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As boring as it may seem, analytic geometry is extremely useful in the aircraft industry, specifically when dealing with the shape of an airplane's fuselage. Conic curves are used to describe the fuselage cross sections and their curvature is modified along the length of the fuselage to render a smooth yet producible surface which at the same time should allow for an efficient internal arrangement.
Conic curves are nothing more than 2nd degree curves, i.e, parabolas, ellipses and hyperbolas. The cross sections for an aircraft may be produced graphically, provided the side and top view of the aircraft have been already laid out, but given the advances in computer-aided design, along with the necessity to iterate the shape of the fuselage during the design process, it is preferrable to determine the equations that define such cross sections. The side and top view of the aircraft may also be defined using conics, making it also possible to mathematically define the fuselage in its entirety to allow for parametrization.
More advanced techniques for the definition of aircraft shapes are now available, the most popular being Bézier curves. However, anyone who wishes to define the shape of an aircraft should begin by learning the use of conic curves since many of the concepts used for these are also applicable to Bézier curves.
That is one application, but I'm pretty sure there are many more...
Source: Aircraft Analytic Geometry, J.J. Apalategui, L.J. Adams, 1944
Regards
Analytic geometry is used in various fields such as engineering, physics, computer graphics, and economics. It helps in solving real-world problems involving lines, curves, and shapes by providing a systematic way to represent and manipulate geometric objects using algebraic equations. This allows for the analysis of geometric relationships and visualization of data, leading to practical applications like designing structures, modeling physical phenomena, and optimizing processes.
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