A large number of the geometric types in the ISO 19107 standard are defined parametrically, that is they are represented by functions from a set of parameters (in a parametric space, usually a subset of some Euclidean n-dimensional coordinate space) into a coordinate space of some larger dimension. The first few dimensions (up to 3) representing geographic space, the next possibly time, and any remainder representing whatever the application needs, such as distributed attributes or some other measures. The type of geometry is usually determined by the dimension of the parameter space, which will normally be equal to the topological dimension of the resulting geometry. So a 0- parameter geometric object is a point, 1-parameter geometric object is a curve, a 2-parameter geometric object is a surface, a 3-parameter geometric object is a solid.
An n-dimensional coordinate space consists of all n-long arrays of numbers; each array represents a point in the space. In particular situations, this may be restricted to a subset of such points, called the extent of validity, usually based on a set of constraints on values of the various offsets within the array. Each point is associated to a spatial or spatial-temporal location, but a single location may be the target of multiple coordinate space points. Locations given by such structures are called direct positions.
All locations in a list or array shall use the same coordinate system and
shall reference reality in a manner representable by continuous functions from the coordinate
DirectPositions) to reality in such a manner that “nearby”
coordinates in the
DirectPositions map to “nearby” positions in reality. The ISO 19107
standard does not assume that these functions maintain topological dimension. See for example
homogeneous direct position.
Interface Summary Interface Description AffinePlacementA placement defined by linear transformation from the parameter space to the target coordinate space. ArcArc of the circle determined by 3 points, starting at the first, passing through the second and terminating at the third. ArcByBulgeEquivalents to the
Arc, except the bulge representation is maintained.
ArcStringSimilar to a line string except that the interpolation is by circular arcs. ArcStringByBulgeA variant of the arc that stores the parameters of the second constructor of the component arcs and recalculates the other attributes of the standard arc. BezierPolynomial splines that use Bezier or Bernstein polynomials for interpolation purposes. BicubicGridA gridded surface that uses cubic polynomial splines as the horizontal and vertical curves. BilinearGridA gridded surface that uses line strings as the horizontal and vertical curves. BSplineCurveA piecewise parametric polynomial or rational curve described in terms of control points and basis functions. BSplineSurfaceA rational or polynomial parametric surface that is represented by control points, basis functions and possibly weights. CircleSame as an arc, but closed to form a full circle. ClothoidThe clothoid (or Cornu's spiral), a plane curve whose curvature is a fixed function of its length. Cone ConicAny general conic curve. CoordinateSystemOrganizes the manner in which the direct positions are described. CubicSplineCubic splines. CylinderA gridded surface given as a family of circles whose positions vary along a set of parallel lines, keeping the cross sectional horizontal curves of a constant shape. GenericCurve GenericSurface GeodesicTwo distinct positions joined by a geodesic curve. GeodesicStringSequence of geodesic segments. GeometryFactoryA factory of geometries. GriddedSurfaceA parametric curve surface defined from a rectangular grid in the parameter space. HomogeneousDirectPositionA direct position adding another element to the coordinate array which carries a non-zero “weight”, and multiplies all other columns in the coordinate array by that weight. KnotControls the constructive parameter space for spline curves and surfaces. LineSegment LineStringA sequence of line segments, each having a parameterization like the one
OffsetCurveA curve at a constant distance from the basis curve. Parameterization ParametricCurveSurfaceThe surface patches that make up the parametric curve surfaces. ParamForPointThe curve parameter for a point. PermutationRepresents the rearrangement of a list, or a projection. PlacementTakes a standard geometric construction and places it in geographic space. PointArrayA sequence of points. PointGridA grid of points. PolygonA surface patch that is defined by a set of boundary curves and an underlying surface to which these curves adhere. PolyhedralSurfaceA surface composed of polygon surfaces connected along their common boundary curves. PolynomialSplineA polynimal spline. SphereA gridded surface given as a family of circles whose positions vary linearly along the axis of the sphere, and whose radius varies in proportion to the cosine function of the central angle. SplineCurveRoot for subtypes of curve segment using some version of spline, either polynomial or rational functions. TinA triangulated surface that uses the Delaunay algorithm or a similar algorithm complemented with consideration for breaklines, stoplines and maximum length of triangle sides. TriangleA planar polygon defined by 3 corners. TriangulatedSurfaceA polyhedral surface that is composed only of triangles.
Class Summary Class Description BSplineSurfaceFormIndicates a particular geometric form represented by a
HandedLabels coordinate systems as being right or left handed as commonly defined in Mathematics. KnotTypeThe type of a B-spline. SplineCurveFormIndicates which sort of curve may be approximated by a particular B-spline.