7.3 Parametric Points and Lines

Topic Version	1	Published	09/11/2015	Topic Change History
For Standard	RESQML v2.0.1

A parametric line is essentially a vector-valued function. Given a parametric value P and the specification of a parametric line, what are the resulting values of X(P), Y(P), and Z(P) (or T(P)) in the Local 3D CRS?

This is a parametric description of the points on the line. The use of a parametric line provides a more compact means of specifying point geometry. As importantly, it provides interpolated geometric information at higher resolution than the spacing of the control points on a piecewise linear line, which supports more accurate intersection calculations and geometric refinement.

Table 7.3-1 lists and describes the available types of parametric lines and the control points needed for each type of line.

Table 7.3-1 Description of RESQML Parametric Lines
Parametric Line	No. of Control Point(s)	Description
Vertical	One control point: (X,Y,-)	Parametric value is Z => (X,Y,Z).
Linear Spline1 or more intervals	Two or more knots, each with a control point: (P,X,Y,Z)i, i=1,2,…	Piecewise linear interpolation in (X,Y,Z) as a function of the parametric value P.
Natural Cubic Spline1 or more intervals	Two or more knots, each with a control point: (P,X,Y,Z)i, i=1,2,…	Piecewise cubic interpolation in (X,Y,Z) as a function of the parametric value P, subject to the additional constraints of continuous first and second derivatives at the knots, and vanishing second derivative at the edge knots.
Cubic Spline1 or more intervals	Two or more knots, each with a control point and a tangent vector. Tangent vectors are defined as the derivative of position with respect to the parameter P.	Piecewise cubic interpolation in (X,Y,Z) as a function of the parametric value P, given the additional constraint of specified tangent vectors at the knots.
Z Linear Cubic Spline1 or more intervals	Two or more knots, each with a control point: (P,X,Y,Z)i, i=1,2,…	Linear spline interpolation in Z as a function of the parametric value P. Natural cubic spline interpolation in (X,Y) as a function of the parametric value P.
Minimum-Curvature Spline1 or more intervals(knots = stations)	One control point for the first knot. Two or more knots, each with a tangent vector. Tangent vectors are defined as the derivative of position with respect to the parameter P.	Piecewise minimum curvature interpolation in (X,Y,Z) as a function of the parametric value P, given the constraint of specified tangent vectors at the knots. Curvature is defined in the units of measure of the local 3d CRS, without unit conversion, so care must be taken when using mixed units.

With the exception of the vertical parametric line, the parameter itself has no specific interpretation, nor is there any requirement that the same parametric values be used along different lines in a parametric line array. However, it is important that the parametric values associated with the control points and the parametric values used for interpolation are consistent. When used to describe the geometry of a wellbore trajectory, the parameter is interpreted as the measured depth along the wellbore. In this case, the specification of the measured depth units of measure and datum location are part of the wellbore trajectory representation, and not the geometry itself.

NOTE: As a companion to this documentation on cubic splines, a spreadsheet (titled 20131223 RESQML Cubic Splines.xslx, included in the download and available here: https://docs.energistics.org/#EO_Resources/RESQML_Cubic_Splines.xlsx) allows you to see the equations described here and experiment with different input values and examine the results.