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We fit k-spheres optimally to n-D point data, in a geometrically total least squares sense. A specific practical instance is the optimal fitting of 2D-circles to a 3D point set.
Among the optimal fitting methods for 2D-circles based on 2D (!) point data compared in Al-Sharadqah and Chernov (Electron. J. Stat. 3:886–911, 2009), there is one with an algebraic form that permits its extension to optimally fitting k-spheres in n-D. We embed this ‘Pratt 2D circle fit’ into the framework of conformal geometric algebra (CGA), and doing so naturally enables the generalization. The procedure involves a representation of the points in n-D as vectors in an (n+2)-D space with attractive metric properties. The hypersphere fit then becomes an eigenproblem of a specific symmetric linear operator determined by the data. The eigenvectors of this operator form an orthonormal basis representing perpendicular hyperspheres. The intersection of these are the optimal k-spheres; in CGA the intersection is a straightforward outer product of vectors.
The resulting optimal fitting procedure can easily be implemented using a standard linear algebra package; we show this for the 3D case of fitting spheres, circles and point pairs. The fits are optimal (in the sense of achieving the KCR lower bound on the variance).
We use the framework to show how the hyperaccurate fit hypersphere of Al-Sharadqah and Chernov (Electron. J. Stat. 3:886–911, 2009) is a minor rescaling of the Pratt fit hypersphere.

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... It is well known that, consequently, the minimisation problem may be linearised by its transformation to the search for the eigenvector problem, which is simple to solve in short time. The algebraic fits are further divided with respect to the normalisation of coefficients in (2) that prevents the trivial minimum existence, i.e. eliminates the trivial solution A = ... = F = 0, see Sect. 3 for more details. The aim of this paper is to formulate and solve the fitting problem in a simple covariant GA setting. ...

... The aim of this paper is to formulate and solve the fitting problem in a simple covariant GA setting. Note that for circle fitting, this is well known, see [2], Chapter 3. More precisely, in Compass Ruler Algebra (CRA), [7], i.e. the conformal geometric algebra for two Euclidean dimensions, both circles and points are vectors in R 3,1 . ...

... More precisely, in Compass Ruler Algebra (CRA), [7], i.e. the conformal geometric algebra for two Euclidean dimensions, both circles and points are vectors in R 3,1 . If we consider a distance in CRA given by dist(S, p i ) = (P i · S)/ √ S 2 , which is so-called tangent distance of a point P i from the circle S, then the minimisation (1) is clearly equivalent to the minimisation of the cost function 2 under the assumption that S 2 = 1. In [2], the author shows by GA differentiation that such problem may be reformulated to the search for the eigenvectors of the operator ...

We present an algorithm for a conic fitting based on a generalization of planar version of conformal geometric algebra to geometric algebra for conics (GAC). We introduce a novel normalization condition that follows naturally from this setting and which is invariant with respect to rotations and scaling. Finally, we provide a comparison to standard methods demonstrated on examples in MATLAB.

... That solves the first few steps along the molecular backbone in DMDGP. We end the paper with a suggestion of how a similar characterization based on a decomposition into orthogonal spheres (from [4]) may be employed for the 'arcs of arcs', 'arcs of arcs of arcs' et cetera that one encounters further down the backbone, albeit in a probabilistic manner. ...

... For the next node v 6 in the example, the choice of one of the two discrete point solutions for v 4 makes the treatment still rather precise: the given exact distance d (4,6) constrains the possibilities to lie onto the sphere σ 4,6 ≡ v 4 − 1 2 d 2 (4, 6)∞. For each point on the arc of possibilities for v 5 , the interval distance d (3,6) allows an arc of possibilities on this sphere, bounded by two circles for the lower and upper bound, parallel to the circle of v 5 possibilities. ...

... With this in mind, it should be no surprise that in CGA there is a better possibility for characterizing covariance, which is a consequence of results from optimal sphere fitting [4]. We can set up a quadratic form for the 'conformal covariance', which leads to a conformal orthonormal basis of spheres: the best sphere fitting the data, the next best orthogonal sphere (together giving the best circle) and the next best orthogonal sphere (with all three then providing the best point pair to describe the data). ...

An orthogonal sphere representation of arcs on spatial circles can be used to compactly perform Boolean combinations of such arcs. We formulate this using conformal geometric algebra, of which the oriented nature allows both minor and major arcs to be treated. Easily computable quantities discriminate the cases of relative positions. An application in the first stages of a problem in Discretizable Molecular Distance Geometry is included. We give a suggestion on how to extend this characterization by orthogonal spheres to the manifolds of arcs in the subsequent stages, using probabilistic eigenspheres of the distributions.

... Fischler and Bolles (1981) propose RANdom SAmple Consensus (RANSAC) to robust model fitting, but RANSAC is sensitive to thresholding errors (Fischler and Bolles (1981); Nurunnabi et al (2015a)). Dorst (2014) embeds circle fitting of Pratt (1987) into the conformal geometric algebra for hypercircle fitting, which is an extension of the hypersphere approach. It performs as an approximate least-squares ap-proach and has strong results for circle fitting without outliers. ...

The shape of circle is one of fundamental geometric primitives of man-made engineering objects. Thus, extraction of circles from scanned point clouds is a quite important task in 3D geometry data processing. However, existing circle extraction methods either are sensitive to the quality of raw point clouds when classifying circle-boundary points, or require well-designed fitting functions when regressing circle parameters. To relieve the challenges, we propose an end-to-end Point Cloud Circle Algebraic Fitting Network (Circle-Net) based on a synergy of deep circle-boundary point feature learning and weighted algebraic fitting. First, we design a circle-boundary learning module, which considers local and global neighboring contexts of each point, to detect all potential circle-boundary points. Second, we develop a deep feature based circle parameter learning module for weighted algebraic fitting, without designing any weight metric, to avoid the influence of outliers during fitting. Unlike most of the cutting-edge circle extraction wisdoms, the proposed classification-and-fitting modules are originally co-trained with a comprehensive loss to enhance the quality of extracted circles.Comparisons on the established dataset and real-scanned point clouds exhibit clear improvements of Circle-Net over SOTAs in terms of both noise-robustness and extraction accuracy. We will release our code, model, and data for both training and evaluation on GitHub upon publication.

... The methods of [VA11] and [Per00] made use of multivector differentiation in geometric calculus [HS84] to compute gradients. Circle and sphere estimation in conformal geometric algebra using linear least squares methods is presented in the works of Dorst, Fontijne, and Mann [DFM07], Rockwood and Hildenbrand [RH10] and Dorst [Dor14]. ...

This thesis is on the estimation of rigid body motions from observations of points, lines, and planes in conformal geometric algebra using nonlinear least-squares optimization. Distance measures based on the ratio and difference of points, lines, and planes are formulated and analyzed. Then, the properties of these distance measures, such as translation invariance and scale dependence are analyzed and discussed. A novel approach to motor estimation using the retraction-based framework of Riemannian optimization is presented. This approach is based on computing the update at each iteration in the tangent space of the motor manifold at the current iterate, and then mapping the solution to the motor manifold using a retraction map. Two retraction maps are presented. The first retraction is based on the exponential map, while the other is based on orthogonal projection onto the manifold. Cost functions and Jacobian matrices based on the distance measures for points, lines, and planes are presented, and it is shown how Jacobian matrices can be derived and computed efficiently. This includes a new approach for computing gradients and Jacobian matrices of multivector valued functions in Euclidean and conformal geometric algebra. The novelty of this approach is the use of automatic differentiation in conformal geometric algebra, which computes derivative values up to machine precision. Implementation details and experimental results of automatic differentiation of multivectors and motor estimation are presented. Performance evaluations of combinations of geometric algebra and automatic differentiation libraries show that there are significant performance differences, and recommendations on how to select the best solutions are made. Experimental results from motor estimation show that the proposed method has similar performance as state-of-the-art methods when estimating motors from observations of points, lines, and planes. The problem related to distance measures that are not translation invariant or scale invariant is discussed, and it is shown that the results are useful from an engineering standpoint although translation invariance is certainly a drawback of the distance measures that are used. Finally, the proposed method can be used for a wide range of optimization problems in conformal geometric algebra, including different distance measures, and data from different geometric objects.

... To find the turn radius, the least-squares regression of a circle is applied to the 2D projections of the 3D positions on a best plane. This plane is also found using the least-squares In [12], the drawbacks of fitting a 2D circle to 3D points are shown. In this paper, we first transform the 3D point to a 2D plane. ...

The Automatic Dependent Surveillance-Broadcast (ADS-B) data has become one of the most popular sources of data for trajectory-based ATM studies. It is can be received in most of the world without restrictions. Extended coverage can be achieved with a network of low-cost receivers and satellites. However, the fact that ADS-B is designed to contain only a low number of aircraft states such as position and velocity poses a challenge for some trajectory-based studies, for example, using ADS-B data to study aircraft turns. To this extent, air traffic controllers commonly rely on ModeS track and turn reports to gather additional information like bank angle and turn rates during turns. Unlike ADS-B, this data has a low update rate and is not always openly available for all researchers. In this paper, we propose methods that allow researchers to extract and analyze aircraft turn parameters from ADS-B data during offline flight analysis. The paper first discusses the dynamics of aircraft turns. Then, based on ADS-B trajectory data, several steps are designed to derive turn radius, bank angle, and turn rate of an aircraft. The estimation results are validated with aircraft track and turn reports from Mode S Enhanced Surveillance. The median errors for bank angle and turn rate are found to be less than 2 degrees and 0.1 degrees/s respectively, which reflects the accuracy of the estimation approach.

... Then and P denote the eigenvalues and eigenvectors of the matrix 1 Q J , respectively. According to (9), the equation 0 and its corresponding eigenvector, respectively [33,35,36]. After tackling the minimum value problem, the standard equation of the circular curve is expressed as: In addition, we define the accuracy of the trace curve fitting. ...

Photoelectric non-contact measurement methods have been widely implemented in space target location, micro-displacement measurement and space optical communication. In term of the reliability and precision of measurement systems, it is a great significance to obtain the laser center location accurately. Since the two-dimensional position sensitive detector (2-D PSD) can continuously detects the gravity center position of the laser spot on its photosensitive surface, this paper provides a flexible method with high precision for the nonlinear correction of PSD as well as a new idea for PSD-based moving laser center positioning. Above all, a laser device system that rotating around a fixed-axis is designed, and the rotation angle can be controlled with the rotation accuracy of 0.50°. The trace of the gravity center of the rotating spot generated by the device on the PSD surface is detected, and then the laser center location is obtained after further processing of data. In order to improve the detection accuracy of PSD, this research mainly focuses on the nonlinear correction method of the detector and the fitting method of the gravity center trace of the laser spot. Moreover, the effects of different sizes of light spots projected onto the 2-D PSD on the collected photocurrent and estimated coordinates were analyzed by experiments. From the experimental results, the non-linear error of PSD is corrected by the error curved surface interpolation method, the overall error is reduced by more than 32%, and the linearity of the positioning of the laser moving along the diagonal of PSD is 0.2%.

... The applications of least-squares include robotics, manipulating parts with complex shapes in unstructured dynamic environments, planning end-effect or motions around singularities, writing on a blackboard, and other motion planning problems involving kinematic constraints [9]. Other works involving the use of interpolation in Euclidean space can be found in [10,11,12,13,14,15,16]. [17] made an investigation of the effects of interpolation error, using two error methods. ...

A Generalised Euclidean Least Square Approximation (ELS) is derived in this paper. The Generalised Euclidean Least Square Approximation is derived by generalizing the interpolation of n arbitrary data set to approximate functions. Existence and uniqueness of the ELS schemes are shown by establishing the invertibility of the coefficient matrix using condensation method to reduce the matrix. The method is illustrated for exponential function and the results are compared to the classical Maclaurin’s series.

... The applications of least-squares include robotics, manipulating parts with complex shapes in unstructured dynamic environments, planning end-effect or motions around singularities, writing on a blackboard, and other motion planning problems involving kinematic constraints [9]. Other works involving the use of interpolation in Euclidean space can be found in [10,11,12,13,14,15,16]. [17] made an investigation of the effects of interpolation error, using two error methods. ...

A Generalised Euclidean Least Square Approximation (ELS) is derived in this paper. The Generalised Euclidean Least Square Approximation is derived by generalizing the interpolation of n arbitrary data set to approximate functions. Existence and uniqueness of the ELS schemes are shown by establishing the invertibility of the coefficient matrix using condensation method to reduce the matrix. The method is illustrated for exponential function and the results are compared to the classical Maclaurin's series.

There are many practical applications that require the simplification of polylines. Some of the goals are to reduce the amount of information, improve processing time, or simplify editing. Simplification is usually done by removing some of the vertices, making the resultant polyline go through a subset of the source polyline vertices. If the resultant polyline is required to pass through original vertices, it often results in extra segments, and all segments are likely to be shifted due to fixed endpoints. Therefore, such an approach does not necessarily produce a new polyline with the minimum number of vertices. Using an algorithm that finds the compressed polyline with the minimum number of vertices reduces the amount of memory required and the postprocessing time. However, even more important, when the resultant polylines are edited by an operator, the polylines with the minimum number of vertices decrease the operator time, which reduces the cost of processing the data. A viable solution to finding a polyline within a specified tolerance with the minimum number of vertices is described in this paper.

The standard algebraic model for Euclidean space En
is an n-dimensional real vector space ℝn
or, equivalently, a set of real coordinates. One trouble with this model is that, algebraically, the origin is a distinguished element, whereas all the points of En
are identical. This deficiency in the vector space model was corrected early in the 19th century by removing the origin from the plane and placing it one dimension higher. Formally, that was done by introducing homogeneous coordinates [110]. The vector space model also lacks adequate representation for Euclidean points or lines at infinity. We solve both problems here with a new model for En
employing the tools of geometric algebra. We call it the homogeneous model of En
.

The technique of "renormalization" for geometric estimation attracted much attention when it was proposed in early 1990s for having higher accuracy than any other then known methods. Later, it was replaced by minimization of the reprojection error. This paper points out that renormalization can be modified so that it outperforms reprojection error minimization. The key fact is that renormalization directly specifies equations to solve, just as the "estimation equation" approach in statistics, rather than minimizing some cost. Exploiting this fact, we modify the problem so that the solution has zero bias up to high order error terms; we call the resulting scheme hyper-renormalization. We apply it to ellipse fitting to demonstrate that it indeed surpasses reprojection error minimization. We conclude that it is the best method available today.

Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.