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Model curved surface using polygons

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.

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

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.

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:

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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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