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Superquadrics

Superquadrics[Barr, 1981, Loeffelmann and Groeller, 1994] are recent geometric shapes with useful properties for computer graphics. Superquadrics are an extension of the basic quadratic surfaces defined as a spherical product of two parametric 2D curves, resulting in a parametric 3D shape. Implicit equation for Piet-Hein superelipsis is
equation46
and parametric set of equations is
 equation52
where tex2html_wrap_inline335 is the east-west or roundness/squareness parameter.

  figure55
Figure 2: Examples of superquadric elipsoids

Extending superelipses into 3D space, divides parametric equations into family of superquadric elipsoids (fig. 2) and superquadric toroids. For elipsoid family the parametric equations (3) are defined as:
 eqnarray67
and similarly for superquadric toroids
 eqnarray70
Sub functions used in the above equations (3 and 4) are defined as:
 eqnarray75
tex2html_wrap_inline337 is the hole diameter used only for toroid shapes. tex2html_wrap_inline339 are scale factors in each dimension. Parameter tex2html_wrap_inline335 in Eq.(2) is divided into two parameters e and n. It is evident, that the equations (3) and (4) can be combined together to form one set of equations.

Several other properties such as normal vectors, volume, mass, and inertia tensor can be easily derived for superquadrics. For detailed description of superquadric properties see [Barr, 1992].



Leon Kos
Wed May 27 11:00:46 CEST 1998