Least squares fitting with rotated paraboloids

In [1] the problem of estimating the parameters of a rotated parabola fitted to measured points in the plane was examined. The corresponding method, also used in [2,3], is extended here to the case of a rotated paraboloid. Fitting by such a surface occurs in computational metrology e.g. when some parabolic reflector will be checked to be a good one

Least-squares fitting of parametric curves with a linear function of several variables as argument

We discuss fitting of a parametric curve in the plane in the least-squares sense when the independent variable is a linear function of several variables with unknown coefficients. A general numerical method is recommended. For two special models the algorithmic details and numerical examples are given

Fitting affine and orthogonal transformations between two sets of points

Let two point sets P and Q be given in $R^n$. We determine a translation and an affine transformation or an isometry such that the image of Q approximates P as best as possible in the least squares sense

Identifying spatial point sets

Two sets of spatial points are checked whether they can pproximately be transformed into each other by applying some unknown translation and some unknown rotation. This problem occurs at least in two dimensions within computational metrology. Numerical methods for two types of objective functions (weighted sum of squared distances and sum of distances) are developed and numerical examples are given

Single facility minisum location on curves

The minisum location problem is well-known and had extensively been studied in the case of the unknown location being situated somewhere in the plane. Also the accompanying Weiszfeld iteration [3] is nowadays well understood, even for noneuclidean distances [2,3]. We introduce as a side condition that the unknown optimal location point may only lie on some given curve in the plane. For a piece at a straight line and for a circle numerical methods are developed and numerical examples got by them are given

Minisum location in space with restriction to curves and surfaces

The minisum location problem is well-known and has extensively been studied in the case of the unknown location being somewhere in the space. Also the accompanying Weiszfeld iteration method [1,2,3] is well understood nowadays, even for noneuclidean distances. We introduce as a side condition for the unknown optimal location that it lies on some given curve or surface in space. For the straight line, the plane, the sphere, and the circle the corresponding Weiszfeld-like iteration methods are developed and numerical examples are given

Data fitting with a set of two concentric spheres

We consider fitting data points in space by a set of two concentric spheres. This problem ought to occur within computational metrology. A heuristic algorithm is developed and its efficiency is demonstrated by some numerical example

Least squares fitting of spheres and ellipsoids using not orthogonal distances

Berman [1] examined the problem of estimating the parameters of a circle when angular differences between successively measured data points were also measured. Applications were reported. Späth [4] generalized that problem by considering an ellipse. Now we will consider measured data points (x_k,y_k,z_k) in space and also associated measured angles (u_k,v_k) k=1, ..., n>8, for the canonical parametric representation of a sphere or an ellipsoid. The center and the radius or the three half axes, respectively, and two other parameters will be fitted such that some suitable sum of squared not orthogonal distances between the two measurements is minimized. Numerical examples are given. Generalizations are discussed. Another numerical method was proposed by Watson [5]

Fitting data in space by surfaces in parametric representation with polynomial components

We consider fitting measured data points in space in the total least squares sense by surfaces in parametric representation with polynomial components in two variables. A well-known descent algorithm is suitably modified. Numerical examples are given

Least squares fitting of conic sections with type identification by NURBS of degree two

Fitting of conic sections is used in reflectometry, aircraft industry, metrology, computer vision, astronomy and propagation of sound waves [5]. So far numerical algorithms assume the type of the conic section to be known in advance. We consider the problem of additionally identifying the type during fitting, i.e. deciding whether the given data are better fitted by an ellipse, a hyperbola or a parabola. To solve this problem we apply a well-known descent algorithm [3,4,6] to NURBS of degree two. Numerical examples will be given