LOUČKA, P. Algorithms for Conics in Geometric Algebras [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2024.
Supervisor’s report on the thesis by Pavel Loučka entitled Algorithms for Conics in Geometric Algebras The thesis prvoides several conic sections-related algortithms in the setting of Geometric Algebra for Conics (abbr. GAC). First, basic introduction of GAC and the necessary theory of conic sections is given. Consequently, this knowledge was used by author to derive the expression of so called improper points (also called ideal points or points at infinity) in GAC, which had not been known before. The notion of the improper points is crucial for the geometry of conic sections, so it is used in other chapters of the thesis as well. The derivation of the concept in GAC setting was published in [4] and its usage appeared in [4, 5]. Second major topic of the thesis consists of the conic fitting problems using GAC forms of the conics. Main focus was on adjusting an ordinary conic fitting problem so that it imposes additional geometric properties on the sought conic. The first subgroup of these additional geometric conditions are, namely, that the conic in question is one of the following types: axes-aligned with the coordinate system, origin-centred or both at the same time. As shown in [1, 3], these conditions are easy to implement in GAC setting. The second subgroup of the adjusting of ordinary conic fitting was imposing a condition that besides fitting as tightly as possible among the data set, the fitted conic should also exactly pass through 1–4 points called waypoints. The way to achieve this was derived in [4] and used in [4, 5]. Moreover, thanks to the mastering of the improper points, the usage of the improper waypoints in conic fitting is both in the thesis and in [4, 5]. Additionally, an iterative algorithm for ordinary conic fitting in GAC was presented, aiming to minimise the consequences of non-invariance with respect to translation of fitted data points. Finally, the third topic of the thesis describes several ways of using the GAC wedge product for construction of conics from points and/or basis vectors of GAC. In the beginning, conics are constructed from five distinct points with great focus on usage of improper points. Then, the thesis give a lot of attention to the concept of pencil of conics (which is a set of all linear combinations of two conics). This is then exploited to show how a conic can be constructed by wedging the intersection of two conics (which is called a four-point) with another point that does not lie on the four-point. After giving some general examples of this type of construction, it is used for construction of special types of conics appearing in a pencil of conics—namely, line-pairs and generalised parabolas. I consider all goals of the thesis to be fulfilled. As for the personality of the candidate, Pavel Loučka showed iniciativity and worked on the topis with enthusiasm. He participated in three international conferences, gave talks at various seminars and successfully published his research in respected journals oriented on geometric algebras, namely AACA, Q2. He was also involved in teaching and other academic procedures and became a scientist who can carry on his own research. Recommendation Based on the arguments above, I hereby recommend the thesis for the defense and approve Pavel Louˇ cka as a candidate for Ph.D. title. References [1] Loučka, P., Vašík, P. (2021). Algorithms for multi-conditioned conic fitting in geometric algebra for Conics. Lecture Notes in Computer Science (pp. 645–657). https://doi. org/10.1007/978-3-030-89029-2_48 [2] Derevianko, A., Loučka, P. (2022). Search for similarity transformation between image point clouds using geometric algebra for conics. Lecture Notes in Computer Science (pp. 215–226). https://doi.org/10.1007/978-3-030-98260-7_13 [3] Loučka, P., Vašík, P. (2023) On multi-conditioned conic fitting in Geometric algebra for conics. Advances in Applied Clifford Algebras, 33(31). https://doi.org/10.1007/ s00006-023-01277-9 [4] Loučka, P. (2023). On Proper and Improper Points in Geometric Algebra for Conics and Conic Fitting Through Given Waypoints. Lecture Notes in Computer Science, 67–79. https://doi.org/10.1007/978-3-031-30923-6_6 [5] Loučka, P., Vašík, P. (2024) Algorithms for Conic Fitting Through Given Proper and Improper Waypoints in Geometric Algebra for Conics. Advances in Applied Clifford Algebras 34, 6 (2024). https://doi.org/10.1007/s00006-023-01308-5
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