@UML(identifier="GM_ParametricCurveSurface", specification=ISO_19107) public interface ParametricCurveSurface extends SurfacePatch
ParametricCurveSurface
are all continuous families of curves,
given by a constructive function of the form:
By fixing the value of either parameter, we have a oneparameter family of curves.surface
(s,t): [a,b]×[c,d] →DirectPosition
c_{t}(s) = c_{s}(t) =
surface
(s,t);
The functions on ParametricCurveSurface
shall expose these two families of curves. The
first gives us the "horizontal" cross sections c_{t}(s), the later the
"vertical" cross sections c_{s}(t). The terms "horizontal" and "vertical"
refer to the parameter space and need not be either horizontal or vertical curves in the coordinate
reference system. The table below lists some possible pairs of types for these surface curves
(other representations of these same surfaces are possible).
Surface type  Horizontal Curve type  Vertical curve type 

Cylinder 
Circle, constant radii  Line Segment 
Cone 
Circle, decreasing radii  Line Segment 
Sphere 
Circle of constant latitude  Circle of constant longitude 
BilinearGrid 
Line string  Line string 
BicubicGrid 
Cubic spline  Cubic spline 
The two partial derivatives of the surface parameterization, i and j are given by:
TODO: copy equations thereand
TODO: copy equations thereThe default upNormal for the surface shall be the vector cross product of these two curve derivatives when they are both nonzero:
k = i × jIf the coordinate reference system is 2D, then the vector k extends the local coordinate system by supplying an "upward" elevation vector. In this case the vector basis (i, j) must be a right hand system, that is to say, the oriented angle from i to j must be less than 180°. This gives a righthanded "moving frame" of local coordinate axes given by <i, j>. A moving frame is defined to be a continuous function from the geometric object to a basis for the local tangent space of that object. For curves, this is the derivative of the curve, the local tangent. For surfaces, this is a local pair of tangents. Parameterized curve surfaces have a natural moving frame and it shall be used as defined in this paragraph to define the upNormal of the surface.
NOTE: The existence of a viable moving frame is the definition of "orientable"
manifold. This is why the existence of a continuous upNormal implies
that the surface is orientable. Nonorientable surfaces, such as the MÃ¶bius band and Klein bottle
are counterintuitive. Surface
forbids their use in application schemas conforming to
the ISO 19107 standard. Klein bottles cannot even be constructed in 3D space, but require 4D
space for nonsingular representations.
Modifier and Type  Method and Description 

CurveInterpolation 
getHorizontalCurveType()
Indicates the type of surface curves used to traverse the surface horizontally
with respect to the parameter s.

CurveInterpolation 
getVerticalCurveType()
Indicates the type of surface curves used to traverse the surface vertically with
respect to the parameter t.

Curve 
horizontalCurve(double t)
Constructs a curve that traverses the surface horizontally with respect to the parameter
s.

DirectPosition 
surface(double s,
double t)
Traverses the surface both vertically and horizontally.

Curve 
verticalCurve(double s)
Constructs a curve that traverses the surface vertically with respect to the parameter
t.

getBoundary, getInterpolation, getNumDerivativesOnBoundary, getSurface
getArea, getPerimeter, getUpNormal
@UML(identifier="horizontalCurveType", obligation=MANDATORY, specification=ISO_19107) CurveInterpolation getHorizontalCurveType()
@UML(identifier="verticalCurveType", obligation=MANDATORY, specification=ISO_19107) CurveInterpolation getVerticalCurveType()
@UML(identifier="horizontalCurve", obligation=MANDATORY, specification=ISO_19107) Curve horizontalCurve(double t)
NOTE: The curve returned by this function or by the corresponding vertical curve function, are normally not part of any complex to which this surface is included. These are, in general, calculated transient values. The exceptions to this may occur at the extremes of the parameter space. The boundaries of the parameter space support for the surface map normally to the boundaries of the target surfaces.
t
 The t value to hold constant.@UML(identifier="verticalCurve", obligation=MANDATORY, specification=ISO_19107) Curve verticalCurve(double s)
s
 The s value to hold constant.@UML(identifier="surface", obligation=MANDATORY, specification=ISO_19107) DirectPosition surface(double s, double t)
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