KURS ISHI

INTRODUCTION . In Uzbekistan, the national investment program for 2023-2025 allocates over 210 trillion UZS to projects aimed at industrial modernization and digital transformation [1]. For instance, the use of sophisticated algorithms for optimizing production processes in the automotive industry, which saw a 34.5% increase in production volumes in 2023 compared to the previous year, hinges on efficient geometric representations and transformations [2]. Globally, the market for CAD/CAM software, which heavily relies on second-order curve mathematics for design and simulation, is projected to reach USD 13.4 billion by 2028, growing at a compound annual growth rate (CAGR) of 6.7% [3]. Despite the established theoretical framework for second-order curves, challenges persist in their efficient application, particularly when integrating complex real-world data and dynamic systems. The transformation of general second-order curve equations to canonical form and the preservation of their invariant properties are critical for simplifying analysis, optimizing computational processes, and ensuring data integrity across transformations. Unresolved issues include the development of more robust numerical methods for handling degenerate cases in practical applications, the optimization of algorithms for real-time data processing in large-scale geometric datasets, and the integration of these advanced mathematical tools into accessible, user-friendly computational platforms that can be readily adopted by engineers and scientists in Uzbekistan. Furthermore, the pedagogical gap in translating complex theoretical concepts into practical, problem-solving skills for local specialists represents a significant area requiring further attention and tailored solutions. International scholars have extensively contributed to the field of second-order curves and their transformations. For example, Salmon (1879) [4] laid much of the foundational work on conic sections and algebraic curves, detailing their classification and properties. Klein (1939) [5] further integrated geometric transformations into a unified framework, emphasizing the role of invariants in his Erlangen Program. More recently, books by Gibson (1998) [6] and Mortenson (2006) [7] have provided comprehensive treatments of computational geometry and curve theory, demonstrating how transformations to canonical forms are ...