Editing Euclidean geometry (section)

==In engineering==

===Design and Analysis===
*'''Stress Analysis''': [[Stress (mechanics)|Stress Analysis]] - Euclidean geometry is pivotal in determining [[Stress–strain analysis|stress distribution]] in mechanical components, which is essential for ensuring [[structural integrity]] and [[durability]].
[[File:Different-types-of-mechanical-stress_EN.svg#%7B%7Bint%3Afiledesc%7D%7D|thumb|Mechanical Stress]]

*'''Gear Design''': [[Gear]] - The design of gears, a crucial element in many [[mechanical system]]s, relies heavily on Euclidean geometry to ensure proper tooth shape and [[engagement]] for efficient [[power transmission]].
[[File:Gear-kegelzahnrad.svg|thumb|150px|Gear]]

* '''Heat Exchanger Design''': [[Heat exchanger]] - In [[thermal engineering]], Euclidean geometry is used to design [[heat exchangers]], where the geometric configuration greatly influences [[thermal efficiency]]. See [[shell-and-tube heat exchanger]]s and [[plate heat exchanger]]s for more details.
[[File:U-tube_heat_exchanger.svg|center|400px|U-Tube Shell and Tube Heat Exchanger]]

* '''Lens Design''': [[Lens]] - In optical engineering, Euclidean geometry is critical in the design of lenses, where precise geometric shapes determine the [[focus (optics)|focusing]] properties. [[Geometric optics]] analyzes the focusing of [[light]] by [[lens]]es and [[mirrors]].
[[File:Lenses en.svg|alt=Types of lenses|center|thumb|450x450px|Types of Lenses]]

=== Dynamics ===
* '''Vibration Analysis''': [[Vibration]] - Euclidean geometry is essential in analyzing and understanding the [[vibrations]] in [[mechanical systems]], aiding in the design of systems that can withstand or utilize these [[vibrations]] effectively.
[[File:Drum vibration mode21.gif|thumb|Vibration - [[Oscillations]]]]

* '''Wing Design''': [[Aerodynamics|Aircraft Wing Design]] - The application of Euclidean geometry in [[aerodynamics]] is evident in [[aircraft]] [[airfoil|wing design]], [[airfoil]]s, and [[hydrofoil]]s where geometric shape directly impacts [[lift (force)|lift]] and [[drag (physics)|drag]] characteristics.
[[File:Wing profile nomenclature.svg|400px|thumb|Airfoil Nomenclature]]

* '''Satellite Orbits''': [[Orbit|Satellite Orbits]] - Euclidean geometry helps in calculating and predicting the [[orbit]]s of [[satellites]], essential for successful [[List of Space Shuttle missions|space missions]] and [[satellite]] operations. Also see [[astrodynamics]], [[celestial mechanics]], and [[elliptic orbit]].
[[File:Animation of Orbital eccentricity.gif|250px|thumb|Animation of Orbit by Eccentricity]]

===CAD Systems===
*'''3D Modeling''': In [[Computer-aided design|CAD (computer-aided design)]] systems, Euclidean geometry is fundamental for creating accurate 3D models of mechanical parts. These models are crucial for visualizing and testing designs before [[manufacturing]].

* '''Design and Manufacturing''': Much of [[Computer-aided manufacturing|CAM (computer-aided manufacturing)]] relies on Euclidean geometry. The design geometry in CAD/CAM typically consists of shapes bounded by [[Plane (geometry)|planes]], [[cylinder]]s, [[cone]]s, [[torus|tori]], and other similar Euclidean forms. Today, CAD/CAM is essential in the design of a wide range of products, from [[car]]s and [[airplane]]s to [[ship]]s and [[smartphone]]s.

* '''Evolution of Drafting Practices''': Historically, advanced Euclidean geometry, including theorems like [[Pascal's theorem]] and [[Brianchon's theorem]], was integral to drafting practices. However, with the advent of modern CAD systems, such in-depth knowledge of these theorems is less necessary in contemporary design and manufacturing processes.
{{see also|History of CAD software}}
[[File:Cad crank.jpg|thumb|150px|3D CAD Model]]

=== Circuit Design ===
* '''PCB Layouts''': [[Printed circuit board|Printed Circuit Board (PCB) Design]] utilizes Euclidean geometry for the efficient placement and routing of components, ensuring functionality while [[optimization|optimizing space]]. Efficient layout of electronic components on PCBs is critical for minimizing [[electrical interference|signal interference]] and optimizing [[electric circuit|circuit performance]].
[[File:SEG DVD 430 - Printed circuit board-4276.jpg|thumb|PCB of a DVD Player]]

=== Electromagnetic and Fluid Flow Fields ===
* '''Antenna Design''': [[Antenna (radio)|Antenna Design]] - Euclidean [[Antenna types|geometry of antennas]] helps in designing antennas, where the spatial arrangement and dimensions directly affect antenna and [[Antenna array|array]] performance in transmitting and receiving [[electromagnetic wave]]s.
[[File:Canberra Deep Dish Communications Complex - GPN-2000-000502.jpg|thumb|NASA [[Cassegrain antenna|Cassegrain]], Extremely high gain ~70&nbsp;dBi.]]

* '''Field Theory''': [[Potential flow|Complex Potential Flow]] - In the study of [[inviscid flow]] fields and [[electromagnetic field]]s, Euclidean geometry aids in visualizing and solving [[potential flow]] problems. This is essential for understanding [[fluid velocity]] field and [[electromagnetic field]] interactions in three-dimensional space. The relationship of which is characterized by an [[irrotational]] [[solenoidal field]] or a [[conservative vector field]].
[[File:Inviscid flow around a cylinder.gif|thumb|200px|Potential Flow Around a Source without [[circulation (physics)|Circulation]]]]

=== Controls ===
* '''Control System Analysis''': [[Control theory|Control Systems]] - The application of Euclidean geometry in [[control theory]] helps in the analysis and design of [[control systems]], particularly in understanding and [[optimization|optimizing]] system stability and [[Request–response|response]].
[[File:Ideal feedback model.svg|thumb|right | Basic feedback loop.]]

* '''Calculation Tools''': [[Jacobian matrix and determinant|Jacobian]] - Euclidean geometry is integral in using Jacobian matrices for transformations and control systems in both [[mechanical engineering|mechanical]] and [[electrical engineering]] fields, providing insights into system behavior and properties. The Jacobian serves as a linearized [[design matrix]] in statistical [[regression analysis|regression]] and [[curve fitting]]; see [[non-linear least squares]]. The Jacobian is also used in [[random matrices]], [[moment (physics)|moment]], [[statistics]], and [[diagnostic]]s.