CFD Utilities Files

CURVATURE3D Description:

Purpose:

   CURVATURE3D serves two purposes:  The first purpose is to estimate
the curvature, k(s), at each point along a given curve in 3-space
using parametric methods, where s is the arc length along the curve.
Four methods are provided for estimating the necessary derivatives:
the finite-differencing method, the conventional cubic spline,
the quasi-Hermite cubic spline, and the monotonic cubic spline.  Both
the quasi-Hermite and monotonic splines prove poor as their lack of
continuous 2nd derivatives would suggest, but they are retained for
illustrative purposes.  The results are saved in QPLOTable form:

             y vs. x,   z vs. x,   z vs. y,
             s vs. x,   s vs. y,   s vs. z,   or
             k vs. s,

any of which may be suppressed.

   The second purpose of CURVATURE3D is to serve as a driving program
for the subroutine CURVDIS3D, which is presented as an alternative to
estimating and plotting local curvatures.  This option redistributes
the points of the input curve with local curve spacing more or less
inversely proportional to the local curvature for any specified number
of output points.  Three scaling options are provided since curvature
magnitudes are sensitive to the units and CURVDIS3D expects data of
order 1.

   This program began as a pedagogical exercise practicing Sterling
Software's programming style.  It has evolved into a convenient utility
for displaying curvature or redistributing points for (say) airframe
sections.

Method:

   The program reads in points from a file with a "SMOOTH2D" format:
[title and number of points in records 1 and 2 now optional, followed by]
columns of (x,y,z) data; any further columns are ignored.

   As the first step in displaying curvature, "PLFIT" computes the
cumulative chord lengths, which serve as the parametric variable, s.
Derivatives with respect to s may be estimated in several ways.  For
the conventional cubic spline option, "CSFIT" calculates the spline
coefficients - first for x vs. s, second for y vs. s, and then z vs. s.
After each "CSFIT" call, "CSDVAL" computes the 1st and 2nd derivatives
(dx/ds, d2x/ds2, dy/ds, d2y/ds2, dz/ds, and d2z/ds2) at each point.
The magnitude of the local curvature is then determined from:

               |k(s)| = | X' *  X" | / | X'| ** 3

where:       X' = (x', y', z') and X" = (x", y", z"),

                x' = dx / ds,     x" = d2x / ds2
                y' = dy / ds,     y" = d2y / ds2
                z' = dz / ds,     z' = d2z / ds2


For the quasi-Hermite spline fit, the same steps are taken with "QHSFIT"
used instead of "CSFIT."  And for the monotonic cubic spline, "MSFIT" is
used in place of "CSFIT."  For the finite differencing method, values of
x vs. s, y vs. s and z vs. s are given to "FD12K", which generates the
derivatives directly.  Then k(s) is evaluated as before.

   To redistribute points according to the local curvature, the user
may first normalize the input coordinates such that: 1) x, y and z are
in the range [0,1],  2) the coordinates are around the range [0,1], but
the geometric shape is retained, or 3) the coordinates are normalized
in precisely the same way as data from another geometric curve.  Or, the
user may elect not to rescale the data at all.   After the optional
rescaling, "CURVDIS3D" distributes the specified number of points along
the curve, with a fractional exponent provided to vary the clustering
effects.  Results are scaled back to the original units if necessary and
saved in plottable form.

History:

   08/16/88   BAN/DAS   Initial design & implementation of CURVATURE
                        for XY data
   09/14/88   BAN       Installed subroutine CURVDIS
   11/02/88   DAS       A few refinements in readiness for an XYZ analog
   12/02/88   MDW       Modularized curve normalization procedures
   12/08/88   MDW       Adapted CURVATURE3D from 2D program CURVATURE
   08/21/91   PJT       Increased MXPTS from 350 to 4000.  First
                        tabulation of redistributed points now contains
                        all three coordinates.
   08/27/91   DAS       Installed CURV3D in place of in-line code.
   10/30/91   DAS       CURV3D now has the loop over N inside it.
                        The PLFIT call had NDIM=2 where it should be 3.
                        Replaced IQHSCU with QHSFIT.
   09/02/08   DAS       Carriage control is a thing of the past now.
                        Make the header records optional.
   03/06/13   DAS       8 (not 7) significant digits match CAPSULE_GRID.

Authors:  Michael Wong, NASA Ames/Sterling Software, Palo Alto, CA.
          Brian Nishida
          David Saunders


Final Note:

   Finite difference derivatives are recommended for most purposes.  The
local spline second derivatives are not guaranteed to be continuous across
the data points, and are an option for illustrative purposes only.