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Show HN: Rust Vector and Quaternion Lib

Posted: 2025-03-06 20:32:47

This project introduces lin-alg, a Rust library providing fundamental linear algebra structures and operations with a focus on performance. It offers core types like vectors and quaternions (with 2D, 3D, and 4D variants), alongside common operations such as addition, subtraction, scalar multiplication, dot and cross products, normalization, and quaternion-specific functionalities like rotations and spherical linear interpolation (slerp). The library aims to be simple, efficient, and dependency-free, suitable for graphics, game development, and other domains requiring linear algebra computations.

Summary of Comments ( 8 )
https://news.ycombinator.com/item?id=43284811

Hacker News users generally praised the Rust vector and quaternion library for its clear documentation, beginner-friendly approach, and focus on 2D and 3D graphics. Some questioned the practical application of quaternions in 2D, while others appreciated the inclusion for completeness and potential future use. The discussion touched on SIMD support (or lack thereof), with some users highlighting its importance for performance in graphical applications. There were also suggestions for additional features like dual quaternions and geometric algebra support, reflecting a desire for expanded functionality. Some compared the library favorably to existing solutions like glam and nalgebra, praising its simplicity and ease of understanding, particularly for learning purposes.

The Hacker News post titled "Show HN: Rust Vector and Quaternion Lib" (https://news.ycombinator.com/item?id=43284811) has a modest number of comments, focusing primarily on practical aspects and alternatives rather than deep dives into the library's design.

One commenter points out the existence of nalgebra, a popular linear algebra library in Rust, and questions the rationale behind creating a new library instead of contributing to existing ones. They acknowledge the educational value of such projects but emphasize the benefits of a shared, well-maintained library for the broader community. This comment sparks a brief discussion about the potential reasons for starting a new library, including personal learning experiences, specific needs not met by existing libraries, and the inherent desire to build things from scratch.

Another comment highlights the use of f32 (32-bit floating-point numbers) and suggests considering f64 (64-bit floating-point numbers) for improved precision in certain applications, particularly those involving physics simulations. This leads to a short exchange about the trade-offs between performance and precision, with the original poster (OP) acknowledging the suggestion and explaining their choice of f32 as sufficient for their immediate needs.

A third comment mentions the inclusion of quaternions, asking about the intended use cases. The OP responds, indicating their interest in 3D graphics and game development, where quaternions are commonly used for rotations.

Further comments briefly touch on related topics such as the learning resources used by the OP, the possibility of SIMD optimizations, and comparisons to other linear algebra libraries in different languages. There is a general consensus that while the project might not be groundbreaking in terms of features or performance, it serves as a valuable learning experience and a potential starting point for a more comprehensive library.

Overall, the comments are constructive and offer practical suggestions, focusing on the context of the library within the existing Rust ecosystem and its potential applications. They don't delve deeply into the specific implementation details or offer extensive code reviews but provide useful feedback from the perspective of experienced Rust developers.

Rotors: A practical introduction for 3D graphics (2023)

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Posted: 2025-03-02 20:10:55

This post introduces rotors as a practical alternative to quaternions and matrices for 3D rotations. It explains that rotors, like quaternions, represent rotations as a single action around an arbitrary axis, but offer a simpler, more intuitive geometric interpretation based on the concept of "geometric algebra." The author argues that rotors are easier to understand and implement, visually demonstrating their geometric meaning and providing clear code examples in Python. The post covers basic rotor operations like creating rotations from an axis and angle, composing rotations, and applying rotations to vectors, highlighting rotors' computational efficiency and stability.

Jacques Heunis's blog post, "Rotors: A Practical Introduction for 3D Graphics (2023)," provides a comprehensive yet accessible exploration of rotors as a powerful alternative to other rotation representations like Euler angles, quaternions, and rotation matrices. The post begins by establishing the motivation for using rotors, highlighting the shortcomings of traditional methods, such as gimbal lock with Euler angles and the potential for ambiguity with quaternions (due to their double-covering nature). It emphasizes that rotors, based on the geometric algebra of 3D space, offer a more intuitive and mathematically elegant approach.

Heunis meticulously constructs the concept of rotors from the ground up, starting with the geometric product, a fundamental operation in geometric algebra. He explains how the geometric product combines the dot product and the wedge product, leading to a unified representation of both scalar and bivector quantities. Bivectors, representing oriented planar subspaces, are then shown to be the key to understanding rotations. The post explicitly details how the geometric product of two vectors produces a scalar and a bivector, illustrating this with clear examples.

The core of the post explains how rotors, which are normalized exponentials of bivectors, perform rotations. It meticulously derives the rotor formula and demonstrates how applying a rotor to a vector effectively rotates that vector within the plane defined by the bivector. The post clarifies that the exponential of a bivector results in a rotor, and this rotor acts as a rotation operator. The connection between rotors and quaternions is also addressed, demonstrating how a rotor can be converted to a quaternion and vice-versa, offering a deeper understanding of the relationship between these two representations. This includes a clear mapping of the bivector components to quaternion components.

Furthermore, the post delves into the practical advantages of rotors. It discusses how rotor composition, achieved through rotor multiplication, provides a simple and efficient way to combine multiple rotations. This contrasts with the more complex operations required when using rotation matrices or quaternions. The post also highlights the efficiency of interpolating between rotors, showcasing how smoothly and intuitively this can be accomplished compared to other rotation representations. Specific examples are given, demonstrating the calculations involved in interpolating between two rotors.

Finally, the post concludes by summarizing the key benefits of using rotors in 3D graphics programming, reinforcing their intuitive geometric interpretation, efficient composition, and smooth interpolation properties. It positions rotors as a powerful and practical tool for anyone working with rotations in 3D space, offering a compelling alternative to more traditional methods. Throughout the post, clear diagrams and code snippets are included to further clarify the concepts and facilitate practical implementation.

Summary of Comments ( 7 )
https://news.ycombinator.com/item?id=43234510

Hacker News users discussed the practicality and intuitiveness of using rotors for 3D rotations. Some found the rotor approach more elegant and easier to grasp than quaternions, especially appreciating the clear geometric interpretation and connection to bivectors. Others questioned the claimed advantages, arguing that quaternions remain the superior choice for performance and established library support. The potential benefits of rotors in areas like interpolation and avoiding gimbal lock were acknowledged, but some commenters felt the article didn't fully demonstrate these advantages convincingly. A few requested more comparative benchmarks or examples showcasing rotors' practical superiority in specific scenarios. The lack of widespread adoption and existing tooling for rotors was also raised as a barrier to entry.

The Hacker News post titled "Rotors: A practical introduction for 3D graphics (2023)" has generated a moderate discussion with several interesting comments. Many commenters praise the article for its clarity and insightful approach to explaining rotors.

One commenter appreciates the visual explanation of rotor interpolation, stating that it finally made the concept click for them. They highlight how the article demonstrates how rotors avoid gimbal lock, a common problem with other rotation representations like Euler angles. This comment emphasizes the practical value of the article for those struggling with 3D rotation concepts.

Another commenter points out the connection between rotors and quaternions, explaining that rotors are essentially a different way of looking at quaternions, specifically using a geometric algebra perspective. They delve a bit into the mathematical background, mentioning how rotors represent rotations as oriented arcs of great circles on a 3-sphere. This adds a layer of theoretical depth to the discussion, connecting the article's content to broader mathematical principles.

Further discussion revolves around the practical applications of rotors. One commenter mentions their use in game development, specifically for character animation and camera control. This highlights the real-world relevance of the topic and the potential benefits of using rotors in practical 3D graphics applications.

Another commenter expresses a preference for rotors over quaternions, arguing that they are easier to understand intuitively and visualize. They appreciate the geometric interpretation of rotations provided by rotors. This comment contributes to a small debate about the relative merits of rotors versus quaternions.

Finally, some commenters mention other resources for learning about rotors and geometric algebra, expanding the scope of the discussion and providing further avenues for exploration. They provide links and suggest books, giving interested readers more opportunities to deepen their understanding.

Overall, the comments section reflects a positive reception of the article, praising its clarity and practical approach to explaining rotors. The discussion touches upon the theoretical underpinnings of rotors, their practical applications, and their relationship to other rotation representations.

Rediscovering Quaternions

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Posted: 2025-02-26 17:17:11

This post explores the complexities of representing 3D rotations, contrasting quaternions with other methods like rotation matrices and Euler angles. It highlights the issues of gimbal lock and interpolation difficulties inherent in Euler angles, and the computational cost of rotation matrices. Quaternions, while less intuitive, offer a more elegant and efficient solution. The post breaks down the math behind quaternions, explaining how they represent rotations as points on a 4D hypersphere, and demonstrates their advantages for smooth interpolation and avoiding gimbal lock. It emphasizes the practical benefits of quaternions in computer graphics and other applications requiring 3D manipulation.

Jason Fantl's blog post, "Rediscovering Quaternions," delves into the fascinating world of representing and manipulating 3D rotations, ultimately championing quaternions as a superior method compared to other approaches. The post begins by acknowledging the common challenges associated with rotation matrices, particularly issues like gimbal lock, where degrees of freedom are lost when two rotational axes align. It highlights the computational expense of matrix multiplication and the difficulty in interpolating smoothly between rotations represented by matrices.

The post then introduces Euler angles, a more intuitive representation using three separate rotations around the principal axes (typically yaw, pitch, and roll). While easier to conceptualize, Euler angles also suffer from gimbal lock and ambiguity in the order of rotations. Fantl illustrates these problems vividly, showcasing how different rotation sequences can lead to the same final orientation, making interpolation unpredictable and potentially leading to undesirable twisting motions.

Having established the limitations of existing methods, the post introduces quaternions. It carefully explains their mathematical structure as 4-dimensional complex numbers with specific properties. Fantl methodically builds up the concept, starting with the idea of rotating a point in 2D using complex numbers, and extending this analogy to 3D using quaternions. He defines a pure quaternion to represent a point in 3D space and then meticulously demonstrates how to perform a rotation around an arbitrary axis using quaternion multiplication. The formula for this rotation operation, involving sandwiching the pure quaternion between a unit quaternion and its conjugate, is derived and thoroughly explained.

Furthermore, the post emphasizes the advantages of quaternions. It shows how they avoid gimbal lock completely, offering a smooth and continuous representation of all possible rotations. Quaternion interpolation, using spherical linear interpolation (slerp), is presented as a computationally efficient and elegant solution for generating seamless transitions between orientations. The compact nature of quaternions, requiring only four components compared to nine for rotation matrices, also contributes to their computational efficiency, particularly in applications involving many rotations or limited memory resources.

Finally, the post briefly touches upon how quaternions are related to axis-angle representation, another method for describing rotations. It demonstrates the mathematical connection between these two representations, highlighting how quaternions can be easily converted to and from axis-angle form. In conclusion, Fantl strongly advocates for the adoption of quaternions in any application involving 3D rotations, emphasizing their robustness, efficiency, and elegance in handling the complexities of spatial orientation.

Summary of Comments ( 16 )
https://news.ycombinator.com/item?id=43185733

HN users generally praised the article for its clear explanation of quaternions and their application to 3D rotations. Several commenters appreciated the visual approach and interactive demos, finding them helpful for understanding the concepts. Some discussed alternative representations like rotation matrices and axis-angle, comparing their strengths and weaknesses to quaternions. A few users pointed out the connection to complex numbers and offered additional resources for further exploration. One commenter mentioned the practical uses of quaternions in game development and other fields. Overall, the discussion highlighted the importance of quaternions as a tool for representing and manipulating rotations in 3D space.

The Hacker News post "Rediscovering Quaternions" linking to an article about 3D rotations has a moderate number of comments, sparking a discussion around the topic.

Several commenters discuss their personal experiences and preferences regarding different methods for representing and working with 3D rotations. Some express a preference for quaternions, highlighting their efficiency and robustness against gimbal lock, a common issue with Euler angles. One commenter mentions using quaternions extensively in game development, praising their smooth interpolation capabilities. Another points out the elegance of quaternions from a mathematical perspective, appreciating their connection to complex numbers and their representation as a 4D hypersphere.

Conversely, some commenters argue for alternative approaches like rotation matrices, citing their intuitive nature and direct applicability in linear algebra calculations. One commenter suggests that rotation matrices, while potentially less efficient, offer a clearer understanding of the underlying transformations. Another mentions using a combination of methods depending on the specific application, highlighting the trade-offs between different representations.

A few comments delve into the mathematical details of quaternions, discussing their properties and operations. One commenter explains the concept of quaternion multiplication and its geometric interpretation as rotations in 3D space. Another discusses the relationship between quaternions and axis-angle representation, offering a different perspective on how rotations can be parameterized.

The discussion also touches upon the learning curve associated with quaternions. Some commenters acknowledge the initial difficulty in grasping the concept, but emphasize the long-term benefits once mastered. Others suggest resources for learning about quaternions, including textbooks, online tutorials, and interactive visualizations. One comment even points out the historical context of quaternions, mentioning their discovery by William Rowan Hamilton and their initial resistance from the mathematical community.

Finally, a couple of comments offer practical advice for working with quaternions in software development. One commenter recommends using existing libraries and frameworks that provide optimized quaternion implementations, while another suggests careful consideration of numerical precision and potential issues with floating-point arithmetic.

The Tensor Cookbook (2024)

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Posted: 2025-01-31 18:47:51

The Tensor Cookbook (2024) is a free online resource offering a practical, code-focused guide to tensor operations. It covers fundamental concepts like tensor creation, manipulation (reshaping, slicing, broadcasting), and common operations (addition, multiplication, contraction) using NumPy, TensorFlow, and PyTorch. The cookbook emphasizes clear explanations and executable code examples to help readers quickly grasp and apply tensor techniques in various contexts. It aims to serve as a quick reference for both beginners seeking a foundational understanding and experienced practitioners looking for concise reminders on specific operations across popular libraries.

The Tensor Cookbook (2024) presents itself as a comprehensive and practical guide to understanding and utilizing tensors, the fundamental mathematical objects underpinning many areas of science and engineering, particularly machine learning and deep learning. The website emphasizes the cookbook's focus on providing clear, concise explanations and executable code examples to facilitate a hands-on learning experience. It aims to bridge the gap between theoretical understanding and practical application, catering to a broad audience, from students just beginning their journey with tensors to seasoned practitioners seeking a quick reference.

The cookbook covers a wide spectrum of tensor operations, starting with foundational concepts such as defining tensors, tensor shapes and dimensions, and basic manipulations like reshaping and transposition. It progresses to more advanced topics including tensor contraction, broadcasting, and the application of various linear algebra operations within the tensor context. The coverage extends to essential techniques for tensor decomposition, including Singular Value Decomposition (SVD) and Principal Component Analysis (PCA), elucidating their significance in dimensionality reduction and feature extraction.

The authors emphasize the practical applicability of tensors within the realm of machine learning, specifically addressing automatic differentiation, a crucial technique for training neural networks. The cookbook provides insights into how tensors are used to represent and manipulate data within machine learning models and how automatic differentiation facilitates the calculation of gradients necessary for optimization algorithms.

Importantly, the cookbook isn't purely theoretical. It integrates practical coding examples using popular Python libraries like NumPy, TensorFlow, and PyTorch, enabling readers to experiment with the concepts directly. This practical approach reinforces learning and allows readers to translate theoretical understanding into working code, furthering their proficiency with tensor manipulation within these widely-used frameworks. The website suggests that the code examples are designed to be readily adaptable and reusable, serving as building blocks for more complex tensor operations and machine learning applications. Finally, the cookbook aims to be a dynamic resource, with plans for continuous updates and expansions to encompass emerging trends and techniques in the field of tensor computation.

Summary of Comments ( 19 )
https://news.ycombinator.com/item?id=42890389

Hacker News users generally praised the Tensor Cookbook for its clear explanations and practical examples, finding it a valuable resource for those learning tensor operations. Several commenters appreciated the focus on intuitive understanding rather than rigorous mathematical proofs, making it accessible to a wider audience. Some pointed out the cookbook's relevance to machine learning and its potential as a quick reference for common tensor manipulations. A few users suggested additional topics or improvements, such as including content on tensor decompositions or expanding the coverage of specific libraries like PyTorch and TensorFlow. One commenter highlighted the site's use of MathJax for rendering equations, appreciating the resulting clear and readable formulas. There's also discussion around the subtle differences in tensor terminology across various fields and the cookbook's attempt to address these nuances.

The Hacker News post for "The Tensor Cookbook (2024)" has generated a modest number of comments, primarily focused on the utility and scope of the resource.

One commenter appreciates the cookbook's focus on providing practical, runnable code examples for common tensor operations, contrasting it with more theoretical or abstract resources. They specifically highlight the value of having readily available code snippets for tasks like calculating Jacobians and Hessians, which can be cumbersome to derive and implement from scratch. This commenter views the cookbook as a helpful quick reference for those needing to perform these operations without delving into the underlying mathematical complexities.

Another commenter expresses a desire for the cookbook to expand beyond NumPy and cover other popular tensor libraries like PyTorch and TensorFlow. They acknowledge the value of a NumPy-focused resource but believe that including examples using these widely used deep learning frameworks would significantly broaden the cookbook's appeal and usefulness. This sentiment suggests a demand for practical, code-focused resources that bridge the gap between foundational tensor operations and their implementation within specific deep learning ecosystems.

One commenter questions the necessity of yet another tensor resource, pointing to the abundance of existing tutorials and documentation. They imply that the cookbook might not offer substantial new insights or perspectives compared to readily available materials. This viewpoint raises a valid concern about the potential redundancy of the resource within the already saturated landscape of tensor-related educational content.

A different commenter concurs with the call for PyTorch/TensorFlow examples. They specifically mention automatic differentiation as a crucial feature of these frameworks, hinting at the potential benefits of leveraging these capabilities within the cookbook. They further suggest incorporating examples demonstrating the computation of higher-order derivatives using these frameworks. This comment reinforces the demand for a more comprehensive resource that addresses the practical implementation of tensor operations within established deep learning environments.

Finally, a commenter expresses appreciation for the cookbook, emphasizing its concise and easy-to-understand nature. They highlight its focus on core tensor concepts, which they believe are sometimes overlooked or obscured by overly complex explanations in other resources. This comment suggests that the cookbook's simplicity and focus on fundamental concepts are valued by some users who seek a clear and straightforward introduction to tensor operations.

In summary, the comments generally appreciate the practical, code-focused approach of the cookbook but suggest expanding its scope to include other tensor libraries and functionalities relevant to deep learning practitioners. There's also some skepticism about its unique value proposition given existing resources.

The Graphics Codex

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Posted: 2025-01-26 05:36:33

The Graphics Codex is a comprehensive, free online resource for learning about computer graphics. It covers a broad range of topics, from fundamental concepts like color and light to advanced rendering techniques like ray tracing and path tracing. Emphasizing a practical, math-heavy approach, the Codex provides detailed explanations, interactive diagrams, and code examples to facilitate a deep understanding of the underlying principles. It's designed to be accessible to students and professionals alike, offering a structured learning path from beginner to expert levels. The resource continues to evolve and expand, aiming to become a definitive and up-to-date guide to the field of computer graphics.

The Graphics Codex, an ambitious online compendium of computer graphics knowledge, presents itself as a comprehensive and evolving resource for both novice learners and seasoned professionals in the field. It distinguishes itself from typical textbooks by adopting a practical, "bottom-up" approach, emphasizing implementation details and demonstrable code examples over abstract theoretical discussions. This approach is facilitated by its integrated coding environment, allowing readers to directly interact with and modify the presented algorithms and visualize the results in real-time. This interactivity aims to bridge the gap between theoretical understanding and practical application, a common hurdle in grasping complex graphical concepts.

The Codex covers a wide spectrum of graphics-related topics, ranging from fundamental concepts like color theory, vector mathematics, and transformations, to more advanced rendering techniques such as ray tracing, path tracing, and rasterization. It delves into the intricacies of different shading models, explaining how light interacts with surfaces to create realistic visual effects. Furthermore, it explores the underlying principles of animation and simulation, providing insights into the creation of dynamic and interactive graphical experiences. The content is structured in a modular fashion, allowing readers to navigate and focus on specific areas of interest, or follow a suggested learning path for a more structured learning experience.

A key feature of the Graphics Codex is its emphasis on mathematical rigor and clarity. While prioritizing practical implementation, it doesn't shy away from the underlying mathematical foundations, providing detailed explanations and derivations where necessary. This ensures that readers develop a robust understanding of the theoretical principles governing the presented algorithms and techniques. The code examples provided are meticulously crafted and well-documented, serving as both illustrative tools and practical templates for real-world applications.

Furthermore, The Graphics Codex is committed to staying current with the rapidly evolving landscape of computer graphics. Its online nature allows for continuous updates and additions, incorporating the latest research and advancements in the field. This ensures that the resource remains relevant and valuable for practitioners seeking to stay at the forefront of graphics technology. It aims to be a living document, constantly expanding and refining its content to reflect the latest innovations and best practices in computer graphics.

Summary of Comments ( 9 )
https://news.ycombinator.com/item?id=42827976

Hacker News users largely praised the Graphics Codex, calling it a "fantastic resource" and a "great intro to graphics". Many appreciated its practical, hands-on approach and clear explanations of fundamental concepts, contrasting it favorably with overly theoretical or outdated textbooks. Several commenters highlighted the value of its accompanying code examples and the author's focus on modern graphics techniques. Some discussion revolved around the choice of GLSL over other shading languages, with some preferring a more platform-agnostic approach, but acknowledging the educational benefits of GLSL's explicit nature. The overall sentiment was highly positive, with many expressing excitement about using the resource themselves or recommending it to others.

The Hacker News post titled "The Graphics Codex" spawned a lively discussion with a variety of comments focusing on the book's content, target audience, pricing model, and comparisons to other resources.

Several commenters praised the book's comprehensive coverage of graphics programming concepts and its focus on foundational knowledge. One user highlighted its valuable insights into topics often overlooked in other resources, particularly its treatment of textures and sampling. Another appreciated the author's deep dive into the underlying mathematics and theory, contrasting it with more superficial treatments found elsewhere. The book's rigorous approach and attention to detail were seen as strengths, offering a valuable resource for those seeking a deeper understanding of graphics principles.

The pricing model, offering both a free online version and a paid print/PDF version, was a point of discussion. Commenters generally supported this approach, viewing it as a fair way to make the content accessible while also allowing the author to be compensated for their work. The ability to access the complete content online for free was praised as being beneficial to students and others with limited resources.

Several comparisons were made to other graphics programming resources. Some commenters drew parallels to "Real-Time Rendering," another highly regarded text, noting that "The Graphics Codex" might serve as a valuable companion or alternative. The perceived target audience for each book was also discussed, with some suggesting that "The Graphics Codex" might be better suited for beginners or those seeking a more foundational understanding.

Some commenters also delved into specific technical aspects discussed in the book, such as the explanation of mipmapping and the treatment of different rendering techniques. These discussions highlighted the depth of the book's content and its potential to spark deeper exploration of specific topics.

A few criticisms were also raised. One commenter expressed concern about the potential for the content to become outdated due to the rapidly evolving nature of graphics programming. Another mentioned some perceived limitations in the book's coverage of certain topics.

Overall, the comments on Hacker News reflect a generally positive reception to "The Graphics Codex," praising its comprehensive content, accessible pricing model, and rigorous approach to graphics programming concepts. While some minor concerns were raised, the overall sentiment suggests that the book is a valuable contribution to the field.

Stories with Tag linear algebra

Show HN: Rust Vector and Quaternion Lib

Summary of Comments ( 8 ) https://news.ycombinator.com/item?id=43284811

Rotors: A practical introduction for 3D graphics (2023)

Summary of Comments ( 7 ) https://news.ycombinator.com/item?id=43234510

Rediscovering Quaternions

Summary of Comments ( 16 ) https://news.ycombinator.com/item?id=43185733

The Tensor Cookbook (2024)

Summary of Comments ( 19 ) https://news.ycombinator.com/item?id=42890389

The Graphics Codex

Summary of Comments ( 9 ) https://news.ycombinator.com/item?id=42827976

Summary of Comments ( 8 )
https://news.ycombinator.com/item?id=43284811

Summary of Comments ( 7 )
https://news.ycombinator.com/item?id=43234510

Summary of Comments ( 16 )
https://news.ycombinator.com/item?id=43185733

Summary of Comments ( 19 )
https://news.ycombinator.com/item?id=42890389

Summary of Comments ( 9 )
https://news.ycombinator.com/item?id=42827976