KURS ISHI

INTRODUCTION . In Uzbekistan, the national investment in science and technology, as a percentage of GDP, reached 0.2% in 2023, reflecting a consistent growth trajectory from 0.15% in 2020, according to the State Statistics Committee of Uzbekistan [1]. This sustained focus underpins the growing demand for specialists proficient in complex geometric analysis, crucial for fields like industrial design and architectural engineering, which collectively contributed approximately 6.5% to the nation’s GDP in 2023 [2]. Furthermore, the education sector in Uzbekistan has seen a 12% increase in STEM graduates over the past five years, totaling over 70,000 specialists annually [3], underscoring the foundational importance of analytical geometry as a core discipline. The ability to precisely manipulate and understand three-dimensional forms is fundamental to optimizing resource utilization and fostering innovation within these burgeoning industries. The current global landscape demands ever-greater precision and computational efficiency in handling complex geometric data, a need that is particularly acute in developing economies striving for technological parity. While significant progress has been made in computational geometry, challenges persist in the intuitive and efficient transformation of second-order surfaces, specifically in the context of real-world engineering constraints and limited computational resources [4]. Unresolved problems include the development of universally applicable, robust algorithms that can handle degenerate cases gracefully and the integration of these methods into user-friendly software tools that cater to the specific needs of local industries, bridging the gap between theoretical understanding and practical application in contexts like Uzbekistan [5]. International scholarship on invariant methods for transforming second-order surfaces has a rich history, with foundational work laying the groundwork for modern applications. X.Y. Lu (2018) demonstrated in their study the efficacy of orthogonal transformations in simplifying quadratic forms, proving that principal axis theorem provides a robust pathway to canonical representation for non-degenerate surfaces, reducing computational complexity by an average of 15% in complex engineering simulations [6]. Z.W. Zhang (2020) proposed a model of generalized eigenvalue decomposition for handlin ...