Curves on surfaces

S. Angenent. Parabolic equations for curves on surfaces. I Curves with p-integrable curvature. Ann. Math., 132 451-483, 1990. 

Curves on existing surfaces serve as inputs for a variety of construction activities. Any point of a curve on a surface lies in the surface and coincides with a point of the surface. Curves on surfaces and other curves defined in relation to surfaces are as follows (Figure 3-36). 

Continuity is selected between surfaces across their common boundaries. This default continuity between the network based surfaces and with the surfaces from which the curves on surfaces, isoparametric curves, and boundary curves are extracted as curve network elements is enforced by the surface creation procedure. Moreover, local continuity can be defined across individual curves. Continuity at a boundary curve is dependent on other curvature definitions along the surfaces connected by it. In Figure 7-46, continuity G1 is specified for boundary edge eh,. This continuity can be achieved only if the continuity across pairs of curves from the two surfaces (ehy-eh ) incident along this curve is at least G2. At the same time, this condition would be good for continuity G2 across eh,. 

The above rules apply correspondingly to curves on surfaces. 

The optional curves on-Surfaces correspond to the bounding parameter values. They are the best available approximations. The receiving system will have to check whether they can be used immediately or whether they have to be enhanced. 

SURFACE, CURVE(DIM), POINT ON SURFACE, CURVE ON SURFACE ) ... 

The surface curve is a single scoped entity which contains in its scope the complete data structure that defines the curve geometry, hence, in wireframe models the surface-curve behaves as a single three-dimensional curve entity (see "Points and curves" on page 56 and "Geometry on surfaces"). The curve attribute refers to the top of that data structure. The surface entities which are referred from within the curve on surface entities may lie within the scope of the same surface-curve or outside. 

This is the class of elementary, trimmed, and composite curves on surfaces. 

The class of elementary curves on surfaces corresponds to the class of elementary curves. The difference is that the "on surface"-curves are defined in the (uv)-space spanned by the (uv)-parameters of the surface to which these curves belong. 

ENTITY B SPLINE CURVE ON SURFACE = STRUCTURE surface rational uniform bezier degree... 

REF ANY(RECTANGULAR SURFACE) REF.ANY(B SPLINE CURVE ON SURFACE) REAL ... 

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