---
title: "Model curved surface using polygons"
subject: "Computer Graphics"
chapter: 6
key: "semester_5/computer_graphics:curved-surface-polygons-tables"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/curved-surface-polygons-tables
site: BCT Engineering
---

# Model curved surface using polygons

Computer Graphics · Chapter 6: Surface Modeling

- Asked in: 2069 Chaitra, 3+5 marks
- Video: [Polygon Surface in Computer Graphics (3D Geometry) ~xRay Pixy](https://www.youtube.com/watch?v=wQRgcCGkIhc) — Ritika xRay Pixy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/curved-surface-polygons-tables
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-6

## Question

How can you model a curved surface using polygons only? Explain the use of polygon tables for boundary representations.

## Short answer

Curved surfaces are approximated by polygon meshes using many small flat polygons whose vertices lie on the true surface (e.g., cylinder with rectangles and n-gons). Boundary representation uses polygon tables: vertex table stores coordinates, edge table stores vertex pairs and adjacent surfaces, surface table stores edge lists, enabling compact storage and efficient visibility, shading, and transformation.

## Answer

#### Modeling a curved surface using polygons
A curved surface (sphere, cylinder, cone) is approximated by a **polygon mesh**: the surface is divided into many small, flat polygons (triangles or quadrilaterals) whose vertices lie on the true surface. The more and smaller the polygons, the closer the approximation. A sphere, for example, is cut by latitude and longitude lines into quadrilaterals, with triangles at the poles. A cylinder is made of rectangles around the side and two n-sided polygons for the top and bottom.

```text
Cylinder (8 sides):
   ___________
  /  /  /  /  /\      top polygon
 |  |  |  |  |  |     rectangles
  \__\__\__\__\/      bottom polygon
```

The flat appearance can be hidden by smooth shading (Gouraud or Phong), which makes the facets look curved.

#### Polygon tables for boundary representation
The boundary representation of the mesh is stored in tables:
- **Vertex table:** (x, y, z) of every vertex, stored once and shared.
- **Edge table:** the two vertices of each edge.
- **Surface table:** the edges (or vertices) of each polygon.
- **Attribute table:** colour, texture, transparency, and the plane equation coefficients.

Example for two triangles sharing an edge:

```text
V: V1 V2 V3 V4
E: E1 (V1 V2)  E2 (V2 V3)  E3 (V3 V1)  E4 (V2 V4)  E5 (V4 V3)
S: S1 = E1 E2 E3     S2 = E2 E4 E5
```

The tables give compact storage, fast transformation (only vertices are transformed), and fast checks of consistency and visibility.

## Check yourself

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

1. A curved surface is modelled with polygons by:
   A. Using one large polygon
   B. Dividing it into many small flat polygons with vertices on the surface
   C. Using only lines
   D. Removing curvature
   Answer: B. More polygons give closer approximation. (Key point: Approximation)

2. How can the flat look of facets be hidden?
   A. Larger polygons
   B. Removing normals
   C. Wireframe only
   D. Gouraud or Phong shading
   Answer: D. Shading interpolation makes facets appear curved. (Key point: Smoothness)

3. A sphere is commonly meshed using:
   A. Only cubes
   B. One circle
   C. Quadrilaterals from latitude/longitude lines, triangles at poles
   D. Random polygons
   Answer: C. Latitude and longitude lines cut it. (Key point: Sphere)

4. Why are vertices stored once in a vertex table?
   A. To store colour
   B. To avoid edges
   C. Shared vertices are transformed once, giving compact storage and fast transformation
   D. To make polygons non-planar
   Answer: C. Only vertices are transformed. (Key point: Tables)

## Related in Computer Graphics

- [Polygon tables to model Dharahara](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Apolygon-vertex-edge-table-dharahara) — Computer Graphics, chapter 6
- [Spatial orientation and polygon tables](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aspatial-orientation-surface-significance) — Computer Graphics, chapter 6
- [Wire-frame model and polygon table](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Awireframe-model-polygon-table) — Computer Graphics, chapter 6
- [Spatial orientation / normal of a polygon](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aspatial-orientation-polygon-calculation) — Computer Graphics, chapter 6, asked 5×
- [Polygon table and normal of S1](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Apolygon-table-surface-normal-s1) — Computer Graphics, chapter 6
- [Polygon as standard graphics object](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Apolygon-standard-graphics-object) — Computer Graphics, chapter 6

## Related in other subjects

- [Hydrostatic force on curved surface](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Ahydrostatic-force-curved-surface) — Hydro Power, chapter 1
- [Resultant force on curved surface BCDEF](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acurved-surface-bcdef-force) — Fluid Mechanics, chapter 3
- [Resultant force on curved surface ABCDE (S = 1.1)](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acurved-surface-abcde-s11-20m) — Fluid Mechanics, chapter 3
- [Resultant force on curved surface ABCDE (S = 1.25)](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acurved-surface-abcde-s125-10m) — Fluid Mechanics, chapter 3