---
title: "Geometry and topology; Euler check of solid"
subject: "Computer Aided Design"
chapter: 2
key: "semester_4/computer_aided_design:geometry-topology-euler"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/geometry-topology-euler
site: BCT Engineering
---

# Geometry and topology; Euler check of solid

Computer Aided Design · Chapter 2: Geometric Modeling Fundamentals

- Asked in: year not given, 6 marks
- Video: [S 3.05 Numerical on Boundary Representation to check Validity of Geometry/model by Euler's Equation](https://www.youtube.com/watch?v=wqv3g_J2VEE) — Dr. Khizar Ahmed Pathan
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/geometry-topology-euler
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-2

## Question

Differentiate between geometry and topology in solid modelling. A rectangular block has a square hole drilled completely through it, from the top face to the bottom face. It has 16 vertices, 24 edges and 10 faces. Use the Euler-Poincare formula to find the number of rings (inner loops) and check that the solid is valid.

## Short answer

Geometry carries metric information such as coordinates, lengths and radii; topology carries connectivity among vertices, edges and faces. For V=16, E=24, F=10, S=1 and H=1, Euler-Poincare gives 16-24+10-R=0, so R=2 inner loops; the solid is valid.

## Answer

#### Geometry and topology
**Geometry** is the metric information of the model: actual coordinates of points, equations of curves and surfaces, lengths, angles, radii. **Topology** is the connectivity information: how vertices, edges and faces are joined, and which side of a face is material. It does not change when the shape is stretched.

| Point | Geometry | Topology |
|---|---|---|
| Describes | Size and shape | Connectivity and adjacency |
| Examples | Coordinates, line equation, radius | Vertex-edge-face relations, loops, shells |
| Under scaling | Changes | Unchanged |
| Used for | Display, measurement | Validity checks, B-rep data structure |

#### Euler-Poincare formula
For a valid solid with holes (genus) and inner loops:

$$
V - E + F - R = 2\,(S - H)
$$

where $V$ = vertices, $E$ = edges, $F$ = faces, $R$ = rings (inner loops on faces), $S$ = shells, $H$ = through-holes (genus).

#### Numerical
Given: $V = 16$, $E = 24$, $F = 10$, one shell $S=1$, one through-hole $H=1$.

$$
\begin{aligned}
16 - 24 + 10 - R &= 2(1-1) = 0 \\
2 - R &= 0 \\
R &= 2
\end{aligned}
$$

The two rings are the square hole boundaries on the top face and on the bottom face. Each of these faces has an outer loop and one inner loop, so $R = 2$, which agrees with the physical block. The formula is satisfied, so the topology is valid.

**Answer: $R = 2$ inner loops (top and bottom faces); $V-E+F-R = 0 = 2(S-H)$, so the solid is valid.**

## Check yourself

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

1. Which statement about topology is correct?
   A. It stores coordinates and radii
   B. It describes connectivity, such as how vertices, edges and faces are joined
   C. It changes when the object is scaled
   D. It is the same as geometry
   Answer: B. Topology is connectivity information and does not change when the shape is stretched or scaled. (Key point: Geometry vs topology)

2. Which form of the Euler-Poincare formula does the answer use?
   A. V - E + F = 2
   B. V + E - F + R = 2(S + H)
   C. V - E + F - R = 2(S - H)
   D. V - E - F + R = S - H
   Answer: C. For solids with holes and rings: V - E + F - R = 2(S - H). (Key point: Euler-Poincare formula)

3. For V=16, E=24, F=10, S=1 and H=1, what is R?
   A. 0
   B. 1
   C. 4
   D. 2
   Answer: D. 16 - 24 + 10 - R = 0 gives 2 - R = 0, so R = 2. (Key point: Rings calculation)

4. Which are the two rings in the drilled block?
   A. The hole boundaries on the top and bottom faces
   B. The four side faces
   C. Two shells
   D. The outer edges of top and bottom
   Answer: A. Each of the top and bottom faces has an outer loop and one inner loop around the hole, so R = 2. (Key point: Physical meaning of R)

## Related in Computer Aided Design

- [Wireframe, surface and solid modelling compared](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Awireframe-surface-solid-modeling) — Computer Aided Design, chapter 2

## Related in other subjects

- [Subnet 10.10.10.0/24 for 5, 16, 23, 27](https://www.bctengineer.com/question.md?key=semester_7%2Fcomputer_networks_and_security%3Asubnetting-10-10-10-0) — Computer Networks and Security, chapter 4, asked 2×
- [3 m wall under wind: fixed top and bottom](https://www.bctengineer.com/question.md?key=semester_5%2Fconcrete_technology_and_masonry_structure%3Awind-wall-3m-fixed-top-bottom) — Concrete Technology and Masonry Structure, chapter 9, asked 2×
- [3 m wall under wind: free top, fixed bottom](https://www.bctengineer.com/question.md?key=semester_5%2Fconcrete_technology_and_masonry_structure%3Awind-wall-3m-free-top) — Concrete Technology and Masonry Structure, chapter 9, asked 2×
- [Completely mixed ASP reactor, 10,000 m³/day](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Aasp-completely-mixed-10000) — Sanitary Engineering, chapter 7