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Rotor

A rotor is the rotating component of a machine, found in electromechanical and fluid devices such as electric motors, generators, turbines, pumps, and compressors.^[1] In electromechanical systems, it is essential for converting energy between electrical and mechanical forms.^[2] It typically consists of a shaft with attached elements like windings, laminations, or blades, supported by bearings within a stationary housing or stator.^[2] In operation, the rotor interacts with magnetic or fluid forces to produce torque, rotation, or power generation, and its design must account for dynamic forces including vibration, imbalance, and high-speed stability to ensure reliable performance.^[1] In electric machines, rotors are classified by their excitation method and construction, including wound rotors with electromagnetic windings connected via slip rings, squirrel-cage rotors featuring conductive bars for induction motors, and permanent magnet rotors for efficient synchronous applications.^[3]^[4] The rotor's materials, often laminated iron cores to minimize eddy current losses, align with the stator's magnetic field to generate rotational motion or electrical output.^[5] Balancing is a critical process for rotors to limit vibration during operation, achieved through techniques like adding or removing mass at specific angular positions. Beyond electrical applications, rotors play vital roles in mechanical systems, such as the main and tail rotors in helicopters for lift and control, or bladed rotors in gas and steam turbines for propulsion and power generation.^[1] Advances in rotor dynamics focus on mitigating issues like whirl, rubbing, and thermal stresses, particularly in high-speed industrial machinery.^[1] These components are fundamental to modern engineering, enabling efficient energy systems across aerospace, power generation, and manufacturing sectors.^[1]

Engineering

Electrical machines

In electrical machines, the rotor is the rotating component that interacts with the stationary stator to generate torque through electromagnetic induction, converting electrical energy into mechanical rotation or vice versa. This fundamental principle applies to both alternating current (AC) and direct current (DC) motors and generators, where the rotor's motion is driven by the interaction between magnetic fields produced by currents in the windings or permanent magnets. Common types of rotors include squirrel-cage, wound, and permanent magnet designs, each optimized for specific applications based on their construction. Squirrel-cage rotors, typically used in induction motors, consist of a cylindrical laminated core with conductive bars shorted by end rings, forming a cage-like structure that induces currents to produce torque without slip rings; the lamination reduces eddy current losses by minimizing magnetic flux paths. Wound rotors, also known as slip-ring rotors, feature windings connected to external resistors or circuits via slip rings, allowing control of starting torque and speed in applications requiring variable performance. Permanent magnet rotors employ rare-earth magnets embedded in or surface-mounted on a laminated core, providing high efficiency and constant flux in synchronous and brushless DC motors by eliminating the need for excitation currents. The torque production in these rotors derives from the Lorentz force acting on current-carrying conductors in a magnetic field. For a basic model, the electromagnetic torque T can be expressed as T = \frac{P \Phi I \sin \theta}{\omega}, where P is the number of pole pairs, \Phi is the magnetic flux per pole, I is the rotor current, \theta is the angle between the current and flux vectors, and \omega is the angular speed. This equation arises from integrating the force \mathbf{F} = I \mathbf{l} \times \mathbf{B} (with \mathbf{B} = \mu_0 \mathbf{H} related to flux) over the rotor's conductors, yielding net rotational torque proportional to the cross-product magnitude; the \sin \theta term maximizes at 90 degrees for optimal alignment. The concept of the rotor traces back to Michael Faraday's 1831 invention of the first electric motor, a homopolar design with a rotating disk in a mercury bath that demonstrated continuous rotation from electromagnetic interaction. This evolved through the 19th century with the development of AC induction motors by Nikola Tesla in 1887, incorporating squirrel-cage rotors for practical polyphase systems, and into the 20th century with wound rotors for industrial drives. Modern brushless DC motors, emerging in the 1960s, advanced rotor design by using electronic commutation to replace mechanical brushes, enhancing reliability in applications like computer drives. As of 2025, recent advancements in rotors for electric vehicles (EVs) emphasize high-efficiency permanent magnet synchronous motors using neodymium-iron-boron magnets to achieve power densities exceeding 5 kW/kg, though supply chain disruptions from 2020–2023 prompted innovations like ferrite-assisted rotors to reduce rare-earth dependency by up to 50% while maintaining torque output. These designs, integrated in vehicles from manufacturers like Tesla and BYD, improve range efficiency to over 4 miles per kWh through optimized flux concentration.

Mechanical and fluid devices

In mechanical and fluid devices, a rotor serves as the rotating component, typically consisting of an impeller or blade assembly mounted on a shaft, that facilitates the transfer of energy between a fluid and mechanical work in turbomachines such as turbines, pumps, and compressors.^[6] In turbines, the rotor extracts kinetic and internal energy from the fluid to produce mechanical power on the shaft, while in pumps and compressors, it imparts energy to the fluid to increase its pressure or velocity.^[7] This energy exchange occurs through the interaction of fluid flow with the rotor blades, enabling applications in power generation and fluid handling systems.^[8] Rotors are classified by flow configuration into axial-flow and radial-flow types. Axial-flow rotors direct fluid parallel to the shaft axis, using propeller-like blades to generate thrust with high flow rates at low pressure increases, suitable for large-volume applications.^[9] In contrast, radial-flow rotors (also known as centrifugal) direct fluid perpendicular to the shaft, entering radially and exiting axially, which produces higher pressure heads at moderate flow rates through centrifugal force.^[10] High-speed rotors in these devices often employ titanium alloys, such as Ti-6Al-4V, for their superior strength-to-weight ratio, corrosion resistance, and fatigue endurance under extreme rotational stresses exceeding 10,000 rpm.^[11] The fundamental principle governing energy transfer in rotors is Euler's turbine equation, derived from the conservation of angular momentum across the rotor. Considering a fluid element entering the rotor at station 1 and exiting at station 2, the torque exerted by the blades changes the fluid's tangential momentum, yielding the specific work transfer Δh = U₂V_{θ2} - U₁V_{θ1}, where U is the blade tangential speed at the respective radii and V_θ is the fluid's absolute tangential velocity component.^[12] This equation arises by applying the moment of momentum theorem: the rate of change in angular momentum equals the torque, so power P = ṁ (r₂V_{θ2} - r₁V_{θ1}) U / r (simplifying to the form above for specific enthalpy change per unit mass), highlighting how blade speed and swirl velocity dictate efficiency.^[12] Rotors find widespread use in steam and gas turbines for power plants, where multi-stage axial rotors convert high-pressure steam or combustion gases into electricity, achieving efficiencies up to 64% in combined-cycle setups. Centrifugal pumps with radial rotors are essential in water supply systems, handling large volumes for municipal distribution, irrigation, and wastewater treatment with heads up to 100 meters.^[13] Common failure modes include cavitation in pumps, where low-pressure zones form vapor bubbles that collapse, causing impeller erosion, vibration, and bearing damage, often mitigated by ensuring net positive suction head margins.^[14] Rotor imbalance, arising from manufacturing tolerances or wear, induces excessive vibrations leading to fatigue; balancing techniques, such as dynamic methods per ISO 21940-11 standards, involve adding or removing mass at high speeds to achieve residual unbalance below G2.5 grades.^[15] Recent advancements through 2025 have integrated additive manufacturing to produce complex rotor geometries for renewable energy, enhancing efficiency in tidal turbines by enabling lightweight, corrosion-resistant blades with internal cooling channels compared to traditional forging.^[16] For instance, 3D-printed composite rotors in tidal stream systems have demonstrated improved hydrodynamic performance and scalability for offshore deployment.^[17]

Aerospace applications

In aerospace, rotors refer to rotating blade assemblies that generate aerodynamic forces, primarily providing lift for vertical flight in helicopters or extracting kinetic energy from wind for power generation in turbines. In helicopters, the rotor acts as an actuator disk, inducing airflow to produce thrust and enable maneuverability through controlled blade pitch. Horizontal-axis wind turbine rotors, typically featuring three blades mounted on a horizontal shaft, capture wind energy by converting rotational motion into electrical power via connected generators. Helicopter rotor systems commonly include a main rotor for primary lift and propulsion, paired with a tail rotor to counteract torque and enable yaw control. Autorotation mechanics allow safe descent without engine power, where upward airflow through the rotor disk drives blade rotation, maintaining lift via adjusted collective pitch to balance energy in driving, driven, and stall regions of the rotor disk. Horizontal-axis wind turbine rotors dominate offshore and onshore installations due to their efficiency in aligning with prevailing wind directions. The lift generated by a helicopter rotor follows the adapted form of the aerodynamic lift equation:

L = \frac{1}{2} \rho A v^2 C_L

where \rho is air density, A is the rotor disk area, v is the induced velocity, and C_L is the lift coefficient influenced by collective pitch angle. This derives from momentum theory, which models the rotor as an actuator disk accelerating air downward to produce thrust. For wind turbine rotors, momentum theory establishes the Betz limit, the theoretical maximum efficiency of 16/27 (approximately 59.3%) for power extraction, as no turbine can slow wind to zero velocity without wake interference.^[18] Key historical milestones include Igor Sikorsky's VS-300, the first practical helicopter, which achieved its initial tethered flight on September 14, 1939, demonstrating stable single main rotor operation with a tail rotor for torque compensation. Post-2010, drone rotor technology surged with the commercialization of multirotor configurations, enabling agile unmanned aerial vehicles for applications like surveillance and delivery, driven by advancements in lightweight materials and battery efficiency.^[19]^[20] Contemporary challenges as of 2025 focus on noise reduction for electric vertical takeoff and landing (eVTOL) rotors in urban air mobility, where techniques like phase synchronization in dual-rotor systems mitigate tonal noise by up to 10 dB through optimized blade timing. In offshore wind farms, variable-speed rotors enhance energy capture in turbulent conditions by adjusting rotational rates via pitch control, improving annual energy production by 5-10% compared to fixed-speed designs while mitigating structural loads.^[21]^[22]

Computing and cryptography

Rotor machines

Rotor machines are electromechanical devices used for encryption, consisting of rotating wheels known as rotors that implement wired permutations to substitute and transpose letters in a polyalphabetic cipher fashion, generalizing earlier systems like the Vigenère cipher by generating vast numbers of substitutions through mechanical rotation.^[23] These machines typically feature multiple rotors, each with 26 electrical contacts corresponding to the alphabet, where internal wiring creates a fixed permutation that shifts with each rotor position, producing a long-period key stream for encoding messages via a keyboard-to-lampboard interface.^[24] The Enigma machine, developed by German engineer Arthur Scherbius and patented in 1918, exemplifies rotor-based encryption and was employed by the German military from 1926 through World War II until 1945.^[24] Its core structure includes an entry wheel that passes input unchanged, three (or more in later variants) selectable rotors wired for unique permutations, a fixed reflector that pairs contacts to reverse the current path without self-encryption, and a plugboard allowing up to 10 letter-pair swaps to further scramble the signal.^[24] Daily key settings, distributed via codebooks, specified rotor order, starting positions, and plugboard connections, while each message began with a unique indicator encrypted twice under the daily key to verify settings.^[24] In operation, pressing a key sends current through the components, where the combined permutations from plugboard, rotors, and reflector yield the output letter, after which a stepper mechanism advances the rightmost rotor one position per keystroke to ensure non-repeating cycles, with subsequent rotors advancing irregularly upon full revolutions via notches, creating irregular stepping akin to a odometer.^[24] Mathematically, encryption relies on composing permutations in the symmetric group S_{26}, where the total transformation is the product of individual rotor wirings shifted by their positions, reflector pairing, and plugboard swaps, yielding approximately 10^{23} possible configurations for the three-rotor model including ring settings and plugboard—vastly more than simpler ciphers due to the 26^3 initial positions, 26^3 ring settings, 60 rotor orders (permutations of 3 out of 5 rotors), and approximately 1.5×10^{14} plugboard arrangements. This design aimed for cryptographic strength through periodicity exceeding message lengths, though vulnerabilities arose from operator errors like repeated indicators. During World War II, Enigma secured German communications, including U-boat operations that sank over 200 Allied ships in early 1942, but its cryptanalysis profoundly influenced the war's outcome.^[25] Polish mathematicians Marian Rejewski, Jerzy Różycki, and Henryk Zygalski broke early versions in 1932 using mathematical cycle analysis and intelligence from traitor Hans-Thilo Schmidt, developing the electromechanical "Bomba" device to test rotor settings efficiently.^[25] Alan Turing, building on Polish work at Bletchley Park, refined these methods and designed the improved Bombe machine—over 200 units built by 1944—that simulated multiple Enigma configurations in parallel to exploit cribs (known plaintext) and indicator weaknesses, decrypting messages in under 20 minutes and enabling Allied interception of vital intelligence, such as U-boat positions, which contributed to sinking 95 German submarines by war's end.^[25] By the 1970s, rotor machines like the U.S. KL-7 and Soviet Fialka had largely been supplanted by electronic and software-based systems for military use, with no widespread active deployment thereafter due to the rise of digital computing.^[26] Their legacy persists in modern simulations via software emulators that replicate historical operations for research and education, and conceptual influences on stream cipher designs, where pseudorandom permutation sequences echo rotor stepping for key generation.^[27]

Software and projects

The Shared Source Common Language Infrastructure (SSCLI), codenamed Rotor, was a 2002 Microsoft project that provided a shared-source implementation of the Common Language Infrastructure (CLI) specification for the .NET Framework, designed to facilitate porting to non-Windows platforms such as FreeBSD and Mac OS X.^[28] Released under the Microsoft Shared Source License, which permitted non-commercial use and modification but restricted redistribution for profit, Rotor aimed to promote educational and research efforts in CLI development while demonstrating .NET's portability.^[29] Rotor included a C# compiler, just-in-time (JIT) compiler, and runtime environment, enabling developers to build and execute .NET applications on supported platforms. It significantly influenced the open-source Mono project, which expanded on Rotor's concepts to create a fully open-source, cross-platform .NET implementation compliant with ECMA standards.^[30] The project released version 2.0 in 2006 and was not actively developed thereafter, with its source code archived for historical reference.^[31] Beyond Microsoft's initiative, "Rotor" has been adopted as a name for various open-source software projects in computing. For instance, ROTOR is a refactoring tool for OCaml codebases, automating the renaming of value bindings across multi-file modules to support safer code evolution.^[32] Similarly, Rotor serves as a lightweight build system for Elixir projects, handling compilation and command execution in response to file changes.^[33] In content management, Rotor is an open-source PHP-based mobile CMS that relies on a MySQL database for site content storage.^[34] Additionally, a Blender add-on named Rotor provides modeling utilities for mesh operations in 3D graphics workflows.^[35] As of 2025, community interest in the original Microsoft Rotor persists through educational revivals, including forks and extensions for studying legacy .NET runtimes in cloud computing environments. A notable recent contribution is an optimizing JIT compiler for Rotor, which enhances its original single-pass design with multi-pass optimizations like loop detection and instruction folding, achieving up to 10 times faster code execution on IA-32 architectures.^[36] These efforts underscore Rotor's enduring role in cross-platform development pedagogy.

Chemistry

Molecular spectroscopy

The rigid rotor model serves as a foundational approximation in quantum chemistry, treating diatomic molecules as rigid, non-vibrating bodies undergoing free rotation to predict discrete rotational energy levels observable in microwave and far-infrared spectra.^[37] This model simplifies the complex dynamics of molecular rotation by assuming fixed internuclear distances, enabling the interpretation of spectral line spacings to infer molecular properties such as bond lengths.^[38] In the quantum mechanical framework, the rotational energy levels for a diatomic rigid rotor are expressed as

E_J = B J(J+1),

where B = \frac{\hbar^2}{2I} is the rotational constant, I = \mu r_e^2 is the moment of inertia with reduced mass \mu and equilibrium bond length r_e, and J = 0, 1, 2, \dots is the rotational quantum number.^[38] For electric dipole-allowed transitions in pure rotational spectroscopy, the selection rule is \Delta J = \pm 1, resulting in equally spaced spectral lines separated by $2B.^[37] This formulation arises from solving the time-independent Schrödinger equation in spherical coordinates for a particle on a rigid sphere, separating variables to yield spherical harmonics as eigenfunctions and the above energy eigenvalues.^[38] The model finds practical application in determining equilibrium bond lengths from observed spectra; for instance, analysis of the microwave rotational spectrum of HCl yields a rotational constant B \approx 10.59 cm^{-1} , corresponding to a bond length of approximately 1.275 Å.^[39] However, the rigid rotor approximation breaks down at higher rotational quantum numbers due to centrifugal forces stretching the bond, introducing a distortion term that modifies the energy levels to

E_J = B J(J+1) - D [J(J+1)]^2,

where D \approx 4B^3 / \omega^2 is the centrifugal distortion constant, with \omega the vibrational frequency. Additionally, pure rotational spectra require a permanent electric dipole moment, which heteronuclear diatomic molecules possess but homonuclear ones (e.g., N_2, O_2) lack, rendering the latter inactive in microwave spectroscopy.^[37] As of 2023, advances have integrated the rigid rotor model with ab initio computational chemistry methods, such as coupled-cluster theory and machine learning-accelerated predictions of rotational constants, to extend its utility to polyatomic nonlinear rotors in astrochemistry, facilitating the detection and assignment of complex interstellar molecules like cyanomethanimine in sources such as Sagittarius B2.^[40] Continuing developments as of 2025 include the tentative interstellar detection of molecules such as 3-hydroxypropanal toward G+0.693−0.027 using rotational spectroscopy supported by laboratory and theoretical predictions.^[41] Similarly, phenalene (c-C_{13}H_{10}) was identified in TMC-1 in 2025 via the QUIJOTE survey, leveraging accurate rotational constants.^[42]

Laboratory equipment

In laboratory settings, rotors serve as the essential rotating components in centrifuges, designed to hold sample tubes or containers and facilitate density-based separation of mixtures through high-speed rotation. These devices exploit centrifugal force to accelerate the sedimentation of particles or molecules differing in density, enabling the isolation of components like cells, proteins, or nucleic acids from complex biological or chemical samples. Centrifuge rotors are broadly classified into fixed-angle and swinging-bucket types, each suited to specific separation needs. Fixed-angle rotors maintain tubes at a constant angle (typically 25–40 degrees) relative to the axis of rotation, promoting rapid pelleting of denser particles along the tube walls and are ideal for high-speed applications such as ultracentrifugation. In contrast, swinging-bucket rotors allow tubes to pivot outward to a horizontal position during operation, enabling isopycnic separations where particles equilibrate at their buoyant densities, though they generally operate at lower speeds due to mechanical constraints. Modern rotors can achieve rotational speeds up to 100,000 rpm, generating relative centrifugal forces (RCF) exceeding 1,000,000 × g, while incorporating safety features like automatic imbalance detection sensors and sealed lids to prevent aerosol formation and ensure operator protection during operation. The underlying principle of rotor-mediated separation relies on sedimentation velocity, governed by the equation v = \frac{\omega^2 r \Delta \rho}{f}, where v is the sedimentation velocity, \omega is the angular velocity, r is the radial distance from the axis of rotation, \Delta \rho is the density difference between the particle and the medium, and f is the frictional coefficient of the particle. This formula, derived from Stokes' law adapted for centrifugal fields, quantifies how rotational speed and geometry drive differential migration, with higher \omega and r amplifying separation efficiency for macromolecules. In chemical and biochemical laboratories, rotors enable key applications in ultracentrifugation, such as purifying proteins by differential centrifugation or isolating DNA through density gradient methods like cesium chloride gradients. Historically, the development of the air-driven ultracentrifuge by Theodor Svedberg in the 1920s revolutionized macromolecular analysis, earning him the 1926 Nobel Prize in Chemistry for demonstrating the molecular weights of proteins via sedimentation equilibrium techniques using early rotor designs. As of 2025, advancements include microfluidic rotors integrated into compact, portable centrifuges for point-of-care testing in biotechnology labs, allowing rapid separation of blood components or exosomes without bulky equipment, as demonstrated in devices achieving 10,000 rpm in handheld formats for diagnostic workflows.

Medicine

Genetic disorders

Rotor syndrome is a rare, benign autosomal recessive disorder characterized by chronic, low-grade conjugated hyperbilirubinemia without associated liver damage or dysfunction.^[43] The condition results from impaired hepatic uptake and storage of bilirubin conjugates, leading to their reflux into the bloodstream, but it does not cause hemolysis or progressive liver disease.^[44] Genetically, Rotor syndrome follows a digenic inheritance pattern, requiring biallelic mutations in both the SLCO1B1 and SLCO1B3 genes, which encode the organic anion-transporting polypeptides OATP1B1 and OATP1B3, respectively; the digenic genetic basis was first identified in 2010.^[45]^[46] These mutations disrupt the reuptake of conjugated bilirubin from plasma into hepatocytes and its intrahepatic storage, causing predominantly elevated direct bilirubin levels (typically 50-80% of total bilirubin).^[47] Over 30 pathogenic variants have been identified across these genes, with common insertions like a 6.1-kb L1 retrotransposon in SLCO1B1 contributing to the phenotype in multiple families.^[46] Clinically, affected individuals often present with mild, intermittent jaundice starting in adolescence or early adulthood, alongside elevated serum direct bilirubin (usually 2-5 mg/dL) but normal liver enzymes, albumin, and prothrombin time.^[48] Diagnosis is confirmed through genetic testing and exclusion of other causes, such as viral hepatitis or biliary obstruction; it is distinguished from the similar Dubin-Johnson syndrome by the absence of coarse brown pigment in centrilobular hepatocytes on liver biopsy and increased total urinary coproporphyrin excretion, with coproporphyrin I comprising 60-80% of the total (in contrast to >80% in Dubin-Johnson syndrome, where total excretion is normal).^[49]^[50] Liver histology in Rotor syndrome shows no abnormalities beyond occasional mild cholestasis.^[44] Epidemiologically, Rotor syndrome is extremely rare, with an estimated prevalence of less than 1 in 1,000,000 worldwide, though it appears more frequent in the Philippines, where it was first described in 1948 by Filipino physician Arturo Belleza Rotor based on observations in six patients from two families.^[44] Cases have since been reported globally, including in Japanese, Irish, and Latin American populations, but no founder effects or elevated incidence in groups like the Amish have been consistently documented.^[51] The disorder's rarity stems from the need for concurrent homozygous or compound heterozygous mutations in two distinct genes.^[47] Management of Rotor syndrome is supportive, as the condition is lifelong but asymptomatic beyond jaundice and carries no risk of liver failure or other complications.^[44] No specific pharmacological interventions are required, though phenobarbital may transiently reduce bilirubin levels in some cases by inducing alternative excretion pathways.^[46] Genetic counseling is recommended for affected families, and diagnosis has been enhanced by advancements in targeted next-generation sequencing panels that identify SLCO1B1 and SLCO1B3 variants with high sensitivity, as demonstrated in recent studies of recurrent mutations in diverse cohorts up to 2025.^[52]

Diagnostic and therapeutic devices

In medical diagnostic and therapeutic devices, rotors play a critical role in enabling precise imaging and fluid delivery through controlled rotational motion. In computed tomography (CT) scanners, the gantry—a large rotating assembly containing the X-ray tube and detectors—encircles the patient at high speeds to acquire cross-sectional images. This rotation, typically completing a full 360-degree cycle in as little as 0.3 seconds in modern systems, allows for rapid data collection while minimizing motion artifacts.^[53] The continuous rotation is facilitated by slip-ring technology, which transmits power and signals without physical contacts that could limit speed or introduce wear.^[54] A key example of rotor application in X-ray generation is the rotating anode within X-ray tubes, essential for both standalone radiography and CT imaging. The anode, often made of tungsten, spins at speeds up to 10,000 revolutions per minute (rpm) to distribute heat generated by the electron beam across a larger surface area, preventing localized overheating and enabling sustained high-power operation. Without this rotation, heat dissipation would be insufficient, limiting scan duration and image quality.^[55] In therapeutic contexts, rotors are integral to peristaltic infusion pumps, where a rotating mechanism with rollers compresses flexible tubing to propel precise volumes of medication or fluids into patients, ensuring sterility and accurate dosing in applications like chemotherapy or anesthesia.^[56] The underlying principles of these rotors often leverage physics such as angular momentum conservation, particularly in advanced surgical robotics. Gyroscopic stabilizers, incorporating spinning rotors, counteract unwanted torques during procedures by precessing to maintain tool stability, thus enhancing precision in minimally invasive surgeries. For instance, in robotic arms or endoscopes, these devices use the gyroscope's angular momentum to resist external disturbances, allowing sub-millimeter accuracy.^[57] While MRI systems rely on gradient coils to spatially encode signals—creating linear magnetic field variations without physical rotation—these coils indirectly support rotational encoding in image reconstruction algorithms.^[58] Historically, the integration of rotors in medical imaging began with the invention of the first CT scanner by Godfrey Hounsfield in 1971, which used a rotating X-ray source and detectors to produce the initial clinical brain images at Atkinson Morley's Hospital. This step-and-shoot approach evolved into helical (spiral) scanning in the early 1990s, where continuous gantry rotation coupled with patient table movement enabled volumetric imaging without gaps, dramatically reducing scan times from minutes to seconds.^[59] By the 2020s, innovations in robotic surgery have incorporated AI algorithms to dynamically optimize rotor speeds in gyroscopic and actuation systems, adapting in real-time to tissue feedback for enhanced precision in procedures like endoscopy. Such advancements, building on Hounsfield's foundational rotor-based design, continue to expand the scope of rotor applications in personalized medicine.

Mathematics

Vector calculus

In vector calculus, the rotor, also known as the curl, of a vector field \mathbf{F} is defined as the vector operator \nabla \times \mathbf{F}, which quantifies the infinitesimal circulation of \mathbf{F} per unit area in three-dimensional Euclidean space. This operator measures the local tendency of the field to rotate or swirl around a point, with the magnitude indicating the rotation strength and the direction aligning with the rotation axis via the right-hand rule.^[60]^[61] The components of the rotor in Cartesian coordinates for a vector field \mathbf{F} = (F_x, F_y, F_z) are explicitly given by

\nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right).

Its physical interpretation arises from Stokes' theorem, which relates the surface integral of the rotor over an oriented surface S to the line integral of \mathbf{F} around the boundary curve \partial S:

\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{A} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}.

This equivalence demonstrates that the rotor captures the net circulation enclosed by the surface, providing a localized measure of rotational behavior.^[60] Key applications of the rotor include fluid dynamics, where the vorticity \boldsymbol{\omega} = \nabla \times \mathbf{v} of the velocity field \mathbf{v} describes the local spinning motion of fluid elements, essential for analyzing turbulence and eddies. In electromagnetism, it features prominently in Ampère's law with Maxwell's correction, expressed as \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}, linking magnetic field rotation to electric currents and time-varying electric fields.^[62]^[63] The mathematical concept underlying the curl was developed by William Rowan Hamilton in the 1850s through his quaternion framework, where the vector part of the quaternion product corresponds to operations foundational to the curl in modern vector analysis. The specific term "rotor" for this operator was introduced later, notably by William Kingdon Clifford in his work on geometric algebras. The more common term "curl" was introduced by James Clerk Maxwell in 1871.^[64]#History) The term "rotor" is used in some mathematical and engineering texts as an alternative to "curl," particularly in contexts influenced by quaternion or geometric algebra traditions.^[65] By 2025, extensions in computational simulations leverage the rotor in computational fluid dynamics (CFD) for climate modeling, where vorticity computations enable accurate forecasting of global atmospheric rotations and storm systems.^[66]

Geometric algebra

In geometric algebra, a rotor is defined as an even-grade multivector R = e^{B/2}, where B is a bivector representing the plane of rotation, and for unit rotors that preserve lengths, it satisfies R \tilde{R} = 1, with \tilde{R} denoting the reverse of R.^[67]^[68] This formulation generalizes the exponential map from Lie algebras to Clifford algebras, allowing rotors to encode rotations as oriented plane elements rather than axis-angle pairs.^[69] Rotors apply rotations to vectors via the sandwich product v' = R v \tilde{R}, which rotates the vector v by twice the angle encoded in the bivector B within its plane.^[70] In three dimensions, this generalizes the quaternion representation of rotations, where a unit quaternion corresponds to a rotor in the even subalgebra of the Clifford algebra \mathrm{Cl}(3,0), but rotors extend naturally to higher dimensions by exponentiating bivectors in arbitrary planes, enabling multi-plane rotations without gimbal lock.^[71]^[72] Compared to rotation matrices, rotors in conformal geometric algebra (CGA) offer advantages for representing affine transformations, as the sandwich product X' = R X \tilde{R} unifies rotations, translations, and dilations under a single operator framework. In CGA, which embeds Euclidean space into a higher-dimensional Minkowski space, rotors (as versors) can model screw motions—combined rotation and translation along a common axis—facilitating compact descriptions of rigid body transformations.^[73]^[74] The modern development of rotors in geometric algebra is primarily attributed to David Hestenes, who from the 1960s to the 1980s reformulated Clifford algebras for physics and geometry, introducing rotors in works like Space-Time Algebra (1966) and Clifford Algebra to Geometric Calculus (1984) to unify rotations across dimensions.^[75]^[76] These tools have since been adopted in computer graphics for efficient interpolation and blending of orientations, and in robotics for forward and inverse kinematics solving via rotor compositions.^[77]^[78] As of 2025, rotors find applications in quantum computing, where even-grade multivectors represent spinors for qubit states and gate operations, as explored in geometric algebra-based Jordan-Wigner transformations and quantum convolutional networks.^[79]^[80] In virtual reality rendering engines, such as those integrated with Unity via packages like GA-Unity, rotors enable real-time handling of 3D transformations and networked synchronization for immersive environments.^[81]^[82]

Other uses

Transportation

In transportation, rotors refer to rotating cylindrical sails employed on ships for wind-assisted propulsion or spinning wheels integrated into experimental rail systems for stability. Rotor ships utilize vertically mounted, motor-driven cylinders that harness the Magnus effect to generate lift perpendicular to the wind direction, providing forward thrust without traditional sails. These systems contrast with conventional propulsion by leveraging aerodynamic forces rather than direct wind capture.^[83] A seminal example is the Flettner rotor ship developed in the 1920s by German engineer Anton Flettner, which employs the Magnus effect for propulsion. The lift force generated is given by the formula

F = \rho v \Gamma L

where \rho is air density, v is the relative wind speed, \Gamma is the circulation around the rotor, and L is the rotor length; this force propels the vessel by creating a pressure differential across the spinning cylinder. The first such vessel, Buckau (later renamed Baden Baden), demonstrated viability by crossing the Atlantic in 1926, achieving speeds up to 10 knots under wind power alone.^[84]^[83] Historically, the rotor ship Barbara, launched in 1926 by A.G. Weser in Bremen, Germany, featured three large rotors and carried up to approximately 3,000 tons of cargo, marking an advancement in scale for the technology. Operated under government auspices, it aimed to reduce fuel consumption by integrating rotors with auxiliary engines, though economic challenges like the Great Depression limited widespread adoption. Interest revived in eco-shipping after 2010, driven by rising fuel costs and environmental regulations, leading to installations on vessels like Enercon's E-Ship 1 in 2010, which used four rotors to cut fuel use by approximately 25% on turbine blade transport routes.^[85]^[86] Beyond ships, rotor wheels appear in experimental monorails for stabilization, as in gyro monorails where high-speed spinning flywheels provide gyroscopic precession to counterbalance tilting forces on a single rail. Early prototypes, such as Louis Brennan's 1910 design, used twin 7-foot-diameter rotors spinning at 1,400 rpm to maintain equilibrium, enabling safe operation at speeds up to 30 mph without derailing. Safety metrics for these systems emphasize redundancy in gyro controls to prevent instability, while efficiency stems from reduced track width and material needs compared to conventional rails; modern simulations indicate potential energy savings of 20-30% in urban transit due to lower friction. Rotor ships similarly offer safety through automated feathering in high winds to avoid overload, with efficiency metrics showing 5-20% average fuel reductions across global routes, depending on wind conditions and vessel size.^[87]^[88] By 2025, hybrid rotor-sail systems on cargo ships have gained traction under International Maritime Organization (IMO) regulations targeting net-zero emissions by 2050, integrating rotors with conventional engines for optimized performance. Installations like those by Norsepower on bulk carriers, such as the 2024 Vale-equipped capesize vessel, achieve 6-10% emissions cuts, while broader fleet analyses project up to 20% fuel reductions through wind assistance, supporting IMO's Energy Efficiency Existing Ship Index (EEXI) and Carbon Intensity Indicator (CII) compliance. In November 2025, Anemoi Marine Technologies installed five 35-meter-tall Rotor Sails on the Vale-chartered 400,000 DWT ore carrier NSU Tubarao, further advancing adoption on large vessels. These systems enhance propulsion without compromising cargo capacity, with over 50 rotors deployed globally as of late 2025.^[89]^[90]^[91]

Entertainment and culture

In entertainment, the Rotor ride, also known as the Gravitron, is a classic amusement park attraction consisting of a large cylindrical chamber that spins rapidly to create a centrifugal force effect, allowing riders to stick to the walls even after the floor drops away. Invented by German engineer Ernst Hoffmeister and first demonstrated at the 1949 Oktoberfest, the ride relies on physics where the normal force exerted by the wall on the rider provides the centripetal acceleration, given by N = m \omega^2 r, with m as the rider's mass, \omega as the angular velocity, and r as the radius of the cylinder. This force must exceed the frictional requirement to counteract gravity, typically achieved at speeds around 33 revolutions per minute, producing an exhilarating "wall-stick" sensation through apparent weightlessness relative to the spinning environment.^[92]^[93] In media, Rotor the Walrus is a prominent fictional character from the Archie Comics' Sonic the Hedgehog series, debuting as an inventive mechanic and Freedom Fighter in issue #¼ published in early 1993. As a level-headed engineer aiding Sonic and his allies against Dr. Robotnik, Rotor embodies ingenuity with gadgets and tools, appearing across over 200 issues until the series' conclusion in 2017. Additionally, FC Rotor Volgograd, a professional Russian soccer club founded in 1929 as Traktor Stalingrad, has garnered cultural significance through its blue-cyan kits and storied history in Soviet and Russian leagues, including runners-up finishes in the Russian Premier League in 1993 and 1997.^[94] Rotor also features in sports equipment, particularly through Rotor Bike Components, a Spanish company established in 1995 by aeronautical engineers to optimize pedaling efficiency by addressing dead spots in the crank cycle. Their patented systems, such as the Q-Rings oval chainrings introduced in the late 1990s, enable variable gearing by altering chain tension throughout the pedal stroke, enhancing power output for cyclists in road and mountain biking. These innovations, protected by multiple patents since 1998, have been adopted by professional teams for improved biomechanics without traditional derailleur shifts.^[95] Culturally, rotor motifs symbolizing perpetual motion appear in modern art, notably Marcel Duchamp's Rotoreliefs from the 1920s onward, where spinning disks create optical illusions of depth and movement, influencing kinetic art movements that explore dynamism and perception. In film, similar rotational elements evoke chaos or energy, as seen in experimental animations blending abstraction and metamorphosis to mimic endless cycles. In New Zealand, Rotorua's geothermal landscapes inspire cultural events blending Māori traditions with natural "rotors" like geysers and hot springs, showcased in experiences at sites such as Te Puia, where festivals highlight volcanic activity alongside haka performances and weaving demonstrations.^[96]^[97] By 2025, rotor-inspired mechanics have emerged in virtual reality gaming, with spinning orbital navigation and momentum-based puzzles that simulate rotational physics for immersive zero-gravity exploration. These VR experiences draw from amusement ride dynamics to heighten disorientation and thrill, reflecting broader cultural fascination with rotational forces in digital entertainment.

References

[1]
Spindle and Rotor Dynamics - UW Mechanical Engineering
Every rotary machine consists of three basic elements: a rotary part (rotor), a stationary part (housing), and multiple bearings that connect the rotary and ...
[2]
ME 6244: Rotordynamics
Rotating shafts are employed in industrial machines such as steam and gas turbines, turbogenerators, internal combustion engines, reciprocating and centrifugal ...
[3]
[PDF] Inductions Motors - Power & Energy Systems
Sep 16, 2016 · Wound rotors have winding schemes similar to stators – typically wound for 3 phases and with a pole count matching the stator. Slip rings are ...
[4]
[PDF] Synchronous Machines – Review Sheet - page for courses.ece.vt.edu
The rotor on the machine is either a permanent magnet or a DC electromagnet. The magnetic field of the rotor follows that of the stator. This causes the ...
[5]
ECE252 Lesson 19 - Speed School of Engineering
Oct 25, 2021 · The stator is the stationary part of the motor or generator; the rotor is the rotational part. Both parts are usually made of iron so that they ...
[6]
Compressor Rotor - an overview | ScienceDirect Topics
A compressor rotor is defined as a component in aircraft gas turbine engines that compresses air to increase thrust, which can be constructed from materials ...
[7]
[PDF] Chapter 12 Pumps and Turbines
❖Turbomachines involve a collection of blades, buckets, flow channels, or passages arranged around an axis of rotation to form a rotor.
[8]
Introduction to turbomachinery
Their characteristic feature is the rotor, which is a shaft with blades around its circumference (called an impeller). The blades form a so-called blade ...
[9]
Axial Flow Pumps vs. Radial Flow Pumps - Pumpworks
Nov 20, 2024 · Axial flow pumps move fluid parallel to the pump shaft. They are characterized by their propeller-like impeller, which generates axial thrust to propel the ...
[10]
Axial Vs Radial Turbines - Choosing Your Turbomachinery - SoftInWay
The primary distinction between axial and radial turbines lies in the way fluid flows through the components (compressor and turbine). These two types of ...
[11]
Applications Of Different Alloys Of Titanium In Aerospace
Apr 3, 2025 · Ti-6Al-4V (Ti-6–4) is the most widely used titanium alloy in aerospace, employed in both rotary and static components, as well as in structural ...
[12]
12.3 The Euler Turbine Equation - MIT
The Euler turbine equation relates the power added to or removed from the flow, to characteristics of a rotating blade row.
[13]
Reliable steam turbines for improved efficiency - Siemens Energy
Generally, state-of-the-art steam turbines can achieve efficiencies ranging from 30% to over 45% in power generation applications. In combined cycle power ...
[14]
Different Types of Centrifugal Pumps and Their Applications | DXP
Oct 20, 2025 · They are commonly used in applications such as wastewater handling, irrigation, and dewatering systems. Vertical vs. Horizontal Centrifugal ...
[15]
https://www.pumpsandsystems.com/following-industry-standard-guidelines-balancing-centrifugal-pumps
[16]
Following Industry Standard Guidelines for Balancing Centrifugal ...
Dec 17, 2011 · Balancing is essential for minimizing vibration, increasing bearing life and minimizing downtime and repair costs.<|separator|>
[17]
Additive Manufacturing Could Turn the Tides for Marine Energy ...
May 3, 2024 · Researchers at the National Renewable Energy Laboratory (NREL) discovered another promising pair: marine energy and additive manufacturing.Missing: rotors 2020-2025
[18]
AMRC Cymru to help develop the next generation of tidal stream ...
May 29, 2025 · “Supporting the design and development of the newest tidal turbine blades, the project will lay the groundwork for a lasting collaboration, ...<|control11|><|separator|>
[19]
[PDF] 1 Momentum Theory - University of Notre Dame
The power extracted by the actuator disc is give by Eq. 21, and the maximum power coefficient, CPmax. = 0.593, is given by the Betz limit given in Eq 24.
[20]
VS-300: The First Practical Helicopter – Igor I Sikorsky Historical ...
The VS-300 was first flown (tethered) on September 14, 1939 and the first untethered flight was on May 13, 1940. Testing continued and on May 6, 1941, Sikorsky ...
[21]
Recent Advances in Unmanned Aerial Vehicles: A Review - PMC
Apr 25, 2022 · UAV inspection has attained numerous growths since 2010 as it is widely implemented in the inspection of various applications such as power ...
[22]
Numerical Study on Dual-Rotor Phase Synchronization for Tonal ...
Sep 23, 2025 · The study shows that the rotor phase synchronization has important potential in the directional noise mitigation for dual-rotor systems.
[23]
[PDF] Harnessing Power from the Wind | AEE
Jul 25, 2025 · The pitch of their blades is adjusted to help regulate the speed of rotor rotation. For horizontal axis wind turbines, increasing the diameter ...
[24]
[PDF] CPSC 467: Cryptography and Computer Security
Sep 9, 2015 · Rotor machines are mechanical polyalphabetic cipher devices that ... History. Several different kinds of rotor machines were built and used ...
[25]
[PDF] Rotor Machines Enigma - CS408 Cryptography & Internet Security
● A machine consists of multiple cylinders. ▪ each cylinder has 26 states, and each state it is a. substitution cipher: the wiring between the contacts. ...
[26]
[PDF] Solving the Enigma: History of Cryptanalytic Bombe
Each rotor on an Enigma can be set in any one of twenty-six positions (1-26 or A-Z). On a three-rotor machine the number of possible settings is 26. 3 or ...
[27]
The Scandalous History of the Last Rotor Cipher Machine
Aug 31, 2021 · The rotor-machine era finally ended around 1970, with the advent of electronic and software encryption, although a Soviet rotor machine called ...
[28]
Rotor machines and bombes - ResearchGate
Although rotor encryptions, like many other encryption techniques, can very easily, cheaply, and quickly be simulated by a modern computer, there is no need to ...Missing: influence | Show results with:influence
[29]
Rotor - Shared Source CLI Provides Source Code for a FreeBSD ...
Jun 18, 2002 · This article discusses some of the things you can learn from the source code facsimile of the CLR, like how JIT compilation works. It will also ...
[30]
https://www.codemag.com/article/2009041/When-Open-Source-Came-to-Microsoft
[31]
https://github.com/SSCLI/sscli20_20060311
[32]
SSCLI/sscli20_20060311: Version: 2.0 Date Published: 3 ... - GitHub
The Shared Source CLI is a compressed archive of the source code to a working implementation of the ECMA CLI and the ECMA C# language specification.
[33]
[PDF] ROTOR: A Tool for Renaming Values in OCaml's Module System
We have built a tool called ROTOR, which is capable of automatically renaming value bindings, as illustrated above, in large multi-file OCaml codebases. It is ...
[34]
HashNuke/rotor: Super-simple build system for Elixir - GitHub
Rotor is a build system for Elixir projects. Use it to compile things, run commands or do anything that needs to be run when files change. Wreckers don't call ...
[35]
visavi/rotor: Rotor - Mobile content management system - GitHub
Rotor mobile cms is a functionally complete open source content management system written in PHP. It uses a MySQL database to store the contents of your site.
[36]
Blender Extensions
No readable text found in the HTML.<|control11|><|separator|>
[37]
[2411.09391] An Optimizing Just-In-Time Compiler for Rotor - arXiv
Nov 14, 2024 · The Shared Source CLI (SSCLI), also known as Rotor, is an implementation of the CLI released by Microsoft in source code. Rotor includes a ...
[38]
Molecular Spectra Vol I : Herzberg,Gerhard. - Internet Archive
Nov 15, 2006 · Molecular Spectra Vol I. by: Herzberg,Gerhard. Publication date: 1950/00/00. Topics: NATURAL SCIENCES, Physics, Mechanics of gases.Missing: seminal paper rigid rotor
[39]
Hunting for Complex Organic Molecules in the Interstellar Medium
Nov 30, 2023 · A new approach to computation at affordable cost of accurate geometrical structures and rotational constants for medium-sized molecules in ...Methods · Pcs/bonds Model · Figure 3
[40]
Rotor syndrome: MedlinePlus Genetics
Mar 1, 2013 · Rotor syndrome is a relatively mild condition characterized by elevated levels of a substance called bilirubin in the blood (hyperbilirubinemia).
[41]
Rotor Syndrome - StatPearls - NCBI Bookshelf - NIH
Feb 19, 2023 · Rotor syndrome, also known as Rotor type hyperbilirubinemia, is an autosomal recessive disease and a rare cause of mixed direct (conjugated) and indirect ( ...Rotor Syndrome · Etiology · Pathophysiology
[42]
237450 - HYPERBILIRUBINEMIA, ROTOR TYPE; HBLRR - OMIM
Rotor type hyperbilirubinemia (HBLRR) is caused by digenic inheritance of homozygous mutations in the SLCO1B1 (604843) and SLCO1B3 (605495) genes.<|control11|><|separator|>
[43]
Complete OATP1B1 and OATP1B3 deficiency causes human Rotor ...
Jan 9, 2012 · The obligatory deficiency in two different, medium-sized genes explains the rarity of RS, with a roughly estimated frequency of about 1 in 106, ...Missing: prevalence | Show results with:prevalence
[44]
Rotor Syndrome - GeneReviews® - NCBI Bookshelf - NIH
Dec 13, 2012 · Rotor syndrome is characterized by mild conjugated and unconjugated hyperbilirubinemia that usually begins shortly after birth or in childhood.Diagnosis · Differential Diagnosis · Management · Genetic CounselingMissing: epidemiology | Show results with:epidemiology
[45]
What Is Rotor Syndrome? - Healthline
Aug 18, 2025 · Rotor syndrome is a rare but harmless genetic liver disorder characterized by mild but lifelong jaundice. It's an autosomal recessive condition, ...What Are The Symptoms Of... · What Causes Rotor Syndrome? · Frequently Asked Questions
[46]
Rotor Syndrome: Symptoms and Treatment | Doctor - Patient.info
Jun 21, 2023 · How common is rotor syndrome? (Epidemiology) ... It occurs sporadically in families and is thought to be inherited in an autosomal recessive ...
[47]
Rotor syndrome | About the Disease | GARD
Rotor syndrome is caused by having genetic changes in both the SLCO1B1 and SLCO1B3 genes and is inherited in an autosomal recessive manner.Rotor syndrome is ...Missing: definition | Show results with:definition
[48]
Recurrent SLCO1B1 and SLCO1B3 mutations identified in three ...
Aug 17, 2025 · Rotor syndrome (RS, OMIM*237450) is a rare and benign genetic disease characterized by low-grade, chronic or fluctuating, predominantly ...
[49]
CT-scan Image Production Procedures - StatPearls - NCBI Bookshelf
The X-ray tube and photon detectors are positioned to face each other and are built to rotate 360 degrees in one direction around the patient. The gantry tilt ...
[50]
[PDF] FMI Medical Systems Inc. Paul McFeely Quality Assurance Engineer ...
Sep 13, 2018 · The gantry is organized into two distinct sections, the stator (stationary elements) and the rotor (rotating elements). Slip rings are utilized ...
[51]
Anode (x-ray tube) | Radiology Reference Article | Radiopaedia.org
Jul 27, 2025 · ... rotating them at the speeds up to 10,000 rpm, with a temperature of 2000°C. The purpose of the rotation is to dissipate heat. Most rotating ...
[52]
How Peristaltic Pumps Work in Medical Applications & Beyond
Nov 23, 2022 · In the case of a mechanical peristaltic pump, a motor rotates rollers over a length of tubing fixed against a curved outer circumferential ...
[53]
US20210338371A1 - Stabilizer for surgical shafts or cannulas
More specifically, gyroscopic stabilizer (210) leverages the principle of conservation of angular momentum to counteract externally applied torques due to ...
[54]
Gradient coils - Questions and Answers in MRI
Each coil set is driven by an independent power amplifier and creates a gradient field whose z-component varies linearly along the x-, y-, and z-directions, ...
[55]
Milestones in CT: Past, Present, and Future - PMC
In 1971, the first patient CT exam by Ambrose and Hounsfield paved the way for not only volumetric imaging of the brain but of the entire body.
[56]
Submillimeter fiber robots capable of decoupled macro-micro motion ...
Nov 22, 2024 · To realize macro-micro manipulation of continuum robots for endoluminal procedures, the associated challenges include miniaturized robot size, ...
[57]
Curl of a Vector - RF Cafe
The curl (or rotor) is a vector operator that describes the rotation of a vector field. At every point in the field, the curl is represented by a vector.
[58]
Vorticity | Applied Mathematics | University of Waterloo
An integral part of fluid dynamics is vorticity. Heuristically, it measures the local rotation of a fluid parcel.Missing: computational | Show results with:computational
[59]
The History of Vectors and Matrices - jstor
The vector part is now called the curl of V. Hamilton was to spend the rest of his life "selling" his quaternions. He believed they were important, not only ...
[60]
The rotational/curl of a vector, scalar or tensor field—Wolfram ...
Curl is also known as rot, rotor, rotational and circulation density. · Curl[f,x] can be input as ∇x f. · An empty template ∇ can be entered as delx , and ...
[61]
[PDF] Applications of Vector Calculus in Fluid Dynamics and Weather ...
Apr 19, 2025 · Vector calculus is an essential mathematical framework used to describe and analyze fluid motion and atmospheric processes.
[62]
(PDF) Rotations in classical mechanics using geometric algebra
Oct 30, 2022 · In geometric algebra, the rotation of a vector is described using rotors. Rotors are phasors where the imaginary number has been replaced by ...
[63]
[PDF] The Supergeometric Algebra - arXiv
Nov 28, 2022 · Rotors comprise the set of even multivectors obtained by exponentiating the orthonormal bivectors of the geometric algebra. One of the core ...
[64]
[PDF] Primer on Geometric Algebra - David Hestenes archive
Jul 14, 2005 · Rotor-vector product = vector: Uθ v = e iθ v= u. Thus rotor algebra represents the algebra of 2d rotations ! The concept of rotor generalizes ...
[65]
Translations - Geometric Algebra Tutorial
v ′ = R 1 v R 1 ~ + d 1 v ″ = R 2 v ′ R 2 ~ + d 2 = R 2 ( R 1 v R 1 ~ + d 1 ) R 2 ~ + d 2 We could multiply this out but we will get a lot of terms and the ...
[66]
Geometric Algebra in Quantum Computational Intelligence
Oct 8, 2025 · In these notes we present preliminary results on quantum-like algorithms where tensor product is replaced by geometric product.
[67]
Introduction to rotors and their properties | Geometric Algebra Class ...
A rotor is a special type of multivector that represents rotations in geometric algebra · Rotors are the geometric product of an even number of unit vectors, ...
[68]
Conformal geometric algebra - Wikipedia
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space.
[69]
[PDF] A New Approach to Screw Theory using Geometric Algebra - HAL
Oct 22, 2023 · The new approach uses Geometric Algebra (G3,0) to generalize screw theory, conciliate geometric and algebraic aspects, and is coordinate-free.Missing: sandwich | Show results with:sandwich
[70]
(Conformal) Geometric Algebra (GA, CGA) - Duet3D Documentation
Conformal Geometric algebra (CGA) can describe transformations (rotation, reflection, translation, dilation) by so-called rotors with the help of versors. Screw ...Missing: sandwich | Show results with:sandwich
[71]
[PDF] Clifford Algebra to Geometric Calculus - MIT Mathematics
Hestenes, David, 1933-. Clifford algebra to geometric calculus. (Fundamental theories of physics). Includes bibliographical references and index. 1. Clifford ...
[72]
[PDF] A Survey of Geometric Algebra and Geometric Calculus - PhilArchive
Mar 21, 2018 · The American physicist and mathematician David Hestenes initiated the modern development of geometric algebra with the publication of Space-Time.
[73]
[PDF] A Guided Tour to the Plane-Based Geometric Algebra PGA
Plane-Based Geometric Algebra (PGA) uses planes as primitive elements to unify objects and operators for Euclidean geometry.
[74]
[PDF] Geometric Computing in Computer Graphics and Robotics using ...
Our inverse kinematics algorithm is based on the embedding of quaternions in Con- formal Geometric Algebra. This is why we investigated how quaternions and some.
[75]
Geometric Algebra Jordan–Wigner Transformation for Quantum ...
Spinors for Quantum Computing. The WB can be used to build spinor spaces and also to represent qubit states. Definition 6. Spinor space. A spinor space is a ...Missing: rotors | Show results with:rotors
[76]
The Power of Geometric Algebra Computing: For Engineering and ...
The optimized code has no more Geometric Algebra operations and is ready to be run efficiently on various platforms, particularly on parallel computing ...
[77]
GA-Unity: A Production-Ready Unity Package for Seamless Integration
Feb 25, 2025 · This paper introduces GA-Unity, the first Unity package specifically designed for seamless integration of Geometric Algebra (GA) into ...
[78]
Less is More: Efficient Networked VR Transformation Handling ...
Dec 8, 2022 · Why use Geometric Algebra? Algebras such as 3D PGA and 3D CGA are showing rapid adaptation to VR implementations due to their ability to ...
[79]
Flettner Rotor For Ships - Uses, History And Problems - Marine Insight
Jan 6, 2021 · The Magnus effect, which is the physical concept behind the Flettner rotor, was first studied by the German physicist Gustav Magnus. He observed ...
[80]
Rotating Sails Help to Revive Wind-Powered Shipping
Dec 1, 2020 · The new rotors need electrical power to spin, but manufacturer Norsepower says they can still save up to 20 percent on fuel consumption and cut ...
[81]
FLETTNER SAILS FOR HOME.; Goes to Direct Operation of New ...
The Barbara has three giant rotors which will have auxiliary motors for the propulsion of the ship. The rotors are expected to effect a saving in fuel and will ...
[82]
The Rotor Sail Revival - Shipping Today & Yesterday Magazine
Apr 11, 2023 · The 21st century has seen a slow revival of Flettner rotors thanks to the commercial and political pressures to reduce fuel costs and CO2 ...
[83]
Gyro monorail - Wikipedia
A gyro monorail, gyroscopic monorail, or gyro-stabilized monorail is a single-rail land vehicle that uses the gyroscopic action of one or more spinning wheels
[84]
Brennan's Gyro Monorail | Amusing Planet
May 19, 2021 · Propelled by electric motors, these twin flywheels rendered stability to the car by resisting any disturbances trying to topple the railcar. The ...
[85]
Simplicity and safety: why rotor sails make sense
Our control system is designed to maximise performance and minimise crew input with automated speed and direction setting, equipment monitoring, safety features ...Missing: metrics | Show results with:metrics
[86]
World's 1st Installation of Rotor Sail on a Capesize Bulk Carrier for ...
Aug 20, 2024 · This is the world's first case of rotor sails on a capesize bulk carrier (Note 1). The vessel is expected to achieve about 6-10% fuel and GHG emissions ...<|separator|>
[87]
[PDF] MEPC 81/INF.39 - International Windship Association
carrying deck cargo require new concepts for arranging sail systems efficiently. ... 20% reduction in fuel usage across the fleet, as almost all ships can ...
[88]
Carnival ride - Physics
There is an amusement park ride called The Gravitron or The Rotor, which is essentially a big cylinder. You get on, lean against the wall, and it starts to spin ...
[89]
The Rotor Ride: An Unsafe Human Spinning Blender, 1950-1970
The Human Spinning Rotor or simply “The Rotor” was invented by a German scientist, Ernst Hoffmeister, and was financially supported by businessman Carl Friese.
[90]
Sonic the Hedgehog Issue # 1/4 (Archie Comics Group)
Archie Sonic the Hedgehog Issue 1/4 was the very first issue of the Archie Sonic the Hedgehog comics to be released, predating Sonic Miniseries #0.
[91]
https://www.ship-technology.com/news/anemoi-rotor-sails-nsu-tubarao/
[92]
Looping the Loop – Brian Hatton proposes a theory of ... - Art Monthly
Brian Hatton proposes a theory of the loop that predates its popular usage in film and video art theory. Marcel Duchamp <em>Rotorelief – Spirale Blanche ...Missing: Rotor motifs symbolizing
[93]
Te Puia, Rotorua Geothermal Park, Māori Culture and Restaurant
Te Puia Rotorua geothermal park is a wonderland of dramatic geysers, bubbling mud, Māori cultural experiences & kiwi conservation centre.Experiences · Minecraft Aotearoa New Zealand · Air New Zealand Board... · Te Puia