---
title: "Boundary representation of 3D objects"
subject: "Computer Graphics"
chapter: 6
key: "semester_5/computer_graphics:boundary-representation-technique"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/boundary-representation-technique
site: BCT Engineering
---

# Boundary representation of 3D objects

Computer Graphics · Chapter 6: Surface Modeling

- Asked in: 2073 Shrawan, 8 marks; 2069 Asar, 8 marks; 2071 Chaitra, 8 marks; 2075 Chaitra, 3 marks (asked 4 times)
- Video: [Boundary Representation (B-Rep) - Techniques For Geometric Modeling - CAD/CAM/CAE](https://www.youtube.com/watch?v=aafsBz50O-Q) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/boundary-representation-technique
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-6

## Question

Explain boundary representation technique to represent three dimensional objects with suitable example.

## Short answer

B-rep describes a solid by its bounding faces, edges, vertices. Data in tables: vertex coordinates, edges as vertex pairs, faces as ordered edge lists. Example: pyramid with square base and apex. Each edge belongs to two faces for closed solids. Enables efficient rendering.

## Answer

**Boundary representation (B-rep)** describes a 3D solid by its surface: the set of faces, edges and vertices that separate the interior of the object from the outside. The most common form is the polygon mesh, where each face is a planar polygon.

The data are held in tables:
- **Vertex table:** coordinates of every vertex.
- **Edge table:** each edge as a pair of vertices.
- **Surface table:** each face as a list of edges.

**Example: a pyramid** with square base V1 V2 V3 V4 and apex V5

```text
 Top view (apex V5 is above the centre of the base)
   V4 -------- V3
   |  \      /  |
   |    \  /    |
   |     V5     |
   |    /  \    |
   |  /      \  |
   V1 -------- V2
```

```text
Vertex table       Edge table       Surface table
V1 (0,0,0)         E1: V1 V2        S1 base: E1 E2 E3 E4
V2 (2,0,0)         E2: V2 V3        S2: E1 E6 E5   (V1 V2 V5)
V3 (2,2,0)         E3: V3 V4        S3: E2 E7 E6   (V2 V3 V5)
V4 (0,2,0)         E4: V4 V1        S4: E3 E8 E7   (V3 V4 V5)
V5 (1,1,2)         E5: V1 V5        S5: E4 E5 E8   (V4 V1 V5)
                   E6: V2 V5
                   E7: V3 V5
                   E8: V4 V5
```

Features:
- Easy to render, since most graphics hardware works on polygons.
- Curved surfaces are approximated by many small polygons or by spline patches.
- A valid solid must be closed: each edge is shared by exactly two faces.
- Only the surface is stored, so volume properties (mass, interior) are not directly available, unlike in CSG or octree representations.

## Check yourself

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

1. Boundary representation describes a 3D solid by:
   A. Dividing space into voxels
   B. Combining primitive solids with set operations
   C. Its volume only
   D. The surfaces (faces, edges, vertices) separating its interior from outside
   Answer: D. B-rep stores the boundary. (Key point: B-rep definition)

2. In a valid closed B-rep polygon mesh, each edge is shared by:
   A. One face
   B. Exactly two faces
   C. Three faces
   D. No face
   Answer: B. A closed solid has no open edges. (Key point: Valid solid)

3. A square pyramid has how many vertices, edges and faces?
   A. 5, 8, 5
   B. 5, 5, 8
   C. 4, 8, 5
   D. 5, 8, 4
   Answer: A. 4 base + apex = 5 vertices; 4 base + 4 slant = 8 edges; base + 4 sides = 5 faces. (Key point: Pyramid)

4. A limitation of B-rep is:
   A. It cannot be rendered fast
   B. It stores too little geometry for display
   C. Interior and volume properties are not directly available
   D. It works only for spheres
   Answer: C. Only the surface is stored. (Key point: Limitation)

## Related in Computer Graphics

- [Polygon meshes for 3D objects](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Apolygon-meshes-3d-objects) — Computer Graphics, chapter 6
- [Geometric tables for 3D object](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ageometric-tables-3d-error-free) — Computer Graphics, chapter 6
- [3D shearing and matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Awhat-is-3d-shearing) — Computer Graphics, chapter 4, asked 3×
- [3D world to viewing coordinate matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aworld-to-viewing-3d-matrix) — Computer Graphics, chapter 4, asked 2×
- [3D scaling about arbitrary point](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ascale-3d-arbitrary-point) — Computer Graphics, chapter 4
- [OpenGL syntax: 3D vertex](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aopengl-3d-float-vertex-syntax) — Computer Graphics, chapter 9

## Related in other subjects

- [Mobile boundary design: three approaches](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Adesign-three-methods-mobile-boundary) — Hydraulics, chapter 10
- [Parametric and non-parametric curve representation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Aparametric-vs-nonparametric-curves) — Computer Aided Design, chapter 2
- [Steps for free body diagram; equilibrium 2D/3D](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Afbd-steps-equilibrium-2d-3d) — Applied Mechanics, chapter 2, asked 2×
- [Polynomial representation and addition algorithm](https://www.bctengineer.com/question.md?key=semester_4%2Fdata_structure_and_algorithm%3Apolynomial-representation-addition-algorithm) — Data Structure and Algorithm, chapter 4