---
title: "Visibility of triangle ABC from (5,5,5)"
subject: "Computer Graphics"
chapter: 7
key: "semester_5/computer_graphics:back-face-triangle-abc-observer"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/back-face-triangle-abc-observer
site: BCT Engineering
---

# Visibility of triangle ABC from (5,5,5)

Computer Graphics · Chapter 7: Visible Surface Determination

- Asked in: 2069 Asar, 10 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-triangle-abc-observer
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-7

## Question

Given a triangular object ABC having A(1,0,0), B(0,1,0) and C(0,0,1), if the observer is at point (5,5,5), is the face ABC visible?

## Short answer

Compute normal N=(1,1,1) and viewer vector V=(4,5,5). Dot product is 14 >0, so the face is a front face and visible from (5,5,5).

## Answer

Vertices A(1,0,0), B(0,1,0), C(0,0,1); observer at P(5,5,5).

**Step 1: normal of ABC** (vertex order A → B → C, so the normal points away from the origin)

AB = B − A = (−1, 1, 0), AC = C − A = (−1, 0, 1)

N = AB × AC = (1·1 − 0·0, 0·(−1) − (−1)·1, (−1)·0 − 1·(−1)) = (1, 1, 1)

**Step 2: plane equation**

x + y + z + D = 0; at A: 1 + D = 0, so D = −1. Plane: **x + y + z − 1 = 0** (A = B = C = 1).

**Step 3: vector to the observer**

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

**Step 4: test**

V · N = 4·1 + 5·1 + 5·1 = **14 > 0**

Since V · N > 0, the face points towards the observer, so it is a **front face and is visible**.

Check using the plane: substituting the observer's position gives 5 + 5 + 5 − 1 = 14 > 0, so the eye is on the outside (positive) side of the plane, and the face is visible.

**Answer:** face ABC is visible from (5, 5, 5).

## Check yourself

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

1. With A(1,0,0), B(0,1,0), C(0,0,1), what is N = AB x AC?
   A. (1, 1, 1)
   B. (-1, -1, -1)
   C. (1, -1, 1)
   D. (0, 0, 1)
   Answer: A. AB = (-1,1,0), AC = (-1,0,1); cross product gives (1,1,1). (Key point: Normal of ABC)

2. What is the plane equation of ABC?
   A. x + y + z + 1 = 0
   B. x + y + z - 1 = 0
   C. x + y + z = 0
   D. x - y + z - 1 = 0
   Answer: B. A = B = C = 1 and at A(1,0,0), 1 + D = 0 so D = -1. (Key point: Plane equation)

3. For observer P(5,5,5) and A(1,0,0), what is V = P - A?
   A. (6, 5, 5)
   B. (4, 5, 4)
   C. (4, 5, 5)
   D. (5, 5, 5)
   Answer: C. (5-1, 5-0, 5-0) = (4,5,5). (Key point: Vector to observer)

4. V . N = 14. What is the conclusion?
   A. Back face, hidden
   B. Edge-on, invisible
   C. Needs z-buffer to decide
   D. Front face, visible
   Answer: D. V . N > 0 means the face points toward the observer. (Key point: Conclusion)

## Related in Computer Graphics

- [Back-face visibility of BED and ABCD](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aback-face-bed-abcd-observer) — Computer Graphics, chapter 7
- [Scan-line method of visible surface](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ascan-line-visible-surface) — Computer Graphics, chapter 7, asked 3×
- [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×
- [Image space versus object space methods](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aimage-space-vs-object-space) — Computer Graphics, chapter 7, asked 9×
- [Reflect triangle about 3X-4Y+8=0](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Areflect-triangle-3x-4y-8) — Computer Graphics, chapter 3, asked 2×
- [Rotate triangle 45° about (3,3)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Arotate-triangle-45-pivot-3-3) — Computer Graphics, chapter 3

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- [Resection point O inside triangle N, P, G](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aresection-nag-triangle-2078c) — Surveying II, chapter 5
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- [Singleton class and visibility types](https://www.bctengineer.com/question.md?key=semester_6%2Fobject_oriented_analysis_and_design%3Asingleton-and-visibility-types) — Object Oriented Analysis and Design, chapter 3