Freeform Surfaces
et al. 2007] Another interesting type of ruled surfaces are developable surfaces which bear special potential in architectural applications. [Pottmann et al. 2008] Developable surfaces will be discussed in detail in chapter 5. r
3.1.4 Pipe Surfaces
c
A pipe surface is the envelope of spheres of equal radius r whose centres lie on a curve c, called the spine curve or central curve c. The pipe surface can also be seen as a family of circles with the radius r lying in the normal planes of a spatial curve. [Pottmann et al. 2007] c
3.1.5 Offset Surfaces An offset surface Sd of the surface S is the surFigure 3.4: A pipe surface as (a) an envelope of
face with a constant normal distance d to the spheres, and (b) a collection of circles lying normal to original surface S. The offset surface Sd and the
the curve c.
surface S share their normals. Further, the tangent planes of S and Sd in corresponding points are parallel, therefore offset surfaces are also called parallel surfaces. [Pottmann et al. 2007]
3.2 Freeform Surfaces Rotational-, translational-, ruled-, helix-, pipe d
surfaces and surface offsets are not closely sufficient to meet the high demands of today’s
S Sd
state-of-the-art architectural designs. More complex shapes are nowadays designed using freeform surfaces because those surfaces offer more flexibility compared to the traditional
Figure 3.5: The offset surface Sd lies at constant distance d to the surface S.
surfaces. We will discuss three types of surfaces 17