---
title: "Derive oblique projection matrix"
subject: "Computer Graphics"
chapter: 4
key: "semester_5/computer_graphics:oblique-projection-matrix-derive"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/oblique-projection-matrix-derive
site: BCT Engineering
---

# Derive oblique projection matrix

Computer Graphics · Chapter 4: Three-Dimensional Graphics

- Asked in: 2070 Chaitra, 5 marks; 2066 Magh (old course); 2079 Baisakh, 4 marks (asked 3 times)
- Video: [3D Transformation : oblique projection](https://www.youtube.com/watch?v=WzMRNAx9-sM) — Umang Rastogi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/oblique-projection-matrix-derive
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-4

## Question

Derive the oblique projection matrix with necessary assumptions.

## Short answer

Oblique projection: projectors incline at angle α to the view plane. For view plane z=0, xp = x + z·L1·cosφ, yp = y + z·L1·sinφ with L1 = cotα. The transformation matrix has rows [1 0 L1cosφ 0; 0 1 L1sinφ 0; 0 0 0 0; 0 0 0 1]. Cavalier uses L1=1, cabinet L1=0.5.

## Answer

An **oblique projection** is a parallel projection in which the projectors are not perpendicular to the view plane.

**Assumptions:**
- The view plane is the xy plane (z = 0).
- A point P(x, y, z) projects to P'(xp, yp) on the view plane.
- The point (x, y, 0) lies on the view plane directly below P (its orthographic projection).
- The projector makes angle α with the view plane. The line joining (x, y) to (xp, yp) on the view plane has length L and makes angle φ with the x-axis.

```text
   z
   |   P(x,y,z)
   |  /|
   | / |
   |/α |         projector P -> P'
   +---+----> view plane
  (x,y,0) P'(xp,yp)
```

**Derivation:** P, (x, y, 0) and P' form a right triangle with height z and base L, so tanα = z/L, which gives L = z/tanα = z·L1, where L1 = 1/tanα is the value of L for a unit length of z. Taking the components of L along x and y:

- xp = x + z·L1·cosφ
- yp = y + z·L1·sinφ
- zp = 0

**Matrix:**

```text
| xp |   | 1 0 L1cosφ 0 | | x |
| yp | = | 0 1 L1sinφ 0 | | y |
| zp |   | 0 0 0      0 | | z |
| 1  |   | 0 0 0      1 | | 1 |
```

If α = 90° then L1 = 0, and the matrix becomes the orthographic projection.

**Common types**

| Type | α | L1 | Effect |
|---|---|---|---|
| Cavalier | 45° | 1 | Receding lines keep full length |
| Cabinet | 63.4° (tanα = 2) | 1/2 | Receding lines are half length, so it looks more natural |

Typical choices of φ are 30° or 45°.

## Check yourself

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

1. What defines an oblique projection?
   A. Projectors meeting at a centre of projection
   B. Parallel projectors that are not perpendicular to the view plane
   C. Projectors perpendicular to the view plane
   D. Projection onto a curved surface
   Answer: B. It is a parallel projection with slanted projectors. (Key point: Oblique definition)

2. The projected coordinates on the xy view plane are:
   A. xp = x*L1*cosφ, yp = y*L1*sinφ
   B. xp = x + L1*cosφ, yp = y + L1*sinφ
   C. xp = x + z*L1*cosφ, yp = y + z*L1*sinφ
   D. xp = x - z, yp = y - z
   Answer: C. The point is shifted on the plane by z*L1 along direction φ. (Key point: Projected coordinates)

3. L1 is related to the projector angle α by:
   A. L1 = 1/tanα
   B. L1 = tanα
   C. L1 = sinα
   D. L1 = cosα
   Answer: A. L1 is the projected length of a unit z-segment, 1/tanα. (Key point: L1)

4. What do cavalier and cabinet projections have as L1?
   A. Cavalier L1 = 1/2, cabinet L1 = 1
   B. Both have L1 = 1
   C. Cavalier L1 = 0, cabinet L1 = 1
   D. Cavalier L1 = 1, cabinet L1 = 1/2
   Answer: D. Cavalier (α = 45°) keeps receding lines full length; cabinet (tanα = 2) halves them. (Key point: Special cases)

## Related in Computer Graphics

- [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×
- [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×
- [Parallel versus oblique projection](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aparallel-vs-oblique-projection) — Computer Graphics, chapter 4

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