Computational Graphics and Modeling

Parametric modeling is a modeling technology that employs parametric equations to represent geometric curves, surfaces, and solids. From the reverse engineering perspective, parametric equations are a set of mathematics equations that explicitly express the geometric parameters, such as the X and y locations of a circle in a Cartesian coordinate. Equation 2.2a to c is a set of example parametric equations of a circle, where r is the radius of the circle and 0 is the measurement of the angle from the zero reference. 

A quadratic surface is a second-order algebraic surface that can be represented by a general polynominal equation, as described by Equation 2.3, with the highest exponent power up to 2. 

Many common geometric surfaces, such as sphere, cone, elliptic cylinder, and paraboloid, can be represented as quadric surfaces. The quadric surfaces have been employed by engineers for solid model generation from measured point data (Chivate and Jablokow, 1993), and reverse engineering physical modeling (Weir et al., 1996). 

In the 1990s, various techniques were developed to reconstruct implicit surfaces from laser data and other mathematical approaches. Implicit surfaces are two-dimensional, infinitesimally thin geometric contours that exist in three-dimensional space. They are defined by a mathematical function of specific measurable quantity, such as distance. This quantity varies within the space but is constant along the surface. For example, a spherical surface can be represented by an implicit function as 

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