Qapital is a personal finance mobile application (app) for the iOS and Android operating systems, developed by Qapital, LLC. The app is designed to motivate users to save money through a gamification of their spending behavior. It moves money from a user's checking account to a separate Qapital account, when certain rules are triggered. Its database is used by psychology professor Dan Ariely to study consumer behavior. Qapital was released in Sweden in 2013, then in the US in early 2015. The application was later withdrawn from the Swedish market in April 2015, in order to focus on the US market. == History == The idea for Qapital was conceived by ex-bankers in Sweden. The software was designed by twin brothers Daniel and Andreas Källbom of Studio Källbom and released in Sweden in December 2013. The original software was a personal finance dashboard, similar to Mint.com, to show its users how they spent their money. Qapital introduced the app into the US market with a different design in 2014 and started focusing exclusively on the US market. The app was re-designed to focus on building savings rather than managing personal finances. The Swedish version shut down in April 2015. The app was initially restricted to the iOS platform, but an Android version was released at the end of 2015. Shortly after its US launch, Qapital invited psychology professor Dan Ariely to join its team as its "chief behavioral economist". He uses the app's database to conduct research into behavioral economics and Qapital in turn uses Ariely's research in design and programming decisions. In 2017, Qapital added checking and debit card services to the app. == Concept and features == Qapital is a free personal finance app for iOS and Android devices, intended to encourage its users to save money. Qapital directs each of its users to set savings goals, then automatically transfers money from their checking account to an account for savings, when a rule established in the app is met. It uses the "if this then that" (IFTTT) rule-based web-service. For example, one rule could be that if a user purchases a cup of coffee, then the app will round up the charge to the nearest dollar and deposit the difference into savings. Users connect their bank accounts to Qapital, so it knows when purchases are made. When a rule is met, money for savings are transferred to a Qapital account operated in partnership with Lincoln Savings Bank. As of 2015, Qapital can connect to more than 180 other apps, such as Facebook, Twitter, Dropbox and Instagram. For example, connecting to Jawbone allows the user to set a rule that if they take a certain number of steps during the day, a set amount of money is transferred to savings. The app also allows users to monitor activity among their other financial accounts, such as deposits and withdrawals. == Reception == In an October 2015 review, PC Magazine gave Qapital four out of five marks and an editor rating of "excellent." The review praised the app for having a "lovely design" and criticized it for being a, "bit simplistic in some of its rules." Bankrate, in a May 2015 review, gave the app a score of 3/5 for "ease of use," 5/5 for "features," 4/5 for "effectiveness," 4/5 for "value," for a total score of 16/20. The reviewer criticized Qapital's savings account for providing a low-interest rate, but concluded that its numerous features make the app "intriguing" and "it would be difficult to find a standard bank app more fun to use than Qapital."
Cloud-based quantum computing
Cloud-based quantum computing refers to the remote access of quantum computing resources—such as quantum emulators, simulators, or processors—via the internet. Cloud access enables users to develop, test, and execute quantum algorithms without the need for direct interaction with specialized hardware, facilitating broader participation in quantum software development and experimentation. In 2016, IBM launched the IBM Quantum Experience, one of the first publicly accessible quantum processors connected to the cloud. In early 2017, researchers at Rigetti Computing demonstrated programmable quantum cloud access through their software platform Forest, which included the pyQuil Python library. Since the early-2020s, cloud-based quantum computing has grown significantly, with multiple providers offering access to a variety of quantum hardware modalities, including superconducting qubits, trapped ions, neutral atoms, and photonic systems. Major platforms such as Amazon Braket, Azure Quantum, and qBraid aggregate quantum devices from hardware developers like IonQ, Rigetti Computing, QuEra, Pasqal, Oxford Quantum Circuits, and IBM Quantum. These platforms provide unified interfaces for users to write and execute quantum algorithms across diverse backends, often supporting open-source SDKs such as Qiskit, Cirq, and PennyLane. The proliferation of cloud-based access has played a key role in accelerating quantum education, algorithm research, and early-stage application development by lowering the barrier to experimentation with real quantum hardware. Cloud-based quantum computing has expanded access to quantum hardware and tools beyond traditional research laboratories. These platforms support educational initiatives, algorithm development, and early-stage commercial applications. == Applications == Cloud-based quantum computing is used across education, research, and software development, offering remote access to quantum systems without the need for on-site infrastructure. === Education === Quantum cloud platforms have become valuable tools in education, allowing students and instructors to engage with real quantum processors through user-friendly interfaces. Educators use these platforms to teach foundational concepts in quantum mechanics and quantum computing, as well as to demonstrate and implement quantum algorithms in a classroom or laboratory setting. === Scientific Research === Cloud-based access to quantum hardware has enabled researchers to conduct experiments in quantum information, test quantum algorithms, and compare quantum hardware platforms. Experiments such as testing Bell's theorem or evaluating quantum teleportation protocols have been performed on publicly available quantum processors. === Software Development and Prototyping === Developers use cloud-based platforms to prototype quantum software applications across fields such as optimization, machine learning, and chemistry. These platforms offer SDKs and APIs that integrate classical and quantum workflows, enabling experimentation with quantum algorithms in real-world or simulated environments. === Public Engagement and Games === Quantum cloud tools have also been used to create educational games and interactive applications aimed at increasing public understanding of quantum concepts. These efforts help bridge the gap between theoretical content and intuitive learning. == Existing platforms == qBraid Lab by qBraid is a cloud-based platform for quantum computing. It provides software tools for researchers and developers in quantum, as well as access to quantum hardware. qBraid provides cloud based access to Microsoft Azure Quantum and Amazon Braket devices including IQM, QuEra, Pasqal, Rigetti, IonQ, QIR simulators, Amazon Braket simulators, and the NEC Vector Annealer, as of August 2025. qBraid's base version is free, where unlimited hardware and simulator access is available with the purchase of credits. Quandela Cloud by Quandela is the platform to access first cloud-accessible European photonic quantum computer. The computer is interfaced using the Perceval scripting language, with tutorials and documentation available online for free. Xanadu Quantum Cloud by Xanadu is a platform with cloud-based access to three fully programmable photonic quantum computers. Forest by Rigetti Computing is a tool suite for cloud-based quantum computing. It includes a programming language, development tools and example algorithms. LIQUi> by Microsoft is a software architecture and tool suite for quantum computing. It includes a programming language, example optimization and scheduling algorithms, and quantum simulators. Q#, a quantum programming language by Microsoft on the .NET Framework seen as a successor to LIQUi|>. IBM Quantum Platform by IBM, providing access to quantum hardware as well as HPC simulators. These can be accessed programmatically using the Python-based Qiskit framework, or via graphical interface with the IBM Q Experience GUI. Both are based on the OpenQASM standard for representing quantum operations. There is also a tutorial and online community. Quantum in the Cloud by The University of Bristol, which consists of a quantum simulator and a four qubit optical quantum system. Quantum Playground by Google is an educational resource which features a simulator with a simple interface, and a scripting language and 3D quantum state visualization. Quantum in the Cloud is an experimental quantum cloud platform for access to a four-qubit nuclear magnetic resonance-NMRCloudQ computer, managed by Tsinghua University. Quantum Inspire by Qutech is the first platform in Europe providing cloud-based quantum computing to two hardware chips. Next to a 5-qubit transmon processor, Quantum Inspire is the first platform in the world to provide online access to a fully programmable 2-qubit electron spin quantum processor. Amazon Braket is a cloud-based quantum computing platform hosted by AWS which, as of June 2025, provides access to quantum computers built by IonQ, Rigetti, IQM, and QuEra. Braket also provides a quantum algorithm development environment and simulator. Forge by QC Ware is a cloud-based quantum computing platform that provides access to D-Wave hardware, as well as Google and IBM simulators. The platform offers a 30-day free trial, including one minute of quantum computing time. Quantum-as-a-Service by Scaleway is a cloud-based platform created in 2022 to access to real quantum hardware from IQM Quantum Computers, Alpine Quantum Technologies, Quandela and Pasqal. It also include access to GPU-powered emulators such as Aer, Qsim and Quandela proprietary emulation.
Amália (LLM)
Amália is a Portuguese large language model (LLM) announced in November 2024 by the Portuguese Prime-Minister Luís Montenegro. Its final version is expected to be launched in 2026. It is being developed by Center for Responsible AI (Centro para a AI Responsável) and by the research centers of NOVA School of Science and Technology and Instituto Superior Técnico. == History == In 2024 it was announced that the Portuguese Agency for Administrative Modernization (Agência para a Modernização Administrativa) transpose this LLM to Portuguese Public Administration. According to Paulo Dimas (CEO of the Center for Responsible AI) the three fundamental points of this LLM project are the linguistic variant (European Portuguese), cultural representation and data protection. In April 2025 it was announced that Amália had entered beta phase with an improved version being expected to be launched in September 2025. The beta version released in September is available only to the Public Administration, but the website launched in October reiterates the final version will be an open model.
Sentence embedding
In natural language processing, a sentence embedding is a representation of a sentence as a vector of numbers which encodes meaningful semantic information. State of the art embeddings are based on the learned hidden layer representation of dedicated sentence transformer models. BERT pioneered an approach involving the use of a dedicated [CLS] token prepended to the beginning of each sentence inputted into the model; the final hidden state vector of this token encodes information about the sentence and can be fine-tuned for use in sentence classification tasks. In practice however, BERT's sentence embedding with the [CLS] token achieves poor performance, often worse than simply averaging non-contextual word embeddings. SBERT later achieved superior sentence embedding performance by fine tuning BERT's [CLS] token embeddings through the usage of a siamese neural network architecture on the SNLI dataset. Other approaches are loosely based on the idea of distributional semantics applied to sentences. Skip-Thought trains an encoder-decoder structure for the task of neighboring sentences predictions; this has been shown to achieve worse performance than approaches such as InferSent or SBERT. An alternative direction is to aggregate word embeddings, such as those returned by Word2vec, into sentence embeddings. The most straightforward approach is to simply compute the average of word vectors, known as continuous bag-of-words (CBOW). However, more elaborate solutions based on word vector quantization have also been proposed. One such approach is the vector of locally aggregated word embeddings (VLAWE), which demonstrated performance improvements in downstream text classification tasks. == Applications == In recent years, sentence embedding has seen a growing level of interest due to its applications in natural language queryable knowledge bases through the usage of vector indexing for semantic search. LangChain for instance utilizes sentence transformers for purposes of indexing documents. In particular, an indexing is generated by generating embeddings for chunks of documents and storing (document chunk, embedding) tuples. Then given a query in natural language, the embedding for the query can be generated. A top k similarity search algorithm is then used between the query embedding and the document chunk embeddings to retrieve the most relevant document chunks as context information for question answering tasks. This approach is also known formally as retrieval-augmented generation. Though not as predominant as BERTScore, sentence embeddings are commonly used for sentence similarity evaluation which sees common use for the task of optimizing a Large language model's generation parameters is often performed via comparing candidate sentences against reference sentences. By using the cosine-similarity of the sentence embeddings of candidate and reference sentences as the evaluation function, a grid-search algorithm can be utilized to automate hyperparameter optimization. == Evaluation == A way of testing sentence encodings is to apply them on Sentences Involving Compositional Knowledge (SICK) corpus for both entailment (SICK-E) and relatedness (SICK-R). In the best results are obtained using a BiLSTM network trained on the Stanford Natural Language Inference (SNLI) Corpus. The Pearson correlation coefficient for SICK-R is 0.885 and the result for SICK-E is 86.3. A slight improvement over previous scores is presented in: SICK-R: 0.888 and SICK-E: 87.8 using a concatenation of bidirectional Gated recurrent unit.
Dynamic Graphics Project
The Dynamic Graphics Project (commonly referred to as DGP) is an interdisciplinary research laboratory at the University of Toronto devoted to projects involving computer graphics, computer vision, human computer interaction, and visualization. The lab began as the computer graphics research group of Department of Computer Science Professor Leslie Mezei in 1967. Mezei invited Bill Buxton, a pioneer of human–computer interaction (HCI) to join. In 1972, Ronald Baecker, another HCI pioneer joined, establishing DGP as the first Canadian university group focused on computer graphics and human-computer interaction. According to csrankings.org, the DGP is the top research institution in the world for the combined subfields of computer graphics, HCI, and visualization. Since then, DGP has hosted many well known faculty and students in computer graphics, computer vision and HCI (e.g., Alain Fournier, Bill Reeves, Jos Stam, Demetri Terzopoulos, Marilyn Tremaine). DGP also occasionally hosts artists in residence (e.g., Oscar-winner Chris Landreth). Many past and current researchers at Autodesk (and before that Alias Wavefront) graduated after working at DGP. DGP is located in the St. George campus of University of Toronto in the Bahen Centre for Information Technology. DGP researchers regularly publish at ACM SIGGRAPH, ACM SIGCHI and ICCV. DGP hosts the Toronto User Experience (TUX) Speaker Series and the Sanders Series Lectures. == Notable alumni == Bill Buxton (MS 1978) James McCrae (PhD 2013) Dimitris Metaxas (PhD 1992) Bill Reeves (MS 1976, Ph.D. 1980) Jos Stam (MS 1991, Ph.D. 1995)
Super-resolution imaging
Super-resolution imaging (SR) is a class of techniques that improve the resolution of an imaging system. In optical SR the diffraction limit of systems is transcended, while in geometrical SR the resolution of digital imaging sensors is enhanced. In some radar and sonar imaging applications (e.g. magnetic resonance imaging (MRI), high-resolution computed tomography), subspace decomposition-based methods (e.g. MUSIC) and compressed sensing-based algorithms (e.g., SAMV) are employed to achieve SR over standard periodogram algorithm. Super-resolution imaging techniques are used in general image processing and in super-resolution microscopy. == Super-resolution principles == Several concepts are fundamental to super-resolution imaging: Diffraction limit: the capacity of an optical instrument to reproduce the details of an object in an image has limits that are imposed by laws of physics: the diffraction equations in the wave theory of light, or the uncertainty principle for photons in quantum mechanics. Information transfer can never be increased beyond this boundary, but packets outside the limits can be cleverly swapped for (or multiplexed with) some inside it. Super-resolution microscopy does not so much “break” as “circumvent” the diffraction limit. New procedures probing electro-magnetic disturbances at the molecular level (in the so-called near field) remain fully consistent with Maxwell's equations. Spatial frequency domain: A succinct expression of the diffraction limit is given in the spatial frequency domain. In Fourier optics light distributions are expressed as superpositions of a series of grating light patterns in a range of fringe widths - these widths represent the spatial frequencies. It is generally taught that diffraction theory stipulates an upper limit, the cut-off spatial-frequency, beyond which pattern elements fail to be transferred into the optical image, i.e., are not resolved. But in fact what is set by diffraction theory is the width of the passband, not a fixed upper limit. No laws of physics are broken when a spatial frequency band beyond the cut-off spatial frequency is swapped for one inside it: this has long been implemented in dark-field microscopy. Nor are information-theoretical rules broken when superimposing several bands, disentangling them in the received image needs assumptions of object invariance during multiple exposures, i.e., the substitution of one kind of uncertainty for another. Information: When the term super-resolution is used in techniques based on the inference of object details using a statistical treatment of the image within standard resolution limits (for example, averaging multiple exposures), it involves an exchange of one kind of information (extracting signal from noise) for another (the assumption that the target has remained invariant). Recent breakthroughs incorporate quantum-transformer hybrids into super-resolution, such as QUIET‑SR, a 2025 model that employs shifted quantum window attention within a transformer to enhance image detail while respecting diffraction and information-theory limits Similarly, frequency-integrated transformers (e.g., FIT) enrich super-resolution by explicitly combining spatial and frequency-domain information via FFT-based attention, improving reconstruction across scales Resolution and localization: True resolution involves the distinction of whether a target, e.g. a star or a spectral line, is single or double, ordinarily requiring separable peaks in the image. When a target is known to be single, its location can be determined with higher precision than the image width by finding the centroid (center of gravity) of its image light distribution. The word ultra-resolution had been proposed for this process but it did not catch on, and the high-precision localization procedure is typically referred to as super-resolution. == Techniques == === Optical or diffractive super-resolution === Substituting spatial-frequency bands: Though the bandwidth allowable by diffraction is fixed, it can be positioned anywhere in the spatial-frequency spectrum. Dark-field illumination in microscopy is an example. See also aperture synthesis. ==== Multiplexing spatial-frequency bands ==== An image is formed using the normal passband of the optical device. Then, some known light structure (for example, a set of light fringes) is superimposed on the target. The image now contains components resulting from the combination of the target and the superimposed light structure, e.g. moiré fringes, and carries information about target detail which simple unstructured illumination does not. The “superresolved” components, however, need disentangling to be revealed. For an example, see structured illumination (figure to left). ==== Multiple parameter use within traditional diffraction limit ==== If a target has no special polarization or wavelength properties, two polarization states or non-overlapping wavelength regions can be used to encode target details, one in a spatial-frequency band inside the cut-off limit the other beyond it. Both would use normal passband transmission but are then separately decoded to reconstitute target structure with extended resolution. ==== Probing near-field electromagnetic disturbance ==== Super-resolution microscopy is generally discussed within the realm of conventional optical imagery. However, modern technology allows the probing of electromagnetic disturbance within molecular distances of the source, which has superior resolution properties. See also evanescent waves and the development of the new super lens. === Geometrical or image-processing super-resolution === ==== Multi-exposure image noise reduction ==== When an image is degraded by noise, the resolution may be improved by averaging multiple exposures. See example on the right. ==== Single-frame deblurring ==== Known defects in a given imaging situation, such as defocus or aberrations, can sometimes be mitigated in whole or in part by suitable spatial-frequency filtering of even a single image. Such procedures all stay within the diffraction-mandated passband, and do not extend it. ==== Sub-pixel image localization ==== The location of a single source can be determined by computing the "center of gravity" (centroid) of the light distribution extending over several adjacent pixels (see figure on the left). Provided that there is enough light, this can be achieved with arbitrary precision, very much better than pixel width of the detecting apparatus and the resolution limit for the decision of whether the source is single or double. This technique, which requires the presupposition that all the light comes from a single source, is at the basis of what has become known as super-resolution microscopy, e.g. stochastic optical reconstruction microscopy (STORM), where fluorescent probes attached to molecules give nanoscale distance information. It is also the mechanism underlying visual hyperacuity. ==== Bayesian induction beyond traditional diffraction limit ==== Some object features, though beyond the diffraction limit, may be known to be associated with other object features that are within the limits and hence contained in the image. Then conclusions can be drawn, using statistical methods, from the available image data about the presence of the full object. The classical example is Toraldo di Francia's proposition of judging whether an image is that of a single or double star by determining whether its width exceeds the spread from a single star. This can be achieved at separations well below the classical resolution bounds, and requires the prior limitation to the choice "single or double?" The approach can take the form of extrapolating the image in the frequency domain, by assuming that the object is an analytic function, and that we can exactly know the function values in some interval. This method is severely limited by the ever-present noise in digital imaging systems, but it can work for radar, astronomy, microscopy or magnetic resonance imaging. More recently, a fast single image super-resolution algorithm based on a closed-form solution to ℓ 2 − ℓ 2 {\displaystyle \ell _{2}-\ell _{2}} problems has been proposed and demonstrated to accelerate most of the existing Bayesian super-resolution methods significantly. == Aliasing == Geometrical SR reconstruction algorithms are possible if and only if the input low resolution images have been under-sampled and therefore contain aliasing. Because of this aliasing, the high-frequency content of the desired reconstruction image is embedded in the low-frequency content of each of the observed images. Given a sufficient number of observation images, and if the set of observations vary in their phase (i.e. if the images of the scene are shifted by a sub-pixel amount), then the phase information can be used to separate the aliased high-frequency content from the true low-frequency content, and the full-resolution image can be accurate
Superquadrics
In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the superellipses. The term may refer to the solid object or to its surface, depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs. The superquadrics include many shapes that resemble cubes, octahedra, cylinders, lozenges and spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation. Some authors, such as Alan Barr, define "superquadrics" as including both the superellipsoids and the supertoroids. In modern computer vision literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics. Comprehensive coverage of geometrical properties of superquadrics and methods of their recovery from range images and point clouds are covered in several computer vision literatures. == Formulas == === Implicit equation === The surface of the basic superquadric is given by | x | r + | y | s + | z | t = 1 {\displaystyle \left|x\right|^{r}+\left|y\right|^{s}+\left|z\right|^{t}=1} where r, s, and t are positive real numbers that determine the main features of the superquadric. Namely: less than 1: a pointy octahedron modified to have concave faces and sharp edges. exactly 1: a regular octahedron. between 1 and 2: an octahedron modified to have convex faces, blunt edges and blunt corners. exactly 2: a sphere greater than 2: a cube modified to have rounded edges and corners. infinite (in the limit): a cube Each exponent can be varied independently to obtain combined shapes. For example, if r=s=2, and t=4, one obtains a solid of revolution which resembles an ellipsoid with round cross-section but flattened ends. This formula is a special case of the superellipsoid's formula if (and only if) r = s. If any exponent is allowed to be negative, the shape extends to infinity. Such shapes are sometimes called super-hyperboloids. The basic shape above spans from -1 to +1 along each coordinate axis. The general superquadric is the result of scaling this basic shape by different amounts A, B, C along each axis. Its general equation is | x A | r + | y B | s + | z C | t = 1. {\displaystyle \left|{\frac {x}{A}}\right|^{r}+\left|{\frac {y}{B}}\right|^{s}+\left|{\frac {z}{C}}\right|^{t}=1.} === Parametric description === Parametric equations in terms of surface parameters u and v (equivalent to longitude and latitude if m equals 2) are x ( u , v ) = A g ( v , 2 r ) g ( u , 2 r ) y ( u , v ) = B g ( v , 2 s ) f ( u , 2 s ) z ( u , v ) = C f ( v , 2 t ) − π 2 ≤ v ≤ π 2 , − π ≤ u < π , {\displaystyle {\begin{aligned}x(u,v)&{}=Ag\left(v,{\frac {2}{r}}\right)g\left(u,{\frac {2}{r}}\right)\\y(u,v)&{}=Bg\left(v,{\frac {2}{s}}\right)f\left(u,{\frac {2}{s}}\right)\\z(u,v)&{}=Cf\left(v,{\frac {2}{t}}\right)\\&-{\frac {\pi }{2}}\leq v\leq {\frac {\pi }{2}},\quad -\pi \leq u<\pi ,\end{aligned}}} where the auxiliary functions are f ( ω , m ) = sgn ( sin ω ) | sin ω | m g ( ω , m ) = sgn ( cos ω ) | cos ω | m {\displaystyle {\begin{aligned}f(\omega ,m)&{}=\operatorname {sgn}(\sin \omega )\left|\sin \omega \right|^{m}\\g(\omega ,m)&{}=\operatorname {sgn}(\cos \omega )\left|\cos \omega \right|^{m}\end{aligned}}} and the sign function sgn(x) is sgn ( x ) = { − 1 , x < 0 0 , x = 0 + 1 , x > 0. {\displaystyle \operatorname {sgn}(x)={\begin{cases}-1,&x<0\\0,&x=0\\+1,&x>0.\end{cases}}} === Spherical product === Barr introduces the spherical product which given two plane curves produces a 3D surface. If f ( μ ) = ( f 1 ( μ ) f 2 ( μ ) ) , g ( ν ) = ( g 1 ( ν ) g 2 ( ν ) ) {\displaystyle f(\mu )={\begin{pmatrix}f_{1}(\mu )\\f_{2}(\mu )\end{pmatrix}},\quad g(\nu )={\begin{pmatrix}g_{1}(\nu )\\g_{2}(\nu )\end{pmatrix}}} are two plane curves then the spherical product is h ( μ , ν ) = f ( μ ) ⊗ g ( ν ) = ( f 1 ( μ ) g 1 ( ν ) f 1 ( μ ) g 2 ( ν ) f 2 ( μ ) ) {\displaystyle h(\mu ,\nu )=f(\mu )\otimes g(\nu )={\begin{pmatrix}f_{1}(\mu )\ g_{1}(\nu )\\f_{1}(\mu )\ g_{2}(\nu )\\f_{2}(\mu )\end{pmatrix}}} This is similar to the typical parametric equation of a sphere: x = x 0 + r sin θ cos φ y = y 0 + r sin θ sin φ ( 0 ≤ θ ≤ π , 0 ≤ φ < 2 π ) z = z 0 + r cos θ {\displaystyle {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \qquad (0\leq \theta \leq \pi ,\;0\leq \varphi <2\pi )\\z&=z_{0}+r\cos \theta \end{aligned}}} which give rise to the name spherical product. Barr uses the spherical product to define quadric surfaces, like ellipsoids, and hyperboloids as well as the torus, superellipsoid, superquadric hyperboloids of one and two sheets, and supertoroids. == Plotting code == The following GNU Octave code generates a mesh approximation of a superquadric: