Definitions
A,
B,
C,
D,
E,
F,
G,
H,
I,
J,
K,
L,
M,
N,
O,
P,
Q,
R,
S,
T,
U,
V,
W,
X,
Y,
Z
- algebraic surface
- A surface created from algebraic functions,
such as polynomials, cubic splines, and so
forth.
- approximation
- A process for generating estimated functional values at
arbitrary locations beginning with a set of known functional
values. At the known values, the approximated values need not
equal the known values. This distinguishes an approximation
from an interpolation.
- aspect
- At a given point on an
interpolated surface, the
aspect at that point is the direction of steepest descent.
Aspects are angles measured clockwise from north.
- basis
- In the context of cubic splines, a basis is a set of
cubic polynomials each of which is
non-zero on a finite domain and such that any cubic spline
can be calculated as a linear combination of cubics in the
basis. Any cubic polynomial in the basis is referred to as
a basis function.
- bounding polygon
- The bounding polygon of a set of points in the plane is the
polygon of smallest area that contains all of the given
points.
- circumcircle
- A circumcircle is any one of the circles in a set of
circles satisfying the
empty circumcircle
criterion.
- collinear
- Coordinates in 2-space or 3-space are collinear if they
lie on a straight line.
- control parameter
- A parameter that controls the behavior of how a procedure
does its gridding, such as whether negative values should
be allowed.
- convex hull
- The convex hull of a bounded subset of a 2D plane is the
convex set of smallest area that
contains the original set.
If one thinks of the points of the original set as pegs on
a board, then the convex hull would be those points interior
to a rubber band stretched around the pegs.
- convex set
- A bounded subset of a 2D plane is convex if, for any two
points in the set, all points on the line segment between the
two points are in the set.
- cubic spline
- A finite sequence of cubic polynomials defined on non-overlaping
domains and connected at
knots. The left and right derivatives
of the polynomials are equal at the knots.
- data sparse region
- A region in the domain space of a discrete function that contains
a disproportionately small number of data points when compared
to the number of data points expected to be in that region if
the points in the domain space were randomly spaced.
- Delaunay triangulation
- A Delaunay triangulation of a finite
set of points in the plane is a
triangulation
that minimizes the standard deviations of the angles of
the triangles, using 60 degrees (the angle of an
equilateral triangle) as the mean. In this sense the
Delaunay triangulation is the most equi-angular triangulation;
it minimizes long skinny triangles.
- empty circumcircle criterion
- The empty circumcircle criterion is a name
for the theorem that states that for every finite set of points in
the plane, there is a mininal set of circles such that
every point in the
convex hull of the original set of
points lies strictly within one or more of the circles,
but none of the original points lies within a circle, i.e.
each of the original points lies on one or more circles, but
not interior to any.
- extrapolation
- Extrapolation is interpolation
extended to points outside
the convex hull of a dataset.
An interpolated value at
a point outside the convex hull of an input dataset is
referred to as an extrapolated value.
- gradient
- If f is a function of two variables, then the
gradient of f at a given point is the vector
sum of the partial derivative of f with respect
to the first variable times the vector i,
plus the partial derivative of f with respect
to the second variable times the vector j.
The maximum rate of increase in f at a given
point is in the direction of the gradient.
- grid
- A set of points in n-space formed by taking the Cartesian
product of points in the coordinate spaces. For the pacakges
in the ngmath library, the points in the coordinate spaces
must be monotone increasing,
but not necessarily equally-spaced. For example,
suppose you had the X-coordinate values X(1),...,X(N1) and
the Y-coordinate values Y(1),...,Y(N2) and the Z-coordinate
values Z(1),...,Z(N3), then the associated three-dimensional
grid would be the set of all points (X(I),Y(J),Z(K)) with
1 <= I <= N1 and 1 <= J <= N2 and 1 <= K <= N3.
- interpolated surface
- An interpolated surface (single-valued function of two variables)
is a surface that
is derived from a finite set of function values by a
process of interpolation.
- interpolation
- A process for generating estimated functional values at
arbitrary locations beginning with a set of known functional
values. At the original points, the estimated values will
be identical to the interpolated values.
- inverse distance weighted average method
- An interpolation method where the interpolated values are
weighted averages, with the
weights being determined in inverse proportion to the distances
that the known data are from an interpolation point.
- isosurface
- The conceptual volume created by the locus of points in
three space (x,y,z) that satisfy an equation v = f(x,y,z)
for a continuous single-valued function f and a
specific value v (called an iso-value).
- knot
- A point in the domain space of a function where
pieces of a fitted surface join.
- least squares fitted plane
- Given a set of points in three space, the least squares fitted
plane is that plane that minimizes the sum of the squares of
the distances from the coordinate points to the plane.
- monotone sequence
- A sequence of real values is monotone increasing if each
value in the sequence is larger than its predecessor;
similarly for monotone decreasing, monotone non-increasing,
and monotone non-decreasing. Any of these four types of
sequence is called monotone.
- natural neighbor
- Intuitively, two points in the plane are natural neighbors
if they share an interface that is equally close to each
point and all other points are no closer. A precise
definition can be given in terms of the
empty circumcircle
criterion. Any two points in a finite subset of the plane
are said to be natural neighbors if they lie on the
same circumcircle.
Such natural
neighbors are also referred to
as first order natural neighbors [cf.
second order natural neighbor].
- natural neighbor
interpolation method
- An interpolation method that uses
a weighted average of
function values at the
natural neighbors of
an arbitrary point to determine an interpolated value
at that point.
- procedure
- Used as a generic term to refer simultaneously to both Fortran
subroutines and C functions when making a distinction between
the two is unnecessary. For example, the user interfaces
in Natgrid can be said to consist of procedures.
- pseudo data point
- A data point calculated internally in Natgrid used to
augment the original input data for the purpose of
extrapolation. Three pseudo data points are calculated that
lie on the least
squares fitted plane
that are well outside of the convex hull.
All of the input data are contained in the triangle
connecting the three psuedo data points.
- second order natural neighbor
- Two points P and Q are said to be second order natural
neighbors if there is a point R such that R is a
first order natural
neighbor of both P and Q.
- slope
- At a given point on an interpolated surface, the slope
at that point is the value of the partial derivative taken in
the direction of the
aspect at the point. The slope
is measured as an angle that is positive below the
horizontal, i.e., horizontal surfaces have a slope of
zero and a vertical cliff has a slope of 90 degrees.
- spline
- Originally, a pliable strip used by draftsmen to draw curves.
In the context of
approximation and
interpolation theory,
a spline is a mathematical function that interpolates or
approximates a finite sequence of data values.
Cubic spline functions are
the most commonly used.
- surface
- The conceptual object created by the function values of
a continuous single-valued function of two variables.
- tension spline
-
A generalization of cubic splines
effected by the introduction
of a tension factor that can be adjusted so that the
interpolation equals a cubic spline
for a zero value of
the tension factor and approaches piecewise linear
interpolation for large values of the tension factor.
- triangulation
- For a finite set of points in the plane, a triangulation
of these points is any partition of the points into a set
of non-overlapping triangles.
- Voronoi Polygon
- Given a finite set, S, of points in the plane,
the Voronoi Polygon associated with a given point P in
S, is the boundary of the set of all points in the plane
that are closer to P than to any other point in S.
- weighted average
- Given a sequence of function values and a matching sequence
of real numbers, called weights,
such that the sum of all of the weights is unity, then
the sum of all of the products of the weights times the
function values is called a weighted average of the
function values.
Return to main ngmath home page
Return to Natgrid home page
Return to Dsgrid home page
Return to Fitgrid home page
Return to Csagrid home page
Return to Cssgrid home page