This post explores the connection between quaternions and spherical trigonometry. It demonstrates how quaternion multiplication elegantly encodes rotations in 3D space, and how this can be used to derive fundamental spherical trigonometric identities like the spherical law of cosines and the spherical law of sines. Specifically, by representing vertices of a spherical triangle as unit quaternions and using quaternion multiplication to describe the rotations between them, the post reveals a direct algebraic correspondence with the trigonometric relationships between the triangle's sides and angles. This approach offers a cleaner and more intuitive understanding of spherical trigonometry compared to traditional methods.
Terence Tao's blog post, "Quaternions and spherical trigonometry," explores the elegant connection between quaternions and the formulas of spherical trigonometry, offering a fresh perspective on these classical relationships. The post begins by establishing the context of spherical trigonometry, which deals with triangles drawn on the surface of a unit sphere. These triangles, formed by arcs of great circles, have sides measured by the angle they subtend at the sphere's center, and their angles are the dihedral angles between the planes containing the great circle arcs. Tao emphasizes that spherical trigonometry, unlike its planar counterpart, exhibits peculiar properties due to the curvature of the sphere, such as the sum of angles in a spherical triangle exceeding π radians (180 degrees).
The core of the post delves into how quaternions, a four-dimensional number system extending complex numbers, can be utilized to derive fundamental formulas of spherical trigonometry in a streamlined and conceptually satisfying manner. Quaternions are introduced, highlighting their structure as combinations of a scalar and a three-dimensional vector. The crucial concept of the quaternion exponential is then explained, demonstrating how it maps to rotations in three-dimensional space. Specifically, exponentiating a pure imaginary quaternion (one with a zero scalar part) yields a rotation around the axis defined by the vector part, with the angle of rotation determined by the magnitude of the vector.
This connection between quaternion exponentials and rotations forms the bridge to spherical trigonometry. Tao meticulously constructs a spherical triangle using carefully chosen rotations represented by quaternions. By examining the composition of these rotations and leveraging the properties of quaternion multiplication, he derives the spherical law of cosines, a central formula relating the sides and angles of a spherical triangle. The derivation unfolds through a series of algebraic manipulations of quaternion products and exponentials, demonstrating the power and efficiency of this approach. He further elaborates on the spherical law of sines, albeit in a less detailed manner, showing how it can also be obtained within this quaternion framework.
Tao then elucidates the geometric interpretation of the quaternion derivation. The spherical law of cosines, derived using quaternions, reflects the geometric relationships between the sides and angles of the spherical triangle, embedded on the surface of the unit sphere. This geometric interpretation provides a deeper understanding of the formula, going beyond a purely algebraic manipulation.
Finally, Tao briefly touches upon the broader implications of this approach. He suggests that this quaternion-based method could potentially be extended to higher dimensions, offering a pathway to exploring analogous relationships in non-Euclidean geometries. He concludes by emphasizing the surprising effectiveness and elegance of using quaternions to understand and derive the fundamental identities of spherical trigonometry, showcasing the interconnectedness between seemingly disparate areas of mathematics.
Summary of Comments ( 19 )
https://news.ycombinator.com/item?id=42880242
The Hacker News comments on Tao's post about quaternions and spherical trigonometry largely express appreciation for the clear explanation of a complex topic. Several commenters note the usefulness of quaternions in applications like computer graphics and robotics, particularly for their ability to represent rotations without gimbal lock. One commenter points out the historical context of Hamilton's discovery of quaternions, while another draws a parallel to using complex numbers for planar geometry. A few users discuss alternative approaches to representing rotations, such as rotation matrices and Clifford algebras, comparing their advantages and disadvantages to quaternions. Some express a desire to see Tao explore the connection between quaternions and spinors in a future post.
The Hacker News post titled "Quaternions and spherical trigonometry," linking to a blog post by Terence Tao, has generated several comments discussing various aspects of quaternions, their applications, and the relationship to spherical trigonometry.
One commenter highlights the practical value of quaternions in 3D graphics and game development, specifically for representing rotations. They mention how quaternions effectively avoid the gimbal lock problem encountered with Euler angles and provide a computationally efficient way to interpolate between rotations, leading to smoother animations. They further note the historical context of quaternions being used even before the widespread availability of matrix libraries.
Another commenter delves into the mathematical connection between quaternions and rotations, pointing out that unit quaternions form a double cover of the rotation group SO(3). This means every rotation can be represented by two quaternions that are negatives of each other. They also touch upon the relationship between quaternion multiplication and the composition of rotations.
A different commenter expands on the geometrical interpretation of quaternions and their relation to spherical trigonometry. They explain how rotations in 3D space can be visualized as rotations on a sphere, and how quaternions provide a concise algebraic way to represent these spherical rotations. This commenter also mentions the connection to stereographic projection and how it relates quaternions to complex numbers.
The discussion also branches into the historical aspect of quaternions, with one comment mentioning Hamilton's discovery and his carving of the fundamental formula into Broom Bridge. This comment also draws a parallel to the discovery of complex numbers and their initial resistance before becoming widely accepted.
Further comments explore alternative representations of rotations, such as axis-angle representation and rotation matrices, comparing their advantages and disadvantages to quaternions. One comment specifically mentions the use of rotation matrices in physics simulations due to their efficiency in certain computations.
Overall, the comments on Hacker News provide a rich discussion surrounding quaternions, covering their mathematical properties, practical applications, historical context, and comparison to other rotation representations. They offer valuable insights and perspectives on the topic for those interested in delving deeper into the subject.