3D computer graphics or three-dimensional computer graphics (in contrast to 2D computer graphics), are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for the purposes of performing calculations and rendering 2D images. Such images may be stored for viewing later or displayed in real-time. 3D computer graphics rely on many of the same algorithms as 2D computer vector graphics in the wire-frame model and 2D computer raster graphics in the final rendered display. In computer graphics software, 2D applications may use 3D techniques to achieve effects such as lighting, and 3D may use 2D rendering techniques. 3D computer graphics are often referred to as 3D models. Apart from the rendered graphic, the model is contained within the graphical data file. However, there are differences: a 3D model is the mathematical representation of any three-dimensional object. A model is not technically a graphic until it is displayed. A model can be displayed visually as a two-dimensional image through a process called 3D rendering or used in non-graphical computer simulations and calculations. With 3D printing, 3D models are similarly rendered into a 3D physical representation of the model, with limitations to how accurate the rendering can match the virtual model.

3D rendering is the 3D computer graphics process of automatically converting 3D wire frame models into 2D images on a computer. 3D renders may include photorealistic effects or non-photorealistic rendering.

Angry Birds is a video game franchise created by Finnish company Rovio Entertainment. The series focuses on multi-colored birds who try to save their eggs from green-colored pigs. Inspired by Crush the Castle, the game has been praised for its successful combination of fun gameplay, comical style, and low price. Its popularity led to many spin-offs, versions of Angry Birds being created for PCs and gaming consoles, a market for merchandise featuring its characters, a televised cartoon series, and a feature film. In January 2014 there had been over two billion downloads across all platforms, including both regular and special editions: as of July 2015, the series’ games have been downloaded more than three billion times collectively, making it the most downloaded freemium game series of all time. The original Angry Birds has been called "one of the most mainstream games out right now", "one of the great runaway hits of 2010", and "the largest mobile app success the world has seen so far". An animated feature film based on the series was released by Columbia Pictures on 20 May 2016, and the first main-series sequel, Angry Birds 2, was released on 30 July 2015. The first game in the series was initially released in December 2009 for Apple iOS. The company then released ports of the game for other touchscreen smartphone operating systems, including Android, Symbian, and Windows Phone, then video game consoles and PCs.

Animation is a method in which pictures are manipulated to appear as moving images. In traditional animation, images are drawn or painted by hand on transparent celluloid sheets to be photographed and exhibited on film. Today, most animations are made with computer-generated imagery (CGI). Computer animation can be very detailed 3D animation, while 2D computer animation can be used for stylistic reasons, low bandwidth or faster real-time renderings. Other common animation methods apply a stop motion technique to two and three-dimensional objects like paper cutouts, puppets or clay figures. Commonly the effect of animation is achieved by a rapid succession of sequential images that minimally differ from each other. The illusion—as in motion pictures in general—is thought to rely on the phi phenomenon and beta movement, but the exact causes are still uncertain. Analog mechanical animation media that rely on the rapid display of sequential images include the phénakisticope, zoetrope, flip book, praxinoscope and film. Television and video are popular electronic animation media that originally were analog and now operate digitally. For display on the computer, techniques like animated GIF and Flash animation were developed. Animation is more pervasive than many people realise. Apart from short films, feature films, animated gifs and other media dedicated to the display of moving images, animation is also heavily used for video games, motion graphics and special effects. Animation is also prevalent in information technology interfaces.The physical movement of image parts through simple mechanics – in for instance the moving images in magic lantern shows – can also be considered animation. The mechanical manipulation of puppets and objects to emulate living beings has a very long history in automata. Automata were popularised by Disney as animatronics. Animators are artists who specialize in creating animation.

Bartholomew JoJo "Bart" Simpson is a fictional character in the American animated television series The Simpsons and part of the Simpson family. He is voiced by Nancy Cartwright and first appeared on television in The Tracey Ullman Show short "Good Night" on April 19, 1987. Cartoonist Matt Groening created and designed Bart while waiting in the lobby of James L. Brooks' office. Groening had been called to pitch a series of shorts based on his comic strip, Life in Hell, but instead decided to create a new set of characters. While the rest of the characters were named after Groening's family members, Bart's name is an anagram of the word brat. After appearing on The Tracey Ullman Show for three years, the Simpson family received its own series on Fox, which debuted December 17, 1989. At ten years old, Bart is the eldest child and only son of Homer and Marge, and the brother of Lisa and Maggie. Bart's most prominent and popular character traits are his mischievousness, rebelliousness and disrespect for authority. He has appeared in other media relating to The Simpsons – including video games, The Simpsons Movie, The Simpsons Ride, commercials, and comic books – and inspired an entire line of merchandise. In casting, Nancy Cartwright originally planned to audition for the role of Lisa, while Yeardley Smith tried out for Bart. Smith's voice was too high for a boy, so she was given the role of Lisa. Cartwright found that Lisa was not interesting at the time, so instead auditioned for Bart, which she thought was a better role. Hallmarks of the character include his chalkboard gags in the opening sequence; his prank calls to Moe; and his catchphrases "Eat my shorts", "¡Ay, caramba!", and "Don't have a cow, man!" Although, with the exception of "Ay, caramba!", they have been retired or not often used. During the first two seasons of The Simpsons, Bart was the show's breakout character and "Bartmania" ensued, spawning Bart Simpson-themed merchandise touting his rebellious attitude and pride at underachieving, which caused many parents and educators to cast him as a bad role model for children. Around the third season, the series started to focus more on the family as a whole, though Bart still remains a prominent character. Time named Bart one of the 100 most important people of the 20th century, and he was named "entertainer of the year" in 1990 by Entertainment Weekly. Nancy Cartwright has won several awards for voicing Bart, including a Primetime Emmy Award in 1992 and an Annie Award in 1995. In 2000, Bart, along with the rest of his family, was awarded a star on the Hollywood Walk of Fame. He has appeared in every Simpsons episode except Four Great Women and a Manicure.

Captain America is a fictional character appearing in American comic books published by Marvel Comics. Created by cartoonists Joe Simon and Jack Kirby, the character first appeared in Captain America Comics #1 (cover dated March 1941) from Timely Comics, a predecessor of Marvel Comics. Captain America was designed as a patriotic supersoldier who often fought the Axis powers of World War II and was Timely Comics' most popular character during the wartime period. The popularity of superheroes waned following the war and the Captain America comic book was discontinued in 1950, with a short-lived revival in 1953. Since Marvel Comics revived the character in 1964, Captain America has remained in publication. Captain America wears a costume that bears an American flag motif, and is armed with a nearly indestructible shield that he throws at foes. The character is usually depicted as the alter ego of Steve Rogers, a frail young man enhanced to the peak of human perfection by an experimental serum to aid the United States government's efforts in World War II. Near the end of the war, he was trapped in ice and survived in suspended animation until he was revived in the present day. Although Captain America often struggles to maintain his ideals as a man out of his time with its modern realities, he remains a highly respected figure in his community which includes becoming the long-time leader of the Avengers. Captain America was the first Marvel Comics character to have appeared in media outside comics with the release of the 1944 movie serial, Captain America. Since then, the character has been featured in other films and television series, more recently in the Marvel Cinematic Universe (MCU) portrayed by Chris Evans in Captain America: The First Avenger, The Avengers, Captain America: The Winter Soldier, Avengers: Age of Ultron, Captain America: Civil War, Spider-Man: Homecoming and the upcoming Avengers: Infinity War (2018) and its untitled sequel (2019). Captain America is ranked sixth on IGN's "Top 100 Comic Book Heroes of All Time" in 2011, second in their list of "The Top 50 Avengers" in 2012, and second in their "Top 25 best Marvel superheroes" list in 2014.

Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures. The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, machine learning, computational mechanics, computational science, biological imaging, neuroscience, physics, probability, and statistics; it also has strong connections with fluid mechanics and geometric mechanics. Additionally, it complements newer, interdisciplinary fields like bioinformatics and neuroinformatics in the sense that its interpretation uses metadata derived from the original sensor imaging modalities (of which Magnetic Resonance Imaging is one example). It focuses on the anatomical structures being imaged, rather than the medical imaging devices. It is similar in spirit to the history of Computational linguistics, a discipline that focuses on the linguistic structures rather than the sensor acting as the transmission and communication medium(s). In computational anatomy, the diffeomorphism group is used to study different coordinate systems via coordinate transformations as generated via the Lagrangian and Eulerian velocities of flow in R 3 {\displaystyle {\mathbb {R} }^{3}} . The flows between coordinates in Computational anatomy are constrained to be geodesic flows satisfying the principle of least action for the Kinetic energy of the flow. The kinetic energy is defined through a Sobolev smoothness norm with strictly more than two generalized, square-integrable derivatives for each component of the flow velocity, which guarantees that the flows in R 3 {\displaystyle \mathbb {R} ^{3}} are diffeomorphisms. It also implies that the diffeomorphic shape momentum taken pointwise satisfying the Euler-Lagrange equation for geodesics is determined by its neighbors through spatial derivatives on the velocity field. This separates the discipline from the case of incompressible fluids for which momentum is a pointwise function of velocity. Computational anatomy intersects the study of Riemannian manifolds and nonlinear global analysis, where groups of diffeomorphisms are the central focus. Emerging high-dimensional theories of shape are central to many studies in Computational anatomy, as are questions emerging from the fledgling field of shape statistics. The metric structures in Computational anatomy are related in spirit to morphometrics, with the distinction that Computational anatomy focuses on an infinite-dimensional space of coordinate systems transformed by a diffeomorphism, hence the central use of the terminology diffeomorphometry, the metric space study of coordinate systems via diffeomorphisms.

Computer animation is the process used for digitally generating animated images. The more general term computer-generated imagery (CGI) encompasses both static scenes and dynamic images, while computer animation only refers to the moving images. Modern computer animation usually uses 3D computer graphics, although 2D computer graphics are still used for stylistic, low bandwidth, and faster real-time renderings. Sometimes, the target of the animation is the computer itself, but sometimes film as well. Computer animation is essentially a digital successor to the stop motion techniques using 3D models, and traditional animation techniques using frame-by-frame animation of 2D illustrations. Computer-generated animations are more controllable than other more physically based processes, constructing miniatures for effects shots or hiring extras for crowd scenes, and because it allows the creation of images that would not be feasible using any other technology. It can also allow a single graphic artist to produce such content without the use of actors, expensive set pieces, or props. To create the illusion of movement, an image is displayed on the computer monitor and repeatedly replaced by a new image that is similar to it, but advanced slightly in time (usually at a rate of 24, 25 or 30 frames/second). This technique is identical to how the illusion of movement is achieved with television and motion pictures. For 3D animations, objects (models) are built on the computer monitor (modeled) and 3D figures are rigged with a virtual skeleton. For 2D figure animations, separate objects (illustrations) and separate transparent layers are used with or without that virtual skeleton. Then the limbs, eyes, mouth, clothes, etc. of the figure are moved by the animator on key frames. The differences in appearance between key frames are automatically calculated by the computer in a process known as tweening or morphing. Finally, the animation is rendered.For 3D animations, all frames must be rendered after the modeling is complete. For 2D vector animations, the rendering process is the key frame illustration process, while tweened frames are rendered as needed. For pre-recorded presentations, the rendered frames are transferred to a different format or medium, like digital video. The frames may also be rendered in real time as they are presented to the end-user audience. Low bandwidth animations transmitted via the internet (e.g. Adobe Flash, X3D) often use software on the end-users computer to render in real time as an alternative to streaming or pre-loaded high bandwidth animations.

Computer graphics are pictures and films created using computers. Usually, the term refers to computer-generated image data created with the help of specialized graphical hardware and software. It is a vast and recently developed area of computer science. The phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes erroneously referred to as computer-generated imagery (CGI). Some topics in computer graphics include user interface design, sprite graphics, vector graphics, 3D modeling, shaders, GPU design, implicit surface visualization with ray tracing, and computer vision, among others. The overall methodology depends heavily on the underlying sciences of geometry, optics, and physics. Computer graphics is responsible for displaying art and image data effectively and meaningfully to the consumer. It is also used for processing image data received from the physical world. Computer graphics development has had a significant impact on many types of media and has revolutionized animation, movies, advertising, video games, and graphic design in general.

The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions and bicomplex numbers. White and Nylander's formula for the "nth power" of the vector v = ⟨ x , y , z ⟩ {\displaystyle {\mathbf {v} }=\langle x,y,z\rangle } in ℝ3 is v n := r n ⟨ sin ⁡ ( n θ ) cos ⁡ ( n ϕ ) , sin ⁡ ( n θ ) sin ⁡ ( n ϕ ) , cos ⁡ ( n θ ) ⟩ {\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(n\theta )\cos(n\phi ),\sin(n\theta )\sin(n\phi ),\cos(n\theta )\rangle } where r = x 2 + y 2 + z 2 {\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}} , ϕ = arctan ⁡ ( y / x ) = arg ⁡ ( x + y i ) {\displaystyle \phi =\arctan(y/x)=\arg(x+yi)} , and θ = arctan ⁡ ( x 2 + y 2 / z ) = arccos ⁡ ( z / r ) {\displaystyle \theta =\arctan({\sqrt {x^{2}+y^{2}}}/z)=\arccos(z/r)} . The Mandelbulb is then defined as the set of those c {\displaystyle {\mathbf {c} }} in ℝ3 for which the orbit of ⟨ 0 , 0 , 0 ⟩ {\displaystyle \langle 0,0,0\rangle } under the iteration v ↦ v n + c {\displaystyle {\mathbf {v} }\mapsto {\mathbf {v} }^{n}+{\mathbf {c} }} is bounded. For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of "lobes" depending on n. Many of their graphic renderings use n = 8. However, the equations can be simplified into rational polynomials when n is odd. For example, in the case n = 3, the third power can be simplified into the more elegant form: ⟨ x , y , z ⟩ 3 = ⟨ ( 3 z 2 − x 2 − y 2 ) x ( x 2 − 3 y 2 ) x 2 + y 2 , ( 3 z 2 − x 2 − y 2 ) y ( 3 x 2 − y 2 ) x 2 + y 2 , z ( z 2 − 3 x 2 − 3 y 2 ) ⟩ {\displaystyle \langle x,y,z\rangle ^{3}=\left\langle \ {\frac {(3z^{2}-x^{2}-y^{2})x(x^{2}-3y^{2})}{x^{2}+y^{2}}},{\frac {(3z^{2}-x^{2}-y^{2})y(3x^{2}-y^{2})}{x^{2}+y^{2}}},z(z^{2}-3x^{2}-3y^{2})\right\rangle } . The Mandelbulb given by the formula above is actually one in a family of fractals given by parameters (p,q) given by: v n := r n ⟨ sin ⁡ ( p θ ) cos ⁡ ( q ϕ ) , sin ⁡ ( p θ ) sin ⁡ ( q ϕ ) , cos ⁡ ( p θ ) ⟩ {\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(p\theta )\cos(q\phi ),\sin(p\theta )\sin(q\phi ),\cos(p\theta )\rangle } Since p and q do not necessarily have to equal n for the identity |vn|=|v|n to hold. More general fractals can be found by setting v n := r n ⟨ sin ⁡ ( f ( θ , ϕ ) ) cos ⁡ ( g ( θ , ϕ ) ) , sin ⁡ ( f ( θ , ϕ ) ) sin ⁡ ( g ( θ , ϕ ) ) , cos ⁡ ( f ( θ , ϕ ) ) ⟩ {\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(f(\theta ,\phi ))\cos(g(\theta ,\phi )),\sin(f(\theta ,\phi ))\sin(g(\theta ,\phi )),\cos(f(\theta ,\phi ))\rangle } for functions f and g.