# 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

## A

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.

## B

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.

## C

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.

## D

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.

## E

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.

## G

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.

## I

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).

## K

knot
A point in the domain space of a function where pieces of a fitted surface join.

## L

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.

## M

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.

## N

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.

## P

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.

## S

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.

## T

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.

## V

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.

## W

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.

## Z 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