Spline interpolation¶
The spline command performs scalar or parametric spline interpolation on data stored in Rbc vectors.
Two interpolation algorithms are available:
natural- natural cubic spline interpolation.quadratic- shape-preserving quadratic spline interpolation.
The command provides two interfaces.
Scalar spline interface¶
The traditional scalar interface treats Y as a function of X:
spline natural x y splX splY
spline quadratic x y splX splY
All four vectors must be real vectors.
x and y contain the source points. splX contains the X coordinates at which the spline is evaluated and
splY receives the corresponding calculated Y values.
Parametric complex interface¶
A two-dimensional parametric spline may instead be generated directly from a complex vector:
spline parametric natural source result samples
spline parametric quadratic source result samples
source and result are complex Rbc vectors. Each source value:
{real imag}
represents one point in the two-dimensional plane.
The parametric form does not require either the real or imaginary coordinate to be monotonic. It can therefore interpolate vertical paths, paths that reverse direction, and loop-like trajectories.
The source points are parameterized by cumulative Euclidean chord length:
t[0] = 0
t[i] = t[i-1] + hypot(real[i]-real[i-1], imag[i]-imag[i-1])
The requested number of samples is distributed uniformly over that cumulative parameter.
The first generated point is exactly the first source point and the final generated point is exactly the final source point.
Parametric interpolation does not implicitly close a path. Repeat the first point as the final source point if a closed trace is required.
catrom is not currently exposed through the standalone spline parametric command. Catmull-Rom parametric
interpolation is available as a graph-element -smooth catrom mode.
Scalar source data¶
Scalar source vectors x and y must have the same length and contain at least three values. All source X and
Y values must be finite.
Values in x must be strictly increasing. Duplicate or decreasing source X coordinates are not permitted.
For example:
vector create x y
x set {0 1 2 3 4}
y set {0 2 1 4 3}
If source data are not already ordered by X coordinate, the vector sort operation can reorder x while
applying the same permutation to y:
x sort y
Scalar evaluation coordinates¶
splX specifies the X coordinates at which a scalar spline is evaluated. Every value must be finite.
splX does not have to be sorted. Evaluation coordinates may appear in any order and splY is produced in
the corresponding order.
For example:
splX set {2.5 0.5 4.5 -1.0 1.5}
spline natural x y splX splY
Evaluation coordinates may lie outside the source X range. Both scalar spline methods extrapolate:
values below the first source X coordinate use the first spline interval;
values above the final source X coordinate use the final spline interval.
No separate extrapolation option is required.
Generating scalar evaluation coordinates¶
VECINST populate is convenient for constructing a denser real evaluation vector:
x populate splX 10
spline natural x y splX splY
populate is not required; splX may contain any finite coordinates appropriate for the application.
Parametric source data¶
A parametric source must be a complex Rbc vector containing at least three points.
All real and imaginary components must be finite.
Consecutive source points must be distinct. A repeated consecutive point has zero chord length and is rejected:
vector create z -type complex
z set {{0 0} {1 1} {1 1} {2 0}}
spline parametric natural z result 100
# error: consecutive duplicate points
Non-consecutive repeated points are permitted. In particular, repeating the first point at the end is a normal way to describe a closed geometric path.
The requested sample count must be at least two.
Parametric natural cubic spline¶
The parametric natural method constructs a natural cubic two-dimensional spline using chord length as the independent parameter.
Distances in the standalone complex-vector command are isotropic: one real-coordinate unit and one imaginary-coordinate unit have equal weight.
For example:
vector create z -type complex
z set {{0 0} {1 1} {0 2} {-1 1} {0 0}}
spline parametric natural z smoothZ 200
The source real coordinate is intentionally non-monotonic; this is valid for the parametric interface.
Parametric quadratic spline¶
The parametric quadratic method uses the same cumulative chord-length parameter for both coordinates. It applies the shape-preserving quadratic spline independently to:
real = f(t)
imag = g(t)
and combines the two results into complex output points.
For example:
spline parametric quadratic z smoothZ 200
Output vectors and transaction behavior¶
In the scalar interface, splY has the same length as splX.
In the parametric interface, result is a complex vector with exactly samples values.
If a result vector does not exist, it is created with the required type. If it already exists, its type must match the requested operation and it is resized when necessary.
Spline generation is transactional. Source data and parameters are validated and the complete result is calculated before an existing result vector is resized or modified. A failed command leaves an existing result vector unchanged.
Input data are copied before the result vector is modified, so output aliasing is supported.
Scalar output may safely be the same vector as x, y, or splX.
Parametric output may safely be the same complex vector as source:
spline parametric natural z z 200
Scalar natural cubic spline¶
A scalar natural spline is a piecewise cubic interpolating function that passes through every source data point. Its second derivative is zero at the first and last source points, providing the natural boundary conditions.
Outside the source range, the cubic polynomial belonging to the first or last source interval is continued beyond its endpoint.
Natural cubic splines generally provide continuous first and second derivatives.
Scalar shape-preserving quadratic spline¶
The scalar quadratic method generates an osculatory piecewise quadratic spline. Slopes at the source points are selected to preserve monotonicity and convexity where those properties are compatible with the source data.
Outside the source range, the spline belonging to the first or last source interval is continued beyond its endpoint.
This method can be useful when preserving the qualitative shape of source data is more important than the additional smoothness of a cubic spline.
Errors¶
Scalar spline generation fails if the source vectors contain fewer than three values, have different lengths, contain non-finite values, or have source X coordinates that are not strictly increasing. It also fails if an evaluation X coordinate is non-finite.
Parametric spline generation fails if the source is not a complex vector, contains fewer than three points,
contains a non-finite real or imaginary component, contains consecutive duplicate points, or if samples is
less than two. An existing result vector must also be complex.
Both forms fail if required temporary storage cannot be allocated or the spline calculation produces an invalid numerical result.
An error never partially modifies an existing output vector.
Graph smoothing¶
The standalone scalar and parametric spline commands operate on vector coordinates.
Graph element smoothing is performed after graph-coordinate mapping and should therefore not be assumed to produce byte-for-byte identical points to a precomputed standalone spline.
Polar elements use parametric natural/cubic, quadratic, and catrom smoothing so that arbitrary
two-dimensional complex-plane trajectories can be displayed.
References¶
The natural cubic spline implementation is based on:
Burden, J. D. Faires and A. C. Reynolds, Numerical Analysis, Prindle, Weber & Schmidt, 1981.
The shape-preserving quadratic spline implementation is based on:
McAllister and J. A. Roulier, An Algorithm for Computing a Shape-Preserving Osculatory Quadratic Spline, ACM Transactions on Mathematical Software, Algorithm 574.
Copyright (c) George Yashin