Zhuming Bi

Computer Aided Design and Manufacturing


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a surface cannot be described by analytic or moving curves, they are called free‐form or sculpture surfaces. Control points are used to determine the surface. The mathematic presentation of these surfaces is similar to the spline curves. The parametric surface description uses two independent variables (u, v). image

      Geometric modelling is used to create a virtual geometric representation of a real or imagined object, which includes information of the shape, dimensions, and materials of an object. Many methods have been developed to model geometrics of products. A better understanding of the theoretical basics of geometric modelling helps in (i) improving modelling efficiency and (ii) shortening the learning curves of various CAD systems. While every method has its limitations, no universal solution is available that satisfies all demands for geometric models in itself. To select a modelling method, one must ensure the validity of the geometries.

N‐dimension Manifold example Non‐manifold example
1D (line) image image The intersecting point is not locally homogeneous.
2D (surface) image image The intersecting line is not locally homogeneous.
3D (solid) image image The geometry mixes 2D and 3D entities.
c02f020

      (2.16)equation

      where F, E, V, B, L, and G are the numbers of faces, edges, vertices, bodies, inner loops on faces, and genuses in a geometry, respectively.

c02f021

      Example 2.5

Example F E V FE + V = 2
image 6 12 8 6 − 12 + 8 ≡ 2
image 7 12 7 7 − 12 + 7 ≡ 2
image 18 48 32 18 − 48 + 32 ≡ 2
image 3 3 2 3 − 3 + 2 ≡ 2
image 2 2 2 2 − 2 + 2 ≡ 2
image 2 2 2 2 − 2 + 2 ≡ 2

      Solution