---
title: "Back-face visibility of BED and ABCD"
subject: "Computer Graphics"
chapter: 7
key: "semester_5/computer_graphics:back-face-bed-abcd-observer"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/back-face-bed-abcd-observer
site: BCT Engineering
---

# Back-face visibility of BED and ABCD

Computer Graphics · Chapter 7: Visible Surface Determination

- Asked in: 2076 Chaitra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/back-face-bed-abcd-observer
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-7

## Question

Find the visibility for the surface BED and ABCD where the observer is at P(5, 5, 5). [Figure: pyramid-like object with base vertices A(1,0,1), B(1,0,-1), C(-1,0,1), D(-1,0,-1) and apex E(0,2,0); observer at P(5,5,5).]

## Short answer

Compute outward normal for each face and viewer vector V = P - point. For ABCD: N=(0,-1,0), V·N=-5 <0, so back face. For BED: N=(0,2,-4), V·N=-14 <0, so back face. Both faces are not visible.

## Answer

Vertices: A(1,0,1), B(1,0,−1), C(−1,0,1), D(−1,0,−1), apex E(0,2,0). Observer P(5,5,5). Test: a face is a back face if V · N ≤ 0, where N is the outward normal and V = P − (point on face) is the vector towards the observer.

#### Surface ABCD (base, plane y = 0)
The base is at the bottom of the pyramid, so its outward normal points down: N = (0, −1, 0).

V = P − A = (5−1, 5−0, 5−1) = (4, 5, 4)

V · N = 0·4 + (−1)·5 + 0·4 = **−5**

V · N < 0, so ABCD is a **back face (not visible)**.

#### Surface BED
B(1,0,−1), E(0,2,0), D(−1,0,−1) is the face on the side z = −1.

BE = E − B = (−1, 2, 1), BD = D − B = (−2, 0, 0)

BE × BD = (2·0 − 1·0, 1·(−2) − (−1)·0, (−1)·0 − 2·(−2)) = (0, −2, 4)

This points inward (towards +z, −y); the outward normal is the reverse: **N = (0, 2, −4)**. Check: this face is on the z = −1 side of the pyramid, so its outward normal must have a negative z-component.

V = P − B = (4, 5, 6)

V · N = 0·4 + 2·5 + (−4)·6 = 10 − 24 = **−14**

V · N < 0, so BED is a **back face (not visible)**.

**Answer:** both ABCD and BED are back faces for the observer at (5, 5, 5), so neither is visible. (For comparison, the face AEC on the z = +1 side has outward normal (0, 2, 4), and V·N = 10 + 16 = 26 > 0, so it is visible.)

## Check yourself

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

1. What is the outward normal of base ABCD (plane y = 0) of the pyramid?
   A. (0, -1, 0)
   B. (0, 1, 0)
   C. (0, 0, 1)
   D. (1, 0, 0)
   Answer: A. The base is at the bottom, so its outward normal points down. (Key point: Base normal)

2. For ABCD, V = (4,5,4) and N = (0,-1,0). What is V . N and the conclusion?
   A. 5, front face
   B. -5, front face
   C. -5, back face
   D. 0, edge-on
   Answer: C. V . N = -5 <= 0, so ABCD is hidden. (Key point: ABCD result)

3. Which is the outward normal of face BED (on the z = -1 side)?
   A. (0, -2, 4)
   B. (0, 2, -4)
   C. (0, 2, 4)
   D. (2, 0, -4)
   Answer: B. BE x BD = (0,-2,4) points inward, so it is reversed to (0,2,-4). (Key point: BED normal)

4. For BED, V = (4,5,6) and N = (0,2,-4). What is the result?
   A. V . N = 34, front face
   B. V . N = 14, front face
   C. V . N = 0, edge-on
   D. V . N = -14, back face
   Answer: D. 10 - 24 = -14 <= 0, so BED is hidden. (Key point: BED result)

## Related in Computer Graphics

- [Back-face detection algorithm](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aback-face-detection-method) — Computer Graphics, chapter 7, asked 8×
- [Limitations of back-face detection](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aback-face-limitations-overcome) — Computer Graphics, chapter 7, asked 2×
- [Perspective projection of pyramid](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aperspective-pyramid-zprp-20) — Computer Graphics, chapter 4, asked 2×
- [Back-face test with given normals](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aback-face-normals-frontface) — Computer Graphics, chapter 7
- [Visibility of triangle ABC from (5,5,5)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aback-face-triangle-abc-observer) — Computer Graphics, chapter 7
- [Construct geometric table, 6 vertices](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ageometric-table-construct-6-vertices) — Computer Graphics, chapter 6

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