Fact-checked by Grok 2 weeks ago

Concave

In geometry, concave describes a curve, surface, or shape that bends or curves inward, such that the interior lies on the side toward which the curve bends, opposite to a convex shape that curves outward.^[1] This term originates from the Latin concavus, meaning "hollow," and is commonly illustrated by the inner surface of a bowl or spoon.^[2] Concave forms appear in various natural and artificial contexts, such as the shape of certain lenses, arches in architecture, and biological structures like the human palate.^[3] In the study of polygons, a concave polygon is a simple polygon that is not convex.^[4] In mathematical analysis, the concept extends to functions, where a function is concave down if its second derivative is negative, meaning the graph lies below its tangent lines, the slope decreases over the domain, and it forms a downward-opening curve like y = -x^2.^[5] Conversely, a concave up function has a positive second derivative, with the graph lying above its tangents and increasing slope, such as y = x^2.^[6] Note that terminology can vary: in optimization and economics, a "concave function" refers to one where the graph lies above the chords connecting any two points (satisfying f(tx + (1-t)y) \geq t f(x) + (1-t) f(y) for t \in [0,1]), equivalent to concave down in calculus, and the negative of a convex function.^[7] These properties are crucial for analyzing optimization problems. Beyond pure mathematics, concave principles apply in physics, such as in concave mirrors that converge light rays to a focal point, enabling applications in telescopes and headlights.^[8] The distinction from convex forms is fundamental across disciplines, influencing design in engineering, computer graphics for rendering curved surfaces, and even economics for modeling utility functions.^[9]

Mathematics

Concave polygons

A concave polygon is a simple polygon in which at least one interior angle measures greater than 180 degrees, known as a reflex angle.^[10] This distinguishes it from convex polygons, where all interior angles are less than or equal to 180 degrees. Alternatively, a concave polygon can be identified if there exists at least one pair of points on its boundary such that the line segment connecting them lies partially outside the polygon. Key properties of concave polygons include the presence of at least one "dent" or indentation in their boundary, which causes the shape to fail the convexity test: not every line segment joining two points within the polygon remains entirely inside it.^[11] Despite this, concave polygons remain simple, meaning they do not intersect themselves, and the sum of their interior angles follows the standard formula for simple polygons: (n-2) × 180 degrees, where n is the number of sides.^[12] These shapes must have at least four sides, as triangles are inherently convex.^[13] Examples of concave polygons include the dart or arrowhead shape, a quadrilateral with one reflex angle creating a pointed indentation, and an L-shaped pentagon, formed by attaching two rectangles at a right angle, resulting in a reentrant boundary.^[14] In visualizations, the dart appears as a chevron with an inward spike, while the L-shape resembles a right-angled hook, highlighting the inward reflex angle.^[15] A fundamental result in polygon classification states that a simple polygon is concave if and only if it is not convex, meaning it lacks the property that all internal angles are non-reflex and all boundary line segments stay inside. This theorem underscores the binary nature of simple polygon types in plane geometry.

Concave functions

In mathematical analysis, a function f: D \to \mathbb{R}, where D is a convex subset of \mathbb{R}^n, is defined as concave if for all x, y \in D and all \lambda \in [0, 1],

f(\lambda x + (1 - \lambda) y) \geq \lambda f(x) + (1 - \lambda) f(y).

^[16] Equivalently, f is concave if and only if -f is convex.^[16] For twice continuously differentiable functions on an open interval, f is concave if its second derivative satisfies f''(x) \leq 0 for all x in the domain.^[16] Concave functions exhibit several key properties. Composition with an affine transformation g(x) = Ax + b, where A is a matrix and b a vector, preserves concavity: if f is concave, then f \circ g is concave provided the domain aligns appropriately.^[16] Concavity also implies quasiconcavity, meaning the upper level sets \{x \mid f(x) \geq \alpha\} are convex for all \alpha.^[17] A fundamental property is Jensen's inequality, which states that for a concave function f and a random variable X with finite expectation,

f(\mathbb{E}[X]) \geq \mathbb{E}[f(X)].

This inequality was established by Johan Jensen in 1906. Examples of concave functions include linear functions f(x) = ax + b, which are both concave and convex.^[16] The natural logarithm f(x) = \log x for x > 0 is strictly concave, as its second derivative is negative.^[16] In economics, utility functions are often modeled as concave to capture diminishing marginal utility.^[18] Concave functions play a central role in optimization: maximizing a concave objective over a convex set is a convex optimization problem, which is computationally tractable and guarantees global optima. Related concepts include strict concavity, where the inequality in the definition holds strictly for \lambda \in (0,1) and x \neq y, ensuring at most one maximum.^[19]

Physics

Concave mirrors

A concave mirror is a spherical mirror that has its reflecting surface curved inward toward the incident light, resembling the interior surface of a sphere. This curvature causes parallel rays of light to converge at a focal point after reflection, making concave mirrors converging optics.^[20] The focal length f of a concave mirror is half the radius of curvature R, so f = \frac{R}{2}. This relationship holds under the paraxial approximation, which assumes rays are close to the principal axis and the mirror's curvature is gentle.^[21] Ray diagrams for concave mirrors rely on three principal rays to locate the image: (1) a ray parallel to the principal axis reflects through the focal point; (2) a ray passing through the focal point reflects parallel to the principal axis; and (3) a ray passing through the center of curvature reflects back along the same path.^[20] These rays intersect to form the image, providing a geometric method to predict its position and nature without equations.^[22] Image formation depends on the object's position relative to the focal point. For objects beyond the focal point, a real, inverted image forms on the same side as the incoming light, which can be projected onto a screen.^[22] When the object is inside the focal point, a virtual, upright, and magnified image appears behind the mirror.^[20] The lateral magnification m is given by m = -\frac{v}{u}, where u is the object distance and v is the image distance (using the sign convention where distances are positive for real images and objects).^[23] Concave mirrors find applications in optical instruments due to their light-converging properties. In telescopes, such as the Cassegrain design, a primary concave mirror collects and focuses distant light, with a secondary convex mirror redirecting it to the eyepiece for observation.^[24] They are used in vehicle headlights to produce a parallel beam for illumination and in solar cookers to concentrate sunlight for heating.^[25] Historically, concave mirrors were employed in ancient times for concentrating light, as in the legendary account of Archimedes using them to focus sunlight on enemy ships during the siege of Syracuse in 214 BCE.^[26] Their optical principles were formalized in the 17th century by Isaac Newton, who constructed the first reflecting telescope in 1668 using a concave primary mirror to avoid chromatic aberration in lenses.^[27]

Concave lenses

A concave lens, also known as a diverging lens, is an optical component that is thinner at its center than at its edges, causing parallel rays of light incident upon it to diverge after refraction as if emanating from a virtual focal point on the same side as the incident light.^[28] This diverging property arises from the inward curvature of at least one surface, which bends light rays outward relative to the principal axis.^[29] Unlike convex lenses, which converge light, concave lenses spread it out, making them essential for applications requiring beam divergence or image minification.^[30] The focal length f of a concave lens is negative (f < 0), indicating its diverging nature, and is determined by the lensmaker's formula for a thin lens in air:

\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)

where n is the refractive index of the lens material, and R_1 and R_2 are the radii of curvature of the first and second surfaces, respectively, following the sign convention that R is positive for convex surfaces facing the incident light and negative for concave ones.^[31] For a typical biconcave lens with symmetric surfaces, this yields a negative f, confirming the lens's inability to form real images.^[32] In ray diagrams for concave lenses, three principal rays are commonly traced to locate the image: a ray parallel to the principal axis refracts through the lens and diverges as if originating from the focal point on the incident side; a ray passing through the optical center of the lens proceeds undeviated; and a ray aimed toward the focal point on the opposite side refracts parallel to the principal axis after passing through the lens.^[33] These rays, when extended backward, intersect at the virtual image position, illustrating the lens's behavior for objects placed anywhere along the axis.^[34] For real objects, concave lenses invariably form virtual images that are upright and diminished in size, located between the lens and its focal point on the same side as the object.^[35] The image distance is always shorter than the object distance, and no real images can be projected onto a screen due to the diverging rays.^[36] This consistent image type makes concave lenses unsuitable for magnification but ideal for correction and divergence tasks. Concave lenses find widespread applications in vision correction, particularly in eyeglasses for myopia (nearsightedness), where they diverge incoming light to shift the virtual image backward onto the retina.^[37] The power P of such lenses is given by P = 1/f (with f in meters), expressed in diopters (D), and is negative for diverging lenses—for instance, a -3.00 D lens corrects moderate myopia by effectively reducing the eye's converging power.^[38] They are also used in compound microscopes as components of objective lens assemblies to correct aberrations and expand beam widths, and in camera systems to achieve wide-angle effects by diverging light for broader fields of view.^[39] Concave lenses for vision correction were developed in 15th-century Italy, with records indicating their production in Florence by the mid-1400s to address nearsightedness, building on earlier convex lenses invented around 1286.^[40] This advancement marked a significant step in optical history, enabling clearer distant vision for those with refractive errors.^[41]

Other applications

In architecture

In architecture, concave elements refer to surfaces or forms that curve inward, forming depressions or hollows that generate negative space and visual depth, in contrast to convex elements that project outward. These inward curves often serve to enclose spaces, foster a sense of enclosure, and manipulate perceptions of scale and movement within built environments.^[42]^[43] Historically, concave forms appeared prominently in ancient Roman basilicas through apses, which were semi-circular, inward-curving projections at the end of the rectangular hall, designed to accommodate judicial or ceremonial functions while enhancing spatial focus. For instance, basilicas like the Basilica Julia in the Roman Forum featured such apses to create a contained area for magistrates, emphasizing hierarchy and intimacy within larger public structures.^[44] In Gothic cathedrals, concave vaults, particularly fan vaults with ribs radiating in concave patterns, contributed to an illusion of greater height by drawing the eye upward through layered, curving compartments that distributed light and weight efficiently. Examples include the fan vaults at King's College Chapel in Cambridge, where the concave arrangement amplifies verticality and ethereal quality.^[45] In modern applications, architects like Zaha Hadid employed fluid concave facades to evoke organic dynamism and cultural fluidity, as seen in the Heydar Aliyev Center in Baku, Azerbaijan, completed in 2012, where sweeping inward curves integrate the building with its landscape and avoid rigid geometries. These forms also support sustainable design by diffusing natural light across interior surfaces, reducing glare and promoting even illumination, as concave profiles scatter incoming rays more uniformly than flat or convex ones.^[46]^[47]^[48] Structurally, inward-curving concave forms present challenges in load-bearing, as they can concentrate stresses and risk buckling under compression without adequate reinforcement, often necessitating tensile materials such as high-strength fabrics or thin concrete shells to maintain stability. Engineers address these by using catenary-inspired tensile membranes that distribute forces through tension rather than bending, or by fabricating prefabricated concrete shells that exploit curvature for self-support while minimizing material use.^[49]^[50] Aesthetically, concave elements enhance spatial perception by creating intimacy in expansive public areas, drawing viewers inward to foster emotional engagement and a sense of shelter, a technique influenced by 17th-century Baroque architecture where undulating concave surfaces generated dramatic movement and immersion. In Baroque designs, such as those by Francesco Borromini, inward curves in facades and interiors manipulated light and shadow to heighten theatricality and psychological depth, laying groundwork for later explorations in perceptual architecture.^[43]^[51]^[52]

In engineering

In engineering, concave profiles refer to curved surfaces that bend inward, often employed to optimize stress distribution, enhance fluid dynamics, or facilitate precise component fitting in various structures and mechanisms. These designs help mitigate stress concentrations by distributing loads more uniformly across surfaces, as seen in contact geometries where curved profiles reduce peak pressures compared to flat interfaces. In fluid dynamics applications, concave curvatures influence boundary layer behavior, promoting the formation of counter-rotating vortices that can enhance heat transfer or mixing in turbulent flows.^[53]^[54]^[55] In mechanical engineering, concave elements such as cams and gears are utilized to achieve smooth, controlled motion in machinery. Concave globoidal cams with swinging roller followers enable precise indexing and high-speed operations by minimizing vibrations and ensuring continuous contact during rotation. Similarly, internal gears feature concave tooth profiles that mesh with external gears to transmit power efficiently while accommodating radial adjustments for alignment. To further reduce stress concentrations in machine parts like shafts and fillets, engineers apply concave fillet radii, which lower peak stresses under bending or torsion loads; Roark's Formulas for Stress and Strain provides tabulated factors showing that increasing the fillet radius-to-shoulder height ratio can reduce stress concentration factors by up to 50% in typical configurations.^[56]^[57]^[58]^[59] Civil engineering leverages concave profiles in bridge and arch designs to manage tensile forces effectively. Suspension bridges like the Akashi Kaikyō Bridge (completed in 1998) employ parabolic cable shapes—approximating a concave upward curve under self-weight—to distribute tension across the main span of 1,991 meters, allowing the structure to withstand seismic and wind loads up to 80 m/s. This configuration transforms the bridge deck into a tension member supported by parabolic cables, optimizing material use and stability in long-span applications.^[60]^[61] In aerospace engineering, concave nozzles are critical for rocket propulsion, where bell-shaped designs expand exhaust gases efficiently to maximize thrust. The Bell nozzle, developed in the mid-20th century, features a concave divergent section that accelerates flow supersonically while minimizing losses; its contour is optimized using the method of characteristics, which solves hyperbolic partial differential equations to align shock waves and achieve uniform exit conditions, potentially increasing specific impulse by approximately 2-10% over conical nozzles.^[62]^[63]^[64]^[65] Concave mirrors also find brief application in satellite dishes, focusing incoming radio signals onto receivers for reliable communication links.^[66] Concave designs offer advantages in engineering by improving aerodynamics through reduced drag in curved flows and enhancing acoustics via controlled wave propagation in nozzles, but they present challenges in manufacturing precision, as deviations in curvature can amplify instabilities or require advanced composites integrated since the 1950s for high-temperature resilience. These profiles demand computer-aided design tools and finite element analysis to balance performance gains against fabrication complexities, such as molding concave composites without defects.^[64]^[67]^[68]

References

[1]
https://www.merriam-webster.com/dictionary/concave
[2]
Concave - Definition, Meaning & Synonyms - Vocabulary.com
Concave describes an inward curve; its opposite, convex, describes a curve that bulges outward. They are used to describe gentle, subtle curves, ...<|control11|><|separator|>
[3]
CONCAVE | definition in the Cambridge English Dictionary
CONCAVE meaning: 1. curving in: 2. curving in: 3. curved inward: . Learn more.<|control11|><|separator|>
[4]
Definition of Polygons - Department of Mathematics at UTSA
Dec 11, 2021 · Concave: Non-convex and simple. There is at least one interior angle greater than 180°.
[5]
[PDF] Section 11.3: Curves, Polygons, and Symmetry Intro to Polygons
A polygon is concave if it is not convex. That is, if you can find two points on the interior such that the line segment between them is not entirely ...<|control11|><|separator|>
[6]
Good Definitions as Biconditionals; Polygons - Andrews University
We may use the word concave to describe a nonconvex polygon (but not a nonconvex set of points). Another term is re-entrant polygon. An easy way to remember the ...
[7]
Calculus I - The Shape of a Graph, Part II - Pauls Online Math Notes
Nov 16, 2022 · A function can be concave up and either increasing or decreasing. Similarly, a function can be concave down and either increasing or decreasing.
[8]
Concavity - Penn Math
By definition, a concave upward function lies below its chords and a concave downward function lies above its chords. 🔗 Figure 3.8.
[9]
[PDF] 21. Concave and Quasiconcave Functions
Dec 6, 2022 · A positive scalar multiple of a concave (convex) function is concave. (convex). 3. The sum of two concave (convex) functions is concave (convex) ...
[10]
5.4 Concavity and inflection points
If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture.
[11]
Concave and Convex Mirrors: A Simple Physics Guide - Avantier
Concave mirrors are spherical mirrors with a reflective surface that curves inward. It resembles the inner surface of a sphere.
[12]
Concave vs. Convex—What's the Difference? | Grammarly
May 22, 2019 · Concave describes shapes that curve inward, like an hourglass. Convex describes shapes that curve outward, like a football (or a rugby ball) ...
[13]
Concave Polygon Definition
A Concave polygon is a polygon that has at least one interior angle greater than 180 degrees. It must have at least four sides. The shape of the concave ...Missing: theorem | Show results with:theorem
[14]
Concave Polygons - Definition, Properties, Examples - Cuemath
A polygon that is not convex is a concave polygon. If any interior angle in a polygon measures more than 180°, then that is a concave polygon. A concave polygon ...
[15]
Concave Polygon - GeeksforGeeks
Jul 23, 2025 · A concave polygon is a polygon in which at least one interior angle measures more than 180°. For example, stars can be represented as concave polygons.
[16]
Concave Polygon – Definition, Types, Properties, Examples, FAQs
A concave polygon is a polygon in which at least one interior angle is a reflex angle (angle greater than 180° but less than 360°).Missing: theorem | Show results with:theorem
[17]
Concave Polygon | Definition, Angles & Examples - Study.com
A concave polygon is formally defined as a polygon that has at least one interior angle that measures greater than 180 degrees, but less than 360 degrees.
[18]
Concave Polygon: Definition, Properties, and Examples - Vedantu
Rating 4.2 (373,000) A concave polygon is a closed, flat shape with straight sides in which at least one interior angle is greater than 180°.Missing: theorem | Show results with:theorem
[19]
What is the history of polygons? - Quora
Jun 6, 2016 · Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram which is a non-convex regular polygon~ ...<|separator|>
[20]
Convex Polygon -- from Wolfram MathWorld
A convex polygon contains all line segments connecting any two of its points. A non-convex polygon is concave.
[21]
[PDF] Convex functions
f is concave if −f is convex. ▷ f is strictly convex if dom f is convex and for x, y ∈ dom f, x ≠ y, 0 <𝜃< 1, f (𝜃x + (1 − 𝜃)y) < 𝜃f (x)+(1 − 𝜃)f (y).
[22]
Mathematical methods for economic theory: 3.4 Quasiconcavity and ...
The notion of quasiconcavity is weaker than the notion of concavity, in the sense that every concave function is quasiconcave. Similarly, every convex function ...
[23]
[PDF] Concave functions in economics 1. Preliminaries 1 2. Concave ...
Since ln x is concave the utility function is concave as it is the sum of concave functions. The constraint function ( ). h x is also concave because it is ...
[24]
3.3 Concave and convex functions of many variables - mjo
As for a function of a single variable, a strictly concave function satisfies the definition for concavity with a strict inequality (> rather than ≥) for all x ...
[25]
[PDF] Legendre transforms
Feb 15, 2019 · The Legendre transform exploits a special feature of a convex (or concave) function f(x): its slope f/(x) is monotonic and hence is a single- ...
[26]
Ray Diagrams - Concave Mirrors - The Physics Classroom
A virtual image is formed if the object is located less than one focal length from the concave mirror. To see why this is so, a ray diagram can be used. Watch ...
[27]
Spherical Mirrors - Richard Fitzpatrick
They reflect the incoming radio waves from (very) distant astronomical sources, and bring them to a focus at a single point, where a detector is placed. In this ...
[28]
Image Formed by Concave Mirrors ( Read ) | Physics - CK-12
Dec 16, 2020 · An incident ray of light passing through the principal focus of a concave mirror creates a reflected ray that is parallel to the principal axis ...
[29]
[PDF] The Mirror Equation – Concave Mirrors Example Problem 1:
To obtain this type of information it is necessary to use the mirror equation and the magnification equation. The mirror equation expresses the relationship ...
[30]
https://www.khanacademy.org/science/ap-physics-2/x0e2f5a2c:geometric-optics/x0e2f5a2c:lenses/v/concave-lenses
[31]
Is a solar cooker really a concave mirror? | CK-12 Foundation
Converging mirrors can be used to focus light – headlights, telescopes, satellite TV receivers, and solar cookers all rely on this principle.
[32]
Refuting the Legend - NYU Courant Mathematics
The legend of Archimedes' burning mirrors is too good to die, regardless of how much evidence is presented to refute it.
[33]
Early Reflectors (Cosmology: Tools) - American Institute of Physics
In 1668, Isaac Newton devised a reflecting telescope. Instead of a lens, it used a single curved main mirror, together with a smaller flat mirror.
[34]
16.2.8: Double Concave Lenses - Physics LibreTexts
Aug 29, 2024 · Every concave lens causes all rays to diverge. Rays that approach the lens parallel to the principal axis refract as if they came from the focal ...
[35]
Thin Lens Equations & Calculators - BYU Cleanroom
The signs are reversed for the back surface of the lens: if R2 is positive, the surface is concave, and if R2 is negative, the surface is convex.
[36]
Concave lenses (video) | Geometric optics - Khan Academy
Jun 4, 2016 · Concave lenses, with their inward curve, create divergent light rays. When a ray parallel to the principal axis hits the lens, it refracts outward as if ...
[37]
Lens-Maker's Formula and Thin Lenses - HyperPhysics Concepts
A single purpose calculation which returns the powers and focal lengths based on the values of the radii and indices of refraction.Missing: concave | Show results with:concave
[38]
Thin Lenses - Richard Fitzpatrick
The above formula is usually called the lens-maker's formula, and was discovered by Descartes. Note that the lens-maker's formula is only valid for a thin ...
[39]
Ray Diagrams for Lenses - HyperPhysics Concepts
The ray diagrams for concave lenses inside and outside the focal point give similar results: an erect virtual image smaller than the object. The image is ...
[40]
Thin Lens Tutorial - BYU Cleanroom
Ray Diagrams for Concave Lenses. Now that we understand these basic rules ... lens (this ray will strike the lens before reaching the focal point). Draw ...
[41]
25.6 Image Formation by Lenses – College Physics - UCF Pressbooks
The virtual image produced by a concave lens is always smaller than the object—a case 3 image. We can see and photograph virtual images only by using an ...
[42]
[PDF] Image formation by lenses - Physics Courses
Convex, converging lenses form real images and virtual images like concave mirrors. • Concave, diverging lenses only form virtual images, like convex mirrors.
[43]
Correcting Myopia and Hyperopia
To weaken the lens, a diverging or negative artificial lens is used, that is, a concave lens. A diverging lens is wider at the edges than at the middle. As a ...
[44]
Introduction to Optics and Refractive Errors of the Eye
Oct 19, 2015 · Minus lenses are used in the correction of myopia. Negative, concave lenses are used to diverge the light rays entering the eye to move the ...
[45]
Lenses Have Many Uses | Physics Van | Illinois
Most high quality cameras, telescopes and binoculars use concave lenses to improve the quality of the images they provide. Microscopes and reflecting telescopes ...
[46]
Through the lens of history - BU Bridge Feature Article
And when some clever soul in 15th-century Florence ground the first concave lens to correct nearsightedness, seeing that world became a whole lot easier. The ...
[47]
Eyeglasses and Concave Lenses in Fifteenth-Century Florence and ...
Nov 20, 2018 · New documents reveal eyeglasses with concave lenses were made in Florence from at least the mid-15th century, and Florence was a leading ...
[48]
Concave - Design+Encyclopedia
Concave is a fundamental geometric property describing a surface or line that curves inward, creating a hollow or depression that resembles the interior of a ...
[49]
Arc-itecture: 7 Concave Designs That Curve With the Landscape
Concave designs use bent facades to create dynamism, integrate with terrain, and create enclosed, welcoming spaces, moving beyond 90-degree angles.
[50]
The Early Christian Basilica
These basilicas regularly had an architectural form we call an apse. The apse was a semi-circular projection usually off the short wall of the rectangular ...Missing: concave elements
[51]
Buttress your knowledge! The wonderful world of medieval vaults
Apr 20, 2021 · This is composed of concave compartments with ribs that spread out in the shape of a fan. Myriad elaborate varieties exist from the periods of ...
[52]
Heydar Aliyev Center / Zaha Hadid Architects - ArchDaily
Nov 14, 2013 · The Heydar Aliyev Center principally consists of two collaborating systems: a concrete structure combined with a space frame system.Missing: concave | Show results with:concave
[53]
Heydar Aliyev Cultural Center, Designed by Zaha Hadid Architects
Sep 17, 2013 · The Zaha Hadid–designed complex in Baku, Azerbaijan, challenges notions of geometry and gravity—with nary a right angle in sight.Missing: concave | Show results with:concave
[54]
Beyond Form: How Light and Shadow Define Architectural ...
Mar 21, 2025 · Explore the interplay of light and shadow in architecture, enhancing perception, atmosphere, and emotional responses.
[55]
Fundamentals of tensile architectural fabric structures - ScienceDirect
Tensile structures are pieces in which tension is the main stress to consider. In this chapter, we evaluate how different loads create different shapes and ...Missing: challenges | Show results with:challenges
[56]
(PDF) Challenges in designing and fabrication of a thin concrete shell
This paper, presents the exploration of freeform surface through the use of Rhinoceros and its plug-ins. Three thin concrete shells, namely a continuous, a ...Missing: tensile | Show results with:tensile
[57]
Baroque architecture | Intro to Humanities Class Notes - Fiveable
Convex and concave surfaces create a sense of movement and dynamism · Undulating facades break away from the rigid symmetry of Renaissance architecture · Curved ...Missing: intimacy | Show results with:intimacy
[58]
Baroque Art and Architecture Movement Overview - The Art Story
Aug 28, 2018 · Baroque art and architecture stressed theatrical atmosphere, dynamic flourishes, and myriad colors and textures.Missing: intimacy | Show results with:intimacy<|separator|>
[59]
Concave Contact Geometry for Enhanced Sealing and Structural ...
Curved contact geometry significantly reduces stress concentration and distributes contact pressure more uniformly compared to flat or rectangular contact ...
[60]
[PDF] The Heat Transfer and Fluid Dynamics of Concave Surface Curvature
The original objective of this work was to identify the mechanism whereby concave curvature increases the heat transfer through a turbulent boundary layer. ...
[61]
Concave Surface Boundary Layer Flows in the Presence ... - J-Stage
Concave surface boundary-layer flows are subjected to centrifugal instability which results in the formation of streamwise counter-rotating vortices.
[62]
Modeling Concave Globoidal Cam with Swinging Roller Follower
Jun 20, 2008 · This paper describes a computer-aided design for design of the concave globoidal cam with cylindrical rollers and swinging follower.
[63]
Nonlinear dynamics behavior of cam-follower system using concave ...
A cam with smooth profile is used to control the impact phenomenon and to prevent follower rebounding.
[64]
Know about gear types and relations between the two shafts | KHK
Gears are defined as “machine elements to transmit rotary motion by successively meshing the concave surface on one shaft with the convex surface (teeth) on the ...Missing: mechanical | Show results with:mechanical
[65]
[PDF] Roark's Formulas for Stress and Strain - Jackson Research Group
Fatigue. Brittle Fracture. Stress Concentration. Effect of Form and Scale on Strength; Rupture Factor. Prestressing. Elastic. Stability. References.
[66]
Akashi Kaikyo Bridge | Type, History, Length, Height, & Facts
The suspension bridge can be viewed as an upside-down arch in tension with only the towers in compression. Because the deck is hung in the air, care must be ...Missing: concave inverted
[67]
Akashi Kaikyo Bridge - Institution of Civil Engineers
The Akashi Kaikyo Bridge is a 3.9km-long suspension bridge connecting Kōbe, in mainland Honshu, and Iwaya, on Awaji Island, in west-central Japan.Missing: concave arches inverted catenary
[68]
[PDF] Liquid Rocket Engine Nozzles - NASA Technical Reports Server
rockets developed for applications such as satellite attitude and orbit control require special methods for nozzle design and analysis because of the relatively.
[69]
[PDF] Introduction to the Method of Characteristics and the Minimum ...
MAE 5540 - Propulsion Systems. 25. Using Method of Characteristics to Design a Bell Nozzle. • This approach “prescribes” the expansion section of the nozzle,.
[70]
[PDF] Rocket nozzles: 75 years of research and development
The present article mainly deals with the aerodynamic features of ideal, coni- cal, bell, plug, expansion-deflection, dual bell, and multi nozzle grid type ...
[71]
22.6 Reflection at concave surfaces – the satellite dish
Mar 15, 2022 · Concave mirrors are useful but the more familiar applications of reflection at concave surfaces involve focussing radio waves and microwaves ...
[72]
[PDF] Future Challenges and Opportunities in Aerodynamics
Today, aeronautics can be characterized by analysis and design tools that are relatively mature based on our current understanding of the physics of flight.
[73]
Top 5 Challenges of Composite-Based Aerospace Design
Aerospace composites offer advanced design possibilities but introduce challenges like material selection, design complexity, manufacturing costs, ...