Graphics · Chapter 4: Three-Dimensional Graphics

- Asked in: 2075 Asoj, 7 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/cube-shear-zx-constant-3
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-4

## Question

A unit length cube with diagonal passing through (0, 0, 0) and (2, 2, 2) is sheared with respect to ZX-plane with shear constants = 3 in both directions. Obtain the final coordinates of the cube after shearing.

## Short answer

Apply x' = x+3y, z' = z+3y, y unchanged. Final coordinates: (0,0,0), (0,0,2), (6,2,6), (6,2,8), (2,0,0), (2,0,2), (8,2,6), (8,2,8).

## Answer

**Assumptions:** the question says "unit length", but a diagonal from (0,0,0) to (2,2,2) means the cube has side 2, with corners at 0 and 2 on each axis. Shear "with respect to the ZX plane" keeps y fixed (the ZX plane, y = 0, does not move). x and z are displaced in proportion to y with shear constants sh_x = sh_z = 3.

Equations: x' = x + 3y, y' = y, z' = z + 3y

```text
| x' |   | 1 3 0 0 | | x |
| y' | = | 0 1 0 0 | | y |
| z' |   | 0 3 1 0 | | z |
| 1  |   | 0 0 0 1 | | 1 |
```

| Corner | (x, y, z) | x + 3y | z + 3y | New corner |
|---|---|---|---|---|
| A | (0,0,0) | 0 | 0 | (0,0,0) |
| B | (0,0,2) | 0 | 2 | (0,0,2) |
| C | (0,2,0) | 6 | 6 | (6,2,6) |
| D | (0,2,2) | 6 | 8 | (6,2,8) |
| E | (2,0,0) | 2 | 0 | (2,0,0) |
| F | (2,0,2) | 2 | 2 | (2,0,2) |
| G | (2,2,0) | 8 | 6 | (8,2,6) |
| H | (2,2,2) | 8 | 8 | (8,2,8) |

**Answer:** (0,0,0), (0,0,2), (6,2,6), (6,2,8), (2,0,0), (2,0,2), (8,2,6), (8,2,8). The bottom face (y = 0) stays in place, and the top face (y = 2) is moved by 6 in both x and z.

## Check yourself

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

1. The diagonal from (0,0,0) to (2,2,2) means the cube side is:
   A. 2
   B. 1
   C. 3
   D. sqrt(2)
   Answer: A. The corners are at 0 and 2 on each axis. (Key point: Cube size)

2. Shear with respect to the ZX plane keeps which coordinate fixed?
   A. x
   B. y
   C. z
   D. None
   Answer: B. The ZX plane is y = 0; x and z are displaced in proportion to y. (Key point: Fixed coordinate)

3. Corner (0,2,0) moves to:
   A. (3, 2, 3)
   B. (0, 6, 0)
   C. (6, 2, 6)
   D. (6, 6, 6)
   Answer: C. x' = 0 + 3(2) = 6, z' = 0 + 3(2) = 6, y stays 2. (Key point: Corner)

4. Which face stays fixed and what happens to the top face?
   A. y = 2 stays; y = 0 moves by 6
   B. Neither face moves
   C. y = 0 stays; y = 2 moves by 3 in y
   D. y = 0 stays; y = 2 moves by 6 in x and z
   Answer: D. The displacement is 3*y = 6 for y = 2 and 0 for y = 0. (Key point: Faces)

## Close variants set in other exams

- [Shear unit cube w.r.t. yz plane](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acube-shear-yz-constant-2) — Computer Graphics, chapter 4, asked 4×