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2D

In mathematics and geometry, 2D refers to two-dimensional space or figures, characterized by having only two dimensions—typically length and width—without depth or thickness, allowing points to be described using two coordinates, such as (x, y).^[1] These flat shapes or planes form the foundation for understanding more complex spatial concepts and are ubiquitous in fields like computer graphics, engineering, and physics.^[2] Common examples of 2D shapes include polygons such as squares, circles, triangles, and rectangles, each defined by their boundaries and properties like area and perimeter.^[3] The study of 2D geometry dates back to ancient civilizations, with foundational principles outlined in Euclid's Elements around 300 BCE, emphasizing axioms for lines, angles, and congruence in planar figures.^[4] In modern applications, 2D representations are essential for modeling real-world phenomena, such as topographic maps, architectural blueprints, and digital imaging, where transformations like rotation, scaling, and translation preserve the planar structure. Unlike three-dimensional (3D) objects, 2D shapes lack volume, enabling precise calculations of properties like symmetry.^[5] Key attributes of 2D space include its Euclidean metric, where distance between points is calculated via the Pythagorean theorem,^[6] and its role in vector analysis for operations in the plane.^[7] Notable theorems, such as the Pythagorean theorem for right triangles and properties of parallel lines, underpin much of 2D analytic geometry, facilitating advancements in areas like computer-aided design (CAD) and animation.^[4]

Mathematical Foundations

Definition of Two-Dimensional Space

In geometry, two-dimensional space, often denoted as 2D space, refers to a geometric setting characterized by exactly two independent dimensions, typically formalized as the Euclidean plane \mathbb{R}^2, where any point is represented by an ordered pair of real numbers (x, y).^[8] This structure allows for the specification of positions, directions, and shapes within a flat, infinite expanse without depth or curvature.^[9] To contextualize 2D space, geometric dimensions describe the minimum number of coordinates required to uniquely identify points in a figure: a zero-dimensional (0D) object is a single point with no length, width, or extent; a one-dimensional (1D) object, such as a line, has length but no width or height; two-dimensional (2D) objects, like planes or surfaces, possess both length and width to define area; and three-dimensional (3D) objects incorporate volume through an additional height dimension.^[10] The standard coordinate system for navigating 2D space is the Cartesian coordinate system, which employs two mutually perpendicular axes—the horizontal x-axis and vertical y-axis—intersecting at an origin point (0, 0), enabling precise location of points via their distances along these axes.^[11] An alternative representation is the polar coordinate system, which specifies points using a radial distance r from the origin and an angular measure \theta from a reference direction, with the conversion to Cartesian coordinates given by the formulas:

x = r \cos \theta, \quad y = r \sin \theta.

These transformations facilitate analysis in contexts where rotational symmetry is advantageous.^[12] The rigorous foundation of 2D Euclidean space rests on Hilbert's axioms for plane geometry, a set of 20 postulates divided into groups addressing incidence (points on lines), order (betweenness), congruence (equality of segments and angles), parallelism (unique line through a point parallel to another), and continuity (completeness of the real numbers), collectively ensuring the space's flatness, uniformity, and absence of intrinsic curvature.^[13]

Key Properties and Theorems

The metric properties of two-dimensional Euclidean space are defined by the Euclidean distance between any two points P_1 = (x_1, y_1) and P_2 = (x_2, y_2), given by the formula d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.^[14] This formula arises from the geometry of the plane, where the shortest path between points is a straight line, and the distance corresponds to the length of the vector \vec{v} = (x_2 - x_1, y_2 - y_1). The length of \vec{v} is derived from the dot product: \|\vec{v}\| = \sqrt{\vec{v} \cdot \vec{v}}, since the dot product \vec{v} \cdot \vec{v} = (x_2 - x_1)^2 + (y_2 - y_1)^2 yields the squared magnitude, and the square root provides the positive distance. In Euclidean 2D space, angles are measured relative to the metric, with the angle \theta between two vectors \vec{u} and \vec{v} satisfying \cos \theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|}, preserving the right-angle property in perpendicular lines where the dot product is zero. Parallelism is governed by the Euclidean parallel postulate, which states that through any point not on a given line, exactly one line can be drawn parallel to the given line.^[15] This postulate, the fifth in Euclid's Elements, ensures unique parallels and distinguishes Euclidean geometry from non-Euclidean geometries, where multiple or no parallels may exist through such a point. A fundamental theorem in 2D Euclidean geometry is the Pythagorean theorem, which asserts that in a right-angled triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2.^[16] One proof uses similar triangles: drop an altitude from the right angle to the hypotenuse, dividing the original triangle into two smaller right triangles similar to each other and to the original. The altitude creates segments p and q on the hypotenuse such that c = p + q, and similarity yields a^2 = c p and b^2 = c q. Adding these equations gives a^2 + b^2 = c(p + q) = c^2.^[17] Area formulas for basic shapes in 2D Euclidean space derive from these metric properties. The area of a triangle with base b and height h is \frac{1}{2} b h, established by showing that a parallelogram with the same base and height has double the area of the triangle, as the triangles formed by a diagonal are congruent. For a circle of radius r, the area is \pi r^2, proven by Archimedes by inscribing and circumscribing polygons and showing the circle's area equals that of a right triangle with legs r and the circumference $2\pi r.^[18] Invariances in 2D space include isometries such as rotations and translations, which preserve distances and angles. A rotation by angle \theta around a point maps points while maintaining the Euclidean metric, as the transformation matrix \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} yields a dot product equal to the original for transformed vectors.^[19] Translations shift all points by a fixed vector without altering relative positions, thus preserving the distance formula and angle measures.^[19]

Physical Applications

2D Models in Physics

In physics, two-dimensional (2D) models are employed to approximate three-dimensional (3D) systems by assuming uniformity or negligible variation in one spatial dimension, thereby enhancing mathematical tractability while capturing essential behaviors. This reduction is particularly useful for systems like thin films, where thickness is minimal compared to lateral extents, allowing treatment as planar structures, or for motions confined to a plane, such as projectile trajectories ignoring vertical depth.^[20]^[21] In classical mechanics, 2D models simplify the analysis of planar motion under Newton's laws. For projectile motion, the governing equations derive from Newton's second law, \mathbf{F} = m\mathbf{a}, assuming no air resistance and constant gravitational acceleration \mathbf{g} = -g \hat{y} in the vertical direction, with no horizontal force. This yields constant acceleration components a_x = 0 and a_y = -g. Integrating twice with respect to time, starting from initial velocity \mathbf{v}_0 = v_0 \cos \theta \, \hat{x} + v_0 \sin \theta \, \hat{y} and initial position at the origin, results in the position equations:

x = (v_0 \cos \theta) t

y = (v_0 \sin \theta) t - \frac{1}{2} g t^2

These equations describe the parabolic trajectory in the xy-plane, enabling analytical solutions for range and maximum height without full 3D complexity.^[21] In electromagnetism, 2D electrostatic models arise for configurations with translational invariance along one axis, such as an infinite line charge, reducing the problem to cylindrical symmetry in the perpendicular plane. Using Gauss's law, \oint \mathbf{E} \cdot d\mathbf{A} = q_{\text{enc}} / \epsilon_0, for a cylindrical Gaussian surface of radius r and length L around a line charge with uniform linear density \lambda, the enclosed charge is q_{\text{enc}} = \lambda L. The flux through the curved surface is E \cdot 2\pi r L, leading to the radial electric field magnitude E = \frac{\lambda}{2\pi \epsilon_0 r}, directed perpendicular to the line. This 2D approximation simplifies field calculations compared to finite 3D charge distributions.^[22] In fluid dynamics, 2D potential flow theory models irrotational, incompressible flows in a plane, assuming velocity \mathbf{v} = \nabla \phi where \phi is the velocity potential. The continuity equation \nabla \cdot \mathbf{v} = 0 then implies Laplace's equation \nabla^2 \phi = 0 in two dimensions, \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0. Solutions to this equation, such as uniform flow or sources/sinks, can be superposed to describe complex flows around obstacles, providing an idealized framework for aerodynamic or hydrodynamic problems while neglecting viscosity and three-dimensional effects.^[23]

Real-World Examples and Simulations

In everyday physics, shadows cast by three-dimensional objects onto a flat surface serve as a natural illustration of projecting higher-dimensional structures into two dimensions, where the silhouette captures contours but loses depth information.^[24] Similarly, soap films stretched across wire frames exemplify two-dimensional minimal surfaces that minimize area according to Plateau's laws, which state that films meet in triples at 120-degree angles and along straight lines or circular arcs of equal curvature.^[25] These configurations arise from surface tension seeking equilibrium, providing a tangible demonstration of variational principles in two-dimensional geometry.^[26] A landmark experimental realization of a two-dimensional atomic lattice is graphene, first isolated in 2004 by Andre Geim and Konstantin Novoselov through mechanical exfoliation of graphite, yielding a single layer of carbon atoms in a honeycomb arrangement. This breakthrough, which earned them the 2010 Nobel Prize in Physics, revealed graphene's unique electronic properties, including charge carriers that behave as massless Dirac fermions due to Dirac cones in the band structure—linear dispersions where the effective mass vanishes at the neutrality point.^[27] These fermions enable phenomena like the quantum Hall effect at room temperature, validating two-dimensional models of relativistic particles in condensed matter.^[27] Computational simulations extend these models by numerically solving two-dimensional field equations. In engineering, finite element methods approximate stress analysis in thin plates or membranes by meshing the domain into triangular or quadrilateral elements and iteratively solving for displacements and strains under load, often revealing principal stress contours for design validation.^[28] Historically, early 20th-century analog computers employed Prandtl's membrane analogy, introduced in 1903, to visualize two-dimensional torsion fields: a thin elastic membrane stretched over a scaled cross-sectional model deflects proportionally to the Prandtl stress function, with contours indicating shear stress lines.^[29] This physical setup, using soap films or rubber sheets, predated digital methods and provided qualitative insights into complex geometries like I-beams.^[29] Despite their utility, two-dimensional approximations falter in ultrathin materials where edge effects dominate, such as in graphene nanoribbons where boundary atoms create localized electronic states that disrupt bulk Dirac cone behavior and introduce scattering or magnetism.^[30] These deviations, arising from dangling bonds or reconstructions at edges, necessitate three-dimensional corrections for accurate predictions in nanoscale devices.^[31]

Computing and Graphics

2D Computer Graphics Techniques

2D computer graphics techniques encompass methods for representing, transforming, and rendering visual elements on digital displays using two-dimensional coordinates. These techniques form the backbone of image creation and manipulation in computing, enabling efficient handling of shapes, colors, and spatial relationships. Raster graphics represent images as grids of pixels, where each pixel holds color and intensity values, making them suitable for complex, photorealistic visuals but prone to quality loss upon scaling. For instance, the BMP format stores raster data as a bitmap of pixel values arranged in rows. In contrast, vector graphics define images through mathematical descriptions of paths and shapes, allowing infinite scalability without degradation. A key element in vector graphics is the Bézier curve, a parametric curve used to model smooth contours. The cubic Bézier curve is defined by four control points P_0, P_1, P_2, P_3 and the equation:

\mathbf{B}(t) = (1-t)^3 \mathbf{P}_0 + 3(1-t)^2 t \mathbf{P}_1 + 3(1-t) t^2 \mathbf{P}_2 + t^3 \mathbf{P}_3, \quad t \in [0,1]

This formulation, originally developed by Pierre Bézier for automotive design, facilitates precise curve generation in graphics software.^[32] Affine transformations are fundamental for manipulating 2D graphics, combining linear operations like rotation, scaling, and shearing with translation to preserve parallelism and ratios. These are represented using 3x3 matrices in homogeneous coordinates, where a point (x, y) is augmented to (x, y, 1). For rotation by an angle \theta around the origin, the transformation matrix is:

\begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

Scaling by factors s_x and s_y uses a diagonal matrix \begin{pmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \end{pmatrix}, while translation by (t_x, t_y) employs \begin{pmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{pmatrix}. Composing these matrices allows complex manipulations, such as rotating and then translating an object, essential for animations and user interfaces.^[33] Rendering pipelines convert geometric descriptions into pixel-based output, with the scanline algorithm being a classic method for filling polygons. This approach processes the image from top to bottom, intersecting each horizontal scanline with polygon edges to identify entry and exit points, then filling spans between paired intersections using edge coherence for efficiency. It assumes non-overlapping edges and handles convex or concave polygons by sorting intersections along the scanline. To mitigate aliasing—jagged edges from discrete sampling—anti-aliasing techniques like supersampling are applied. Supersampling renders the scene at a higher resolution (e.g., 4x samples per pixel), averages the results, and downsamples to the display resolution, reducing artifacts by oversampling and filtering high-frequency details. This method, while computationally intensive, improves edge smoothness in rasterized output.^[34]^[35] The evolution of 2D graphics techniques traces back to Ivan Sutherland's 1963 Sketchpad system, the first interactive computer-aided design program, which introduced core concepts like light pens for direct manipulation and constraint-based drawing on the Lincoln TX-2 computer. This pioneering work laid the groundwork for modern vector-based editing. Subsequent advancements led to standardized APIs, such as the HTML5 Canvas element, introduced in 2008 as part of the WHATWG specification, which provides a bitmap canvas for imperative 2D drawing via JavaScript, supporting paths, transformations, and raster operations in web browsers. More recent developments include enhanced 2D rendering in WebGPU, standardized in 2023, which enables GPU-accelerated 2D graphics in web applications for improved performance.^[36]^[37]^[38]

2D in Video Games and Media

In video games, 2D graphics have been foundational to numerous genres, particularly platformers and top-down action-adventure titles. Platformers like Super Mario Bros. (1985), developed by Nintendo for the Nintendo Entertainment System, rely on sprite-based 2D rendering to depict characters, enemies, and environments in a side-scrolling format, enabling precise jump mechanics and level design that defined the genre.^[39] Similarly, top-down RPGs and action-adventure games, such as The Legend of Zelda (1986), also by Nintendo, utilize 2D overhead views to facilitate exploration, puzzle-solving, and combat in expansive worlds like the kingdom of Hyrule.^[40] In animation for media, 2D techniques emphasize hand-drawn or digital sequences to create fluid motion. Traditional cel animation, pioneered by Walt Disney Studios in the 1930s, involved inking character designs onto transparent celluloid sheets placed over painted backgrounds, allowing for efficient production of films like Snow White and the Seven Dwarfs (1937).^[41] Modern digital tools have evolved this process; frame-by-frame animation in software like Adobe Animate enables artists to create individual keyframes that change per frame on the timeline, ideal for complex, non-linear movements in contemporary 2D productions.^[42] The evolution from pure 2D to pseudo-3D marked a transitional phase in gaming, with 2.5D techniques providing depth illusions without requiring full 3D polygonal models. Games like Doom (1993), developed by id Software, employed binary space partitioning (BSP) trees combined with elements of ray casting to render walls, floors, and sprites in a first-person perspective, simulating three-dimensional navigation on 2D maps while limiting vertical complexity. This approach allowed for immersive environments in early first-person shooters, bridging 2D simplicity with emerging 3D aesthetics. As of 2025, trends continue to highlight a revival of 2D styles in indie games, particularly through pixel art, which evokes nostalgia while supporting intricate narratives and mechanics. Titles like Celeste (2018), developed by Maddy Makes Games, and more recent examples such as Animal Well (2024) by Shared-Wires and Prince of Persia: The Lost Crown (2024) by Ubisoft showcase pixel art and 2D platforming's ongoing effectiveness in delivering challenging gameplay and artistic expression.^[43]^[44]^[45] Additionally, 2D games offer accessibility advantages over 3D counterparts, featuring simpler controls and interfaces that make them more approachable for casual players and those with varying skill levels or hardware limitations.^[46]

Other Uses and Interpretations

Acronyms and Abbreviations

In various technical contexts, particularly in engineering and information technology, "2D" serves as an abbreviation for "two-dimensional," referring to representations or structures that operate in two spatial dimensions, such as 2D drafting in design software. ^[47] A prominent example is in barcode technology, where 2D barcodes encode data both horizontally and vertically in a matrix format, enabling higher data density than one-dimensional codes; the QR code, a widely adopted 2D barcode, was developed in 1994 by Denso Wave, a subsidiary of the Japanese automotive supplier Denso Corporation, initially for tracking vehicle parts during manufacturing. ^[48] ^[49] Organizationally, especially in military nomenclature, "2D" or "2d" denotes "second division," shorthand for units like the 2d Marine Division of the United States Marine Corps, an infantry division responsible for expeditionary operations and training, established in its modern form during World War II. ^[50] Commercially, "2D" appears in product branding tied to technical applications, such as 2D barcode systems licensed by Denso Wave for global use in inventory and consumer tracking since the QR code's release. ^[51] To disambiguate "2D" from its primary meaning as an abbreviation for two-dimensional space—a geometric concept involving length and width without depth—the interpretation depends on domain-specific context; for example, in military references, it aligns with ordinal "second" rather than geometric dimensions. ^[52]

Cultural and Artistic References

In literature, Edwin Abbott Abbott's novella Flatland: A Romance of Many Dimensions (1884) employs two-dimensional space as a central metaphor for perceptual limitations and social hierarchies, portraying a world of geometric figures to explore themes of dimensionality, vision, and reality that resonate in postmodern art and philosophy.^[53] The work critiques Victorian-era constraints while symbolizing "flatness" as a conceptual barrier, influencing artistic explorations of abstraction and perspective in visual culture.^[54] Its enduring impact is evident in contemporary exhibitions, such as the 2014 "Flatland" show at the Guadalupe Cultural Arts Center, where artists drew on the novella to examine cultural flattening in a globalized, digital era.^[55] In music, "2D" is the stage name of the fictional lead singer and keyboardist of the virtual band Gorillaz, created by Damon Albarn and Jamie Hewlett in 1998. The character, known for his blue hair and laid-back persona, has become an iconic figure in pop culture, blending animation, music, and multimedia storytelling.^[56] Within otaku culture, the mid-2010s meme "2D > 3D" encapsulates a preference for idealized two-dimensional anime and manga characters over three-dimensional real-life individuals, reflecting a broader romantic and aesthetic attachment to fictional representations.^[57] A 2013 survey of 500 Japanese male otaku found that 23% selected "2D" as their preferred type of "girlfriend," highlighting how this trope underscores the perceived purity and escapism of 2D forms in fan communities.^[57] This sentiment ties into larger discussions of otaku identity, where 2D characters offer emotional fulfillment unbound by physical realities.^[58] The release of James Cameron's Avatar in 2009 sparked debates over 2D versus 3D in cinema, with widespread backlash against rushed post-production conversions of existing 2D films to 3D formats, often resulting in dim visuals and technical flaws.^[59] Cameron himself criticized the trend, noting that such conversions compromised image quality by darkening screens and altering colors, unlike native 3D productions like Avatar.^[60] Films like Clash of the Titans (2010) exemplified the controversy, as their hasty 3D retrofits were derided for poor depth and viewer discomfort, contributing to a broader skepticism about 3D's viability beyond select blockbusters.^[61] This backlash tempered the post-Avatar 3D boom, emphasizing the artistic trade-offs between dimensionality and narrative clarity.^[62]

References

[1]
Two-Dimensional Definition (Illustrated Mathematics Dictionary)
Having only two dimensions, such as width and height but no thickness. Squares, Circles, Triangles, etc are two-dimensional objects. Also known as "2D".
[2]
2D - Math.net
A two-dimensional object is an object in which a point on the object can be specified using 2 coordinates; in other words, the object has 2 separate dimensions ...
[3]
2D Shapes - Names, Definition, Properties - Cuemath
A 2D shape is a flat shape that has only two dimensions - length and width, with no thickness or depth. Some common 2 Dimensional shapes are square, ...
[4]
2D & 3D Shapes | Definition, Differences & Examples - Study.com
A two-dimensional, or 2D, object is a flat object that has length and width but no depth. If a two-dimensional shape is graphed on a coordinate plane, only the ...
[5]
2D (Two Dimensional) Shapes – Definition With Examples
A 2D shape or two-dimensional shape is a flat figure that has two dimensions—length and width. Learn examples, formulas, properties of 2D shapes and much ...
[6]
2D Shapes Definition - BYJU'S
In maths, 2d shapes can be defined as the plane figures that can be drawn on a flat (or plane) surface or a piece of paper. All the 2d shapes have various ...<|control11|><|separator|>
[7]
Euclidean Space -- from Wolfram MathWorld
Euclidean n-space, sometimes called Cartesian space or simply n-space, is the space of all n-tuples of real numbers, (x_1, x_2, ..., x_n).
[8]
Euclidean space - PlanetMath
Mar 22, 2013 · Euclidean space is a Euclidean vector space that has “forgotten” its origin. •. A 2-dimensional Euclidean space is often called a Euclidean ...Missing: two- | Show results with:two-
[9]
Dimensions - Math.net
A line or any of its parts (line segment or ray) is one dimensional (1D). In one-dimensional space, any point can be described with one number, as shown in the ...
[10]
Cartesian coordinates - Math Insight
The Cartesian coordinates (also called rectangular coordinates) of a point are a pair of numbers (in two-dimensions) or a triplet of numbers (in three- ...
[11]
Calculus II - Polar Coordinates - Pauls Online Math Notes
Nov 13, 2023 · ... equations that will convert polar coordinates into Cartesian coordinates. Polar to Cartesian Conversion Formulas. x=rcosθy=rsinθ x = r cos ...Missing: authoritative | Show results with:authoritative
[12]
[PDF] Hilbert's axioms for Euclidean geometry
On every line there exist at least two distinct points. There exist at least three points which are not on the same line. GROUP II: AXIOMS OF ORDER.
[13]
Distance -- from Wolfram MathWorld
In Euclidean three-space, the distance between points (x_1,y_1,z_1) and (x_2,y_2,z_2) is. d=sqrt((x_2-x_1)^2+(y_2-y_1. (2). In general, the distance between ...
[14]
Euclid's Elements, Book I, Postulate 5 - Clark University
This postulate is usually called the “parallel postulate” since it can be used to prove properties of parallel lines. Euclid develops the theory of parallel ...
[15]
Pythagorean Theorem -- from Wolfram MathWorld
For a right triangle with legs a and b and hypotenuse c, a^2+b^2=c^2. (1) Many different proofs exist for this most fundamental of all geometric theorems.<|separator|>
[16]
Pythagoras' Theorem Elaboration on the Similarity Argument
The Pythagorean theorem deals with three pairs of correspondingly similar shapes. The first component in a pair is a right triangle, the second could be any ...
[17]
[PDF] ARCHIMEDES, MEASUREMENT OF A CIRCLE1
The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the ...
[18]
Isometry -- from Wolfram MathWorld
An isometry of the plane is a linear transformation which preserves length. Isometries include rotation, translation, reflection, glides, and the identity map.
[19]
Let's discuss: When can we call a thin film 2-dimensional?
In other words, the energy and movements of electrons in graphene can be mathematically described and calculated using a 2D plane. Moreover, the sp2-bonded ...Missing: motion | Show results with:motion
[20]
[PDF] Chapter 3 Motion in Two and Three Dimensions
Projectile motion is a special case of constant acceleration, so we simply use Eqs. 3.16–. 3.19, with the proper values of ax and ay. 3.1.6 Uniform Circular ...
[21]
An Infinite Line of Charge - Physics
The total charge enclosed is qenc = λL, the charge per unit length multiplied by the length of the line inside the cylinder. To find the net flux, consider the ...
[22]
[PDF] Potential Flow Theory - MIT
We can substitute in the relationship between potential and velocity and arrive at the Laplace Equation, which we will revisit in our discussion on linear waves ...
[23]
Projecting 3D objects - Peter Collingridge
Projecting 3D objects involves converting 3D coordinates to 2D coordinates, like looking at a shadow on a 2D screen. The simplest projection is head-on.
[24]
Soap Films, Minimal Surfaces and Beyond
Through his experimental observations, he formulated a set of laws – known as Plateau's laws – that describe the shape and configuration of all possible soap ...
[25]
Bursting the Bubble: Solution to the Kirchhoff-Plateau Problem
Mar 31, 2017 · Plateau hypothesized that when you dip a rigid wire frame into a soap solution, the surface of the soap film formed on the frame represents a ...
[26]
Two-dimensional gas of massless Dirac fermions in graphene - Nature
Nov 10, 2005 · Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions.
[27]
Two-Dimensional Finite Element Method - ScienceDirect.com
Two-dimensional FEM is defined as a computational technique used to solve differential equations for two-dimensional applications, characterized by its ...
[28]
[PDF] Unit 11 Membrane Analogy (for Torsion)
For a number of cross-sections, we cannot find stress functions. However, we can resort to an analogy introduced by Prandtl (1903). Consider a membrane ...
[29]
Edge effects in graphene nanostructures: From multiple reflection ...
Edge effects in graphene nanostructures influence the electronic density of states, with different edge types, especially zigzag, having a prominent role.
[30]
[2407.14134] On the Edge Roughness of Two-Dimensional Materials
Jul 19, 2024 · This study examines edge roughness in 2D materials, finding ultra-flat edges in graphene and how dynamic effects and crack deflection ...
[31]
[PDF] A Curve Tutorial for Introductory Computer Graphics
Oct 7, 2003 · The two most important types of approximating curves in computer graphics are Bézier. Curves and B-Splines. 6.1 Bézier Curves. Mathematician ...
[32]
[PDF] Affine Transformations
The affines include translations and all linear transformations, like scale, rotate, and shear. Let us first examine the affine transforms in 2D space, where ...
[33]
[PDF] Scanline Fill Algorithm - Computer Science | UC Davis Engineering
1) intersect scanline y with each edge 2) sort interesections by increasing x [p0,p1,p2,p3] 3) fill pairwise (p0 −> p1, p2−> p3, ....) Rule: If the intersection ...
[34]
Antialiasing methods
Supersampling or postfiltering is the process by which aliasing effects in graphics are reduced by increasing the frequency of the sampling grid and then ...
[35]
Sketchpad: a man-machine graphical communication system
The Sketchpad system makes it possible for a man and a computer to converse rapidly through the medium of line drawings.<|control11|><|separator|>
[36]
Canvas API - MDN Web Docs
Jul 17, 2025 · The Canvas API provides a means for drawing graphics via JavaScript and the HTML element. Among other things, it can be used for animation, ...CanvasRenderingContext2D · Canvas tutorial · WebGL: 2D and 3D graphics...
[37]
https://developer.mozilla.org/en-US/docs/Web/API/Canvas_API
[38]
The Legend of Zelda (1986) - MobyGames
A young, pointy-eared boy named Link takes on an epic quest to restore the fragmented Triforce of Wisdom and save the Princess Zelda from the clutches of the ...NES credits (1986) · Screenshots · Releases · Covers<|separator|>
[39]
Cel Animation | The Walt Disney Family Museum
In the 1930s, the Walt Disney Studios used a then-innovative animation method: artists drew on celluloid sheets, or “cels,” with ink, and then placed these ...
[40]
Frame-by-frame animation with Animate - Adobe Help Center
May 24, 2023 · Frame-by-frame animation changes the contents of the Stage in every frame. It is best suited to complex animation in which an image changes in every frame.Frame-by-frame animationAdobe, Inc.
[41]
The Pixel Art Revolution Will Be Televised - WIRED
Jan 4, 2022 · Over the past decade, pixel art has experienced a renaissance thanks to the popularity of indie-developed games like Celeste and Eastward.
[42]
The future of 2D gaming | GamesIndustry.biz
Nov 28, 2018 · 2D interfaces are everywhere, so casual gamers will always find these games simpler to use and more fun to play. The frustration of a 3D ...
[43]
2D Detail Drafting - All Acronyms
The abbreviation 2D stands for Detail Drafting and is mostly used in the following categories: Design, Drafting, Engineering, Technology, Business. Whether ...
[44]
QR Code development story｜Technologies｜DENSO WAVE
In 1994, there was an event that totally changed the concept of code reading, that is, the advent of the QR Code system. It was developed by engineers working ...
[45]
What Is a 2D barcode and How Does It Work? - TechTarget
Dec 14, 2021 · A 2D (two-dimensional) barcode is a graphical image that stores information horizontally as one-dimensional barcodes do, as well as vertically.
[46]
2nd-Degree Burn: What It Looks Like, Treatment & Healing
Dec 13, 2022 · Second-degree burns are a mild type of burn that causes blistering, shiny skin, pain and skin discoloration. They're the most common type of burn.
[47]
2d Marine Division
The mission of the 2d Marine Division (2d MARDIV) is to generate, train, and certify forces in order to conduct expeditionary operations.
[48]
History of QR Code | QRcode.com | DENSO WAVE
In 1994, DENSO WAVE (then a division of DENSO CORPORATION) announced the release of its QR Code. The QR in the name stands for quick response, expressing ...
[49]
2D 2-Dimensional - All Acronyms
2D - 2-Dimensional. The abbreviation 2D stands for 2-Dimensional and is mostly used in the following categories: Technology, Telecom, Coding, Shape, Software.Missing: engineering | Show results with:engineering<|separator|>
[50]
Flatland: A Romance of Many Dimensions by Edwin Abbott Abbott
The story unfolds in a two-dimensional world inhabited by geometric figures, primarily focusing on a Square who narrates his experiences and insights about ...Missing: metaphor postmodern art<|separator|>
[51]
https://www.qrcode.com/en/history/
[52]
Local, International Artists Explore a 'Flatland' at the Guadalupe
Jul 12, 2014 · The presenting artists of “Flatland” are at the intersection of cultural form, process and meaning in this emerging, “flat” world.
[53]
http://www.gutenberg.org/ebooks/201
[54]
[PDF] Defining the Ongoing Relationship between Anime and Otaku ...
The first section of the case study frames the relationship between anime and otaku culture through the history of Astro Boy. Astro Boy, which is commonly ...Missing: meme | Show results with:meme
[55]
What James Cameron and 'Avatar' Did (and Didn't Do) for 3D ...
Dec 13, 2022 · Cameron is still bullish on 3D, and not just in terms of the Avatar franchise; he even says that 3D TVs can make a comeback if designed correctly.
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James Cameron says film-makers are 'not using 3D properly'
Jul 9, 2013 · The Avatar director tells a Mexico technology conference that blockbusters such as Man of Steel and Iron Man 3 do not need the effect.
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Clash of the Titans: Is 'fake 3D' a scam? - The Week
Jan 8, 2015 · The blockbuster remake was hastily converted into 3D in postproduction rather than being filmed in the medium in the first place.<|control11|><|separator|>
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James Cameron Blasts 3D Movies Fad After Avatar's Success
Oct 25, 2022 · James Cameron criticizes the scramble to produce films in 3D that went on after his first Avatar film came out in 2009.