Unformatted text preview: n the intended functionalities, geometrical specifications and physical measurements. 1.2. Classification of Euclidean surfaces A GPS mathematical concept is the classification of three-dimensional surfaces based on their invariance properties [Srinivasan, 1999]. It relies on the definition of connected Lie subgroups of T(3)×SO(3) - the group of rigid motions - where T(3) [SO(3)] denotes the group of translations [rotations] in ℜ3, respectively. Actually only seven subgroups do leave invariant some proper subset of ℜ3, thus obtaining the classes listed in Table I. Class Ci C1 C2 C3 C4 C5 C6 C7 Surface S ⊆ ℜ3 Sphere Cylinder Plane Helical Revolution Rigid motion group Gi SO(3) T(1)xSO(1) T(2)xSO(1) T(1)xSO(1) pitch ε SO(1) Reference element Point Straight line Plane Helix Parameter Radius (ro) Radius (ro), Height (z) Length (x), Wide (y), Height (z) Radius (ro), Height (z), Pitch (ε) Profile (ro(z)), Height (z) Point, Straight line Straight line, Profile (x,y), Height (z) Prismatic T(1) Plane Point, Straight None Complex I3 line, Pla...
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- The Land, ........., Computer-aided design, Joseph K. Davidson, M. GIORDANO