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Eccentricity

Eccentricity has multiple meanings across different fields. In behavioral and cultural contexts, it refers to unconventional or peculiar behavior and perspectives. In mathematics, particularly conic sections, eccentricity is a geometric parameter that defines the shape of a conic, determined by the constant ratio of the distance from any point on the conic to a fixed focus (PF) and to the corresponding directrix (PD), expressed as e = \frac{PF}{PD}. This value classifies the conic: $0 \leq e < 1 for an ellipse (including a circle at e = 0), e = 1 for a parabola, and e > 1 for a hyperbola.^[1] In the standard equation of conic sections, eccentricity is calculated as e = \frac{c}{a}, where c is the distance from the center to the focus (linear eccentricity) and a is the semi-major axis for ellipses or the semi-transverse axis for hyperbolas. For a circle, both foci coincide at the center, yielding e = 0, while as e approaches 1 from below in an ellipse, the shape elongates toward a line segment. In hyperbolas, higher values of e result in more pronounced divergence of the branches from the center. This parameter provides a precise measure of deviation from circularity.^[2]^[3]

Behavioral and cultural meanings

Definition and characteristics

Eccentricity in behavior refers to unusual or unconventional actions, habits, or mannerisms that deviate from established social norms without implying harm or pathology, often manifesting as quirkiness that sets an individual apart from the conventional.^[4] This contrasts with typical behavior, which aligns closely with societal expectations for conformity in dress, speech, and daily routines, whereas eccentricity embraces deviations that are perceived as odd yet benign by observers.^[5] Such behavior is typically non-disruptive and self-aware, distinguishing it from more extreme or involuntary quirks. Key characteristics of eccentricity include a strong propensity for non-conformity, where individuals prioritize personal expression over fitting in, often leading to unique choices in attire, communication styles, or living arrangements, such as adopting an unconventional diet or residing in a customized, unconventional home.^[6] Accompanying traits frequently encompass heightened creativity and curiosity, driving innovative problem-solving and a playful or mischievous sense of humor that challenges routine interactions.^[7] Optimism also plays a role, as eccentrics often maintain a positive outlook amid their deviations, viewing their differences as sources of fulfillment rather than isolation.^[8] The term originates from the Greek roots "ek" meaning "out" and "kentron" meaning "center," originally describing orbital paths off-center from a focal point, which metaphorically extends to behavioral deviation from the societal "center" or norm.^[9] This etymological foundation underscores eccentricity as a deliberate or inherent shift away from the expected core of social standards. In contemporary contexts as of 2025, eccentricity is increasingly valued in creative fields like art and innovation, where non-conformist traits foster originality and breakthrough ideas, enhancing perceived skill and appreciation in professional outputs.^[10] Similarly, in mathematics, eccentricity measures deviation from a central point in conic sections, serving as a loose analogy for how behavioral eccentricity represents a departure from normative centers.^[11] In psychological research, eccentricity is viewed as a personality trait characterized by non-conformity, high creativity, intense curiosity, and a childlike sense of wonder, often associated with elevated intelligence and emotional sensitivity.^[12] Studies from the late 20th century, such as those conducted by neuropsychologist David Weeks, identified eccentrics as individuals who exhibit these traits without delusion or distress, demonstrating robust mental health and a strong sense of self-awareness.^[13] Weeks' longitudinal research on over 1,000 eccentrics revealed that they score higher on measures of originality and divergent thinking compared to the general population, linking their non-conventional behaviors to adaptive psychological strengths rather than deficits.^[14] From a social perspective, eccentricity plays a vital role in fostering innovation and cultural advancement by challenging established norms and encouraging novel ideas in fields like science and the arts.^[6] Eccentrics often exhibit lower levels of social conformity, which correlates with higher life satisfaction and contributions to societal progress; for instance, Weeks' findings showed that eccentrics report greater overall happiness and physical health, attributing this to their curiosity-driven pursuits that yield creative breakthroughs.^[15] This non-conformist stance enables eccentrics to drive change, as seen in historical examples such as inventor Nikola Tesla, whose unconventional habits and visionary ideas revolutionized electrical engineering, or aviator Howard Hughes, whose eccentric behaviors did not hinder his advancements in aviation and film.^[16]^[17] Supported by evidence that such traits enhance problem-solving and adaptability in dynamic environments.^[18] A key distinction in psychological literature separates eccentricity from pathological conditions like schizophrenia, emphasizing that eccentrics remain highly functional, insightful about their behaviors, and free from the impaired reality-testing seen in disorders.^[12] Unlike individuals with mental illnesses who suffer from their symptoms, eccentrics actively choose and embrace their quirks, with Weeks' research confirming lower incidences of familial mental health issues and no evidence of delusion among them.^[19] Recent developments in the neurodiversity movement, particularly in 2025, have further embraced eccentricity as a positive variation in human cognition, promoting its inclusion in workplaces and education to harness diverse thinking for innovation without pathologizing it.^[20] Perceptions of eccentricity vary by gender and culture, with Western individualistic societies generally showing greater tolerance due to their emphasis on personal autonomy and non-conformity.^[21] In contrast, collectivist cultures often view eccentric behaviors more negatively, associating them with disruption to group harmony and imposing stronger social pressures for conformity.^[22] These variations highlight how cultural norms shape the valuation of eccentric traits, with individualistic frameworks prioritizing individual expression over collective uniformity.^[23]

Mathematical definitions

Conic sections

In conic sections, eccentricity e is defined as the constant ratio of the distance from any point on the conic to a fixed point called the focus and the perpendicular distance from that point to a fixed line called the directrix.^[24] This focus-directrix property provides a unifying geometric characterization of all non-degenerate conics. The value of e distinguishes the type of conic: a circle when e = 0, an ellipse when $0 < e < 1, a parabola when e = 1, and a hyperbola when e > 1.^[25] In Cartesian coordinates, for an ellipse centered at the origin with semi-major axis a (along the x-axis) and semi-minor axis b (along the y-axis), the linear eccentricity c is the distance from the center to each focus, given by c = \sqrt{a^2 - b^2}; for a hyperbola, with semi-transverse axis a and semi-conjugate axis b, c = \sqrt{a^2 + b^2}; then, e = c / a.^[24] This concept originated with Apollonius of Perga in the 3rd century BCE, who systematically developed the theory of conic sections in his treatise Conics, introducing the focus and directrix as key elements to describe their properties without relying solely on cone intersections.^[26] For a circle, e = 0 because the focus coincides with the center and there is no distinct directrix, making all points equidistant from the center in a limiting sense.^[25] In this case, the equation simplifies to x^2 + y^2 = r^2, where r = a = b and c = 0. The derivation follows from the ellipse formula as b approaches a, yielding e = \sqrt{1 - (b/a)^2} = 0.^[24] For an ellipse, $0 < e < 1, and the explicit formula is e = \sqrt{1 - (b/a)^2}, where a > b > 0. This arises from the focus-directrix property: for a point P(x, y) on the ellipse, the distance to the focus F(c, 0) divided by the distance to the directrix x = a/e equals e. Setting the focus-directrix condition PF / \text{distance to directrix} = e and substituting c = ae leads, after algebraic manipulation and simplification, to the standard ellipse equation \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, with b^2 = a^2(1 - e^2).^[24] Geometrically, the two foci and corresponding directrices illustrate how points on the ellipse "hug" the foci more closely as e increases toward 1, deviating further from circular symmetry.^[25] A parabola has e = 1, with standard form (x - h)^2 = 4p(y - k) for a vertical opening, where the focus is at (h, k + p) and the directrix is y = k - p. The derivation directly from the property states that for any point P(x, y), the distance to the focus equals the distance to the directrix: \sqrt{(x - h)^2 + (y - (k + p))^2} = |y - (k - p)|. Squaring both sides and simplifying yields the parabolic equation, confirming e = 1 as the boundary case where the curve extends infinitely in one direction.^[24] Here, a single focus and directrix define the shape, with no second focus. For a hyperbola, e > 1, and the formula is e = \sqrt{1 + (b/a)^2}, where the branches open away from the center. Using the focus-directrix property for one branch, with foci at (\pm c, 0) and directrix x = \pm a/e, the condition PF / \text{distance to directrix} = e (choosing the appropriate focus and directrix per branch) derives the equation \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, where b^2 = a^2(e^2 - 1) and c = ae.^[24] The two foci and directrices highlight the hyperbolic separation, with e quantifying how sharply the asymptotes diverge as e grows.^[25] Overall, eccentricity geometrically measures the conic's deviation from perfect circularity (e = 0), increasing as the shape elongates or opens up; for instance, diagrams typically show the circle as a point-focus limit, the ellipse with foci inside, the parabola tangent to its directrix at the vertex, and the hyperbola with foci outside and branches receding.^[25] This parameter assumes familiarity with the major and minor axes for ellipses or transverse and conjugate axes for hyperbolas but underscores the conic's focal structure as central to its form.

Graph theory

In graph theory, the eccentricity of a vertex v in a connected undirected graph G = (V, E) is defined as the maximum shortest-path distance from v to any other vertex in G, formally expressed as \epsilon(v) = \max \{ d(v, u) \mid u \in V \}, where d(v, u) denotes the length of the shortest path between v and u.^[27] This measure quantifies how "far" a vertex is from the most distant vertex, serving as an indicator of a vertex's position relative to the graph's periphery. For disconnected graphs, the eccentricity is conventionally infinite, though analyses typically focus on connected components.^[27] Related concepts include the graph radius, which is the minimum eccentricity over all vertices \mathrm{rad}(G) = \min \{ \epsilon(v) \mid v \in V \}, representing the smallest possible maximum distance from a central vertex; the graph diameter, \mathrm{diam}(G) = \max \{ \epsilon(v) \mid v \in V \}, which captures the longest shortest-path distance between any pair of vertices; and the graph center, the set of all vertices achieving the radius, i.e., \{ v \in V \mid \epsilon(v) = \mathrm{rad}(G) \}.^[27] These parameters provide insights into the graph's overall spread and central structure. For example, in a complete graph K_n with n \geq 2 vertices, every vertex has eccentricity 1, yielding radius 1 and diameter 1, as all pairs are adjacent.^[27] In contrast, for a path graph P_n on n vertices, the endpoints have eccentricity n-1, while central vertices have eccentricity \lceil (n-1)/2 \rceil, resulting in diameter n-1 and radius \lceil (n-1)/2 \rceil.^[28] Such properties highlight eccentricity's role in assessing vertex centrality, with low-eccentricity vertices indicating central nodes useful in network analysis for identifying hubs.^[29] Eccentricity can be computed exactly for unweighted graphs by performing a breadth-first search (BFS) from each vertex to find all shortest-path distances, with the maximum yielding the eccentricity; the overall time complexity is O(|V|(|V| + |E|)).^[30] This approach is foundational but scales poorly for large graphs, motivating approximation algorithms in practice. Extensions to directed graphs define eccentricity using directed shortest paths, often as the maximum out-distance from the vertex, while weighted graphs incorporate edge weights into distance calculations via algorithms like Dijkstra's instead of BFS.^[31] Recent advancements, such as efficient estimation frameworks for large-scale networks, have enhanced eccentricity's utility in social network analysis by enabling rapid identification of influential or peripheral actors in dynamic structures, as demonstrated in distributed computing models for centrality measures updated through 2025.^[32]^[33]

Scientific applications

Orbital mechanics

In orbital mechanics, eccentricity serves as a key parameter in Kepler's first law, which states that the orbits of planets and other celestial bodies around a central mass are conic sections with the primary body located at one focus.^[34] This law, derived by Johannes Kepler in 1609 using precise observational data from Tycho Brahe, revolutionized understanding of celestial motion by replacing circular orbits with elliptical paths characterized by eccentricity e.^[34] For bound orbits, such as those of planets, $0 \leq e < 1 describes ellipses; e = 1 yields a parabola for marginally unbound trajectories; and e > 1 results in hyperbolas for unbound, escaping paths. The eccentricity directly influences the geometry and dynamics of orbits through relations like the periapsis distance r_{\min} = a(1 - e) and apoapsis distance r_{\max} = a(1 + e), where a is the semi-major axis, defining the closest and farthest points from the central body, respectively.^[35] These distances highlight how higher e elongates the orbit, leading to pronounced variations in radial separation. The vis-viva equation further quantifies speed along the orbit: v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right), where G is the gravitational constant, M the central mass, and r the instantaneous distance; this shows that velocity peaks at periapsis and minima at apoapsis, with greater e amplifying these differences. Representative examples illustrate these effects: Earth's orbit has a low eccentricity of approximately 0.0167, rendering it nearly circular and resulting in minimal seasonal distance-based temperature variations.^[36] In contrast, Halley's Comet follows a highly elliptical path with e \approx 0.967, approaching within 0.59 AU of the Sun at perihelion before retreating to nearly 36 AU at aphelion, which drives extreme velocity changes and periodic visibility from Earth.^[37] Among exoplanets, systems like HD 80606 b exhibit e \approx 0.93, where the planet endures intense heating during close stellar approaches, as confirmed by recent spectroscopic analyses.^[38] Higher eccentricity exacerbates temperature extremes on orbiting bodies by varying insolation: planets or moons experience scorching periapsis passages and frigid apoapsis distances, potentially destabilizing climates over orbital cycles.^[39] Additionally, elevated e heightens susceptibility to perturbations from nearby bodies, such as gravitational tugs that can excite or dampen eccentricity, influencing long-term orbital stability in multi-body systems like planetary rings or exoplanet architectures.^[40]

Geodesy and Earth sciences

In geodesy, eccentricity quantifies the deviation of Earth's reference ellipsoid from a sphere, modeling the planet's oblate spheroid shape due to rotation. The squared eccentricity e^2 is defined as e^2 = 1 - \frac{b^2}{a^2}, where a is the semi-major axis (equatorial radius) and b is the semi-minor axis (polar radius). For the World Geodetic System 1984 (WGS84) ellipsoid, widely used as a global standard, a \approx 6378 km and e^2 \approx 0.0067, corresponding to a polar flattening of about 21 km.^[41] This parameter distinguishes the geodetic eccentricity, which describes the static figure of the planetary body, from orbital eccentricity, which characterizes the dynamic path around another body. The eccentricity plays a crucial role in applications such as GPS positioning and cartographic projections, where accurate ellipsoidal models ensure precise coordinate transformations. In GPS, the WGS84 parameters, including eccentricity, define the reference frame for satellite ephemerides and receiver locations, minimizing errors in navigation to centimeters. Cartographic projections, like the Universal Transverse Mercator (UTM), incorporate eccentricity to compute distortions in scale and convergence on maps derived from the ellipsoid. Additionally, the flattening f, approximated as f \approx \frac{e^2}{2} for small values, links to gravity models by influencing geoid computations, which are essential for orthometric heights and sea-level monitoring. Satellite missions like GRACE (Gravity Recovery and Climate Experiment) refine these values by detecting temporal variations in Earth's oblateness, revealing mass redistributions from ice melt that alter e^2 by fractions of a percent over decades. In Earth sciences, eccentricity also informs paleoclimate studies through Milankovitch cycles, where orbital eccentricity variations over approximately 100,000 years modulate seasonal insolation and drive glacial-interglacial transitions. Earth's current orbital eccentricity is about 0.0167, near its minimum and projected to decrease slowly over the coming millennia.^[42] GRACE data further supports geophysical modeling by linking oblateness changes—tied to eccentricity—to sea-level rise, as polar ice loss reduces Earth's rotational bulge and affects global gravity fields.^[43]

Engineering applications

Mechanical systems

In mechanical systems, eccentricity refers to the intentional offset distance e between the axis of rotation and the geometric center of a circular disk or sheave, enabling the conversion of rotary motion into linear reciprocating motion.^[44] This offset creates a simple harmonic motion profile for the follower, where the displacement y = e (1 - \cos \theta), velocity \dot{y} = e \omega \sin \theta, and acceleration \ddot{y} = e \omega^2 \cos \theta, with \theta as the rotation angle and \omega as angular velocity.^[44] The stroke length, or throw, equals $2e, providing a predictable linear travel without the complexity of profiled cams.^[44] Key applications include eccentric sheaves in 19th-century steam engines for driving valves and pumps, as seen in early locomotive designs from the Industrial Revolution era that improved upon reciprocating mechanisms.^[45] In bicycles, eccentric bottom brackets adjust chain tension by offsetting the crank axis, allowing fixed-gear or single-speed setups with variable geometry.^[46] Modern uses extend to pumps and valves, where eccentric cams drive diaphragms or plungers for fluid displacement, as in miniature eccentric diaphragm pumps that achieve precise metering through the offset-driven reciprocation.^[47] Design principles emphasize stress analysis to mitigate contact stresses at the cam-follower interface, governed by Hertzian theory where maximum stress \sigma_{\max} = 0.564 \sqrt{\frac{P}{t_h \left( \frac{1}{\rho_c} + \frac{1}{\rho_f} \right) \left( \frac{1 - \nu_c^2}{E_c} + \frac{1 - \nu_f^2}{E_f} \right)}}, with P as load, t_h as Hertzian width, \rho as radii of curvature, E as moduli of elasticity, and \nu as Poisson's ratios.^[44] Advantages include compact variable timing, such as in automotive valve trains where eccentric adjustments enable phase shifts for optimized engine performance across RPM ranges. Torque transmission follows T = F \cdot e, where F is the tangential force, highlighting the offset's role as a lever arm.^[44] Historically, eccentric mechanisms evolved from early 19th-century locomotive valve gears to contemporary robotics, where eccentric actuators provide compact, high-force bending in soft pneumatic systems for tasks like soft grippers, achieving strokes up to 50% greater than symmetric designs while minimizing bulk.^[48] By 2025, these actuators integrate into robotic limbs for enhanced dexterity, building on foundational principles to support lightweight, energy-efficient motion in dynamic environments.^[49]

Structural design

In structural engineering, eccentricity denotes the perpendicular distance between the line of action of an applied load and the centroidal axis of a structural member, often arising from construction tolerances, connection geometries, or unintended offsets in load paths. This offset induces a bending moment M = P × e, where P is the axial load and e is the eccentricity, transforming pure compression or tension into combined axial and flexural stresses that compromise stability.^[50]^[51] In columns and beams, eccentricity triggers additional bending, exacerbating the P-delta effect—a second-order phenomenon where axial loads interact with lateral deflections to amplify moments and reduce load-carrying capacity. Design standards, such as ANSI/AISC 360, mandate accounting for these effects through second-order analysis or approximate methods, with provisions to limit eccentricity in connections to prevent excessive stresses in braced and moment-resisting frames. For instance, beam-to-column connections in steel structures often introduce controlled eccentricities that must be evaluated to maintain overall frame integrity under gravity and lateral loads.^[52]^[53] Eccentric connections are routinely employed in steel frames for practical fabrication, while in piping systems, manufacturing tolerances for eccentricity in fittings, such as up to ±3 mm for offset in eccentric reducers, ensure alignment and avoid excessive stresses per ASME B16.9.^[54] Notable failures underscore the hazards of overlooked eccentricity; for example, several girder bridges in the 1980s and later collapsed under eccentric heavy vehicle loads, where unaccounted offsets led to localized overstressing and progressive failure. Stress analysis employs interaction equations like \sigma = \frac{P}{A} \pm \frac{M c}{I}, where \sigma is the combined stress, A is the cross-sectional area, c is the distance to the extreme fiber, and I is the moment of inertia; here, e directly increases M, heightening the risk of yielding or buckling. As of 2025, finite element modeling advancements in seismic design better simulate these amplified stresses, enabling more accurate predictions of dynamic responses in irregular structures.^[55]^[56]^[57] Mitigation strategies focus on minimizing offsets through precise centering during erection, such as aligning loads via temporary bracing, and using shimming under base plates to correct minor deviations and restore concentricity. These techniques ensure compliance with tolerances and enhance long-term durability against unintended eccentricities.^[58]

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P-Delta Theoretical Background - RISA
P-Delta analysis is a methodology to compute structures which are prone to the load-displacement interaction, resulting in second-order effects.
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[PDF] ANSI/AISC 360-16 Specification for Structural Steel Buildings
ANSI/AISC 360-16 is a specification for structural steel buildings, providing design criteria for routine use, and using both ASD and LRFD design methods.
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Eccentric Footings | SkyCiv Engineering
Nov 19, 2024 · Footings that are not concentrically loaded are known as “Eccentric Footings”. In addition, when footings are subjected to an axial load “P” and ...
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Dimension Tolerances of ASME B16.9 Pipe Fittings
The dimension tolerances of ASME B16.9 pipe fittings: general tolerances for all fittings; tolerances for center-to-end dimensions of elbows, tees; ...Missing: eccentricity 0.1
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Failure Analysis of a Girder Bridge Collapse under Eccentric Heavy ...
This study suggests that bridge accidents are caused not only by instability issues of integral overturning but also by strength failures of the components.
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Structural Demands under Loading-Induced Bidirectional Eccentricity
Aug 20, 2025 · This research examines combinations of 4-, 8-, and 12-story moment-resisting frame (MRF) structures. Three-dimensional finite element models ...
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[PDF] STEEL CONSTRUCTABILITY CONSIDERATIONS
A 2 inch grout space should be detailed below the base plate allowing a single shim pack to be set centered under the column base plate. This allows the erector ...<|control11|><|separator|>