3D Modeling

3D Modeling

3D modeling is the process of creating mathematical representations of 3D objects. Models can be defined using primitive shapes, polygon meshes, curves, or parametric surfaces.

Polygon Meshes

The most common representation. A mesh consists of vertices, edges, and faces (usually triangles or quads) that define the surface of an object.

Triangle Mesh:

  Vertex List:         Face List (indices):
  V0 = (0, 0, 0)      F0 = (0, 1, 2)
  V1 = (1, 0, 0)      F1 = (1, 2, 3)
  V2 = (0, 1, 0)
  V3 = (1, 1, 0)

       V2----V3
       |\    |     Two triangles form a quad
       | \   |
       |  \  |
       |   \ |
       V0----V1

Indexed format:
  vertices = [(0,0,0), (1,0,0), (0,1,0), (1,1,0)]
  faces = [[0,1,2], [1,2,3]]

Geometric Primitives

Common primitives:

  Cube:      8 vertices, 12 triangles (6 faces)
  Sphere:    Subdivided icosahedron or UV sphere
  Cylinder:  Circle extruded along axis
  Torus:     Donut shape (revolution surface)
  Plane:     Simple quad or grid

Normal Vectors

Normals define the direction a surface faces at each point. They are essential for lighting calculations and shading.

Face Normal (flat shading):
  n = (V1 - V0) × (V2 - V0)
  n = normalize(n)

Vertex Normal (smooth shading):
  Average of normals of adjacent faces:
  n_v = (n_f1 + n_f2 + ... + n_fn) / n
  n_v = normalize(n_v)

Example: Triangle normal
  V0 = (0,0,0), V1 = (1,0,0), V2 = (0,1,0)
  e1 = V1 - V0 = (1,0,0)
  e2 = V2 - V0 = (0,1,0)
  n = e1 × e2 = (0,0,1)

Vertex Attributes

Each vertex can store:
  - Position (x, y, z)
  - Normal (nx, ny, nz)
  - Color (r, g, b, a)
  - Texture coordinates (u, v)
  - Tangent (for normal mapping)

Vertex Buffer Layout:
  [pos.x, pos.y, pos.z, norm.x, norm.y, norm.z, u, v]
  |-------------|  |-------------|  |--------|
   Position       Normal           TexCoord