SafekipediaGeometric algebra
Adapted from Wikipedia Β· Adventurer experience
In mathematics, a geometric algebra (also known as a Clifford algebra) is a special kind of algebra. It helps us understand shapes and points in space. It uses two main actions: adding and multiplying. When we multiply points using this special way, we get new objects called multivectors.
Geometric algebra was first studied by Hermann Grassmann and later developed by William Kingdon Clifford in 1878. It brings together ideas from different parts of math, such as the Grassmann algebra and Hamilton's quaternion algebra. Today, it is useful in many areas, like relativity, physics, computer graphics, and robotics. Many experts think it is a clear and simple way to solve problems in these subjects.
Definition and notation
Geometric algebra is a type of math that helps us work with shapes and spaces beyond simple points and lines. It adds new ways to combine things, especially vectors, which have both direction and size.
It uses two main operations: addition and the geometric product. When you use the geometric product on vectors, you get new objects called multivectors. These can stand for planes, lines, and other shapes in space.
The geometric product mixes ideas from the dot product and cross product. This gives a single, simple way to talk about how shapes and positions relate to each other. This makes it very helpful for studying physics and building things.
| Subgroup | Definition | GA term |
|---|---|---|
| Ξ {\displaystyle \Gamma } | { S β G Γ β£ S ^ V S β 1 β V } {\displaystyle \{S\in {\mathcal {G}}^{\times }\mid {\widehat {S}}VS^{-1}\subseteq V\}} | versors |
| Pin {\displaystyle \operatorname {Pin} } | { S β Ξ β£ S S ~ = Β± 1 } {\displaystyle \{S\in \Gamma \mid S{\widetilde {S}}=\pm 1\}} | unit versors |
| Spin {\displaystyle \operatorname {Spin} } | Pin β© G [ 0 ] {\displaystyle {\operatorname {Pin} }\cap {\mathcal {G}}^{}} | even unit versors |
| Spin + {\displaystyle \operatorname {Spin} ^{+}} | { S β Spin β£ S S ~ = 1 } {\displaystyle \{S\in \operatorname {Spin} \mid S{\widetilde {S}}=1\}} | rotors |
| Name | Signature | Blades, e.g., oriented geometric objects that algebra can represent | Rotors, e.g., orientation-preserving transformations that the algebra can represent |
|---|---|---|---|
| Hyperbolic numbers | G ( 1 , 0 , 0 ) {\displaystyle {\mathcal {G}}(1,0,0)} | Points | |
| Complex numbers | G ( 0 , 1 , 0 ) {\displaystyle {\mathcal {G}}(0,1,0)} | Points | |
| Dual numbers | G ( 0 , 0 , 1 ) {\displaystyle {\mathcal {G}}(0,0,1)} | Points | |
| Vectorspace GA (VGA), algebra of physical space (APS) | G ( 3 , 0 , 0 ) {\displaystyle {\mathcal {G}}(3,0,0)} | Planes and lines through the origin | Rotations, e.g. S O ( 3 ) {\displaystyle \mathrm {SO} (3)} |
| Projective GA (PGA), Rigid GA (RGA), plane-based GA | G ( 3 , 0 , 1 ) {\displaystyle {\mathcal {G}}(3,0,1)} | Planes, lines, and points anywhere in space | Rotations and translations, e.g., rigid motions, S E ( 3 ) {\displaystyle \mathrm {SE} (3)} aka S O ( 3 , 0 , 1 ) {\displaystyle \mathrm {SO} (3,0,1)} |
| Spacetime algebra, STA | G ( 3 , 1 , 0 ) {\displaystyle {\mathcal {G}}(3,1,0)} | Volumes, planes and lines through the origin in spacetime | Rotations and spacetime boosts, e.g. β S O ( 3 , 1 ) {\displaystyle \mathrm {SO} (3,1)} β , the Lorentz group |
| Spacetime Algebra Projectivized (STAP), | G ( 3 , 1 , 1 ) {\displaystyle {\mathcal {G}}(3,1,1)} | Volumes, planes, lines, and points (events) in spacetime | Rotations, translations, and spacetime boosts (Poincare group) |
| Conformal GA (CGA) | G ( 4 , 1 , 0 ) {\displaystyle {\mathcal {G}}(4,1,0)} | Spheres, circles, point pairs (or dipoles), round points, flat points, lines, and planes anywhere in space | Transformations of space that preserve angles (conformal group β S O ( 4 , 1 ) {\displaystyle \mathrm {SO} (4,1)} β ) |
| Conformal spacetime algebra, CSTA | G ( 4 , 2 , 0 ) {\displaystyle {\mathcal {G}}(4,2,0)} | Spheres, circles, planes, lines, light-cones, trajectories of objects with constant acceleration, all in spacetime | Conformal transformations of spacetime, e.g. transformations that preserve rapidity along arclengths through spacetime |
| Balanced algebra, mother algebra | G ( 3 , 3 , 0 ) {\displaystyle {\mathcal {G}}(3,3,0)} | Unknown | Projective group |
| GA for Conics (GAC) Quadric conformal 2D GA (QC2GA) | G ( 5 , 3 , 0 ) {\displaystyle {\mathcal {G}}(5,3,0)} | Points, point pair/triple/quadruple, conic, pencil of up to 6 independent conics | Reflections, translations, rotations, dilations, others |
| Quadric conformal GA (QCGA) | G ( 9 , 6 , 0 ) {\displaystyle {\mathcal {G}}(9,6,0)} | Points, tuples of up to 8 points, quadric surfaces, conics, conics on quadratic surfaces (such as Spherical conic), pencils of up to 9 quadric surfaces | Reflections, translations, rotations, dilations, others |
| Double conformal geometric algebra (DCGA) | G ( 8 , 2 , 0 ) {\displaystyle {\mathcal {G}}(8,2,0)} | Points, Darboux cyclides, quadrics surfaces | Reflections, translations, rotations, dilations, others |
Geometric interpretation in the vector space model
Geometric algebra helps us understand shapes and directions in space. It uses two main actions: addition and a special kind of multiplication called the geometric product. This multiplication lets us combine vectors (which show direction and size) to create new objects called multivectors.
One important idea in geometric algebra is projection and rejection. Projection tells us how much of one vector goes in the direction of another, while rejection tells us what part of a vector is perpendicular to another direction. These ideas help us break down vectors into parts that are parallel or at right angles to each other.
Another key concept is reflection, which flips a vector over a plane. This can help us create more complex movements like rotations. By combining reflections, we can rotate vectors in space while keeping their lengths and angles the same. This makes geometric algebra useful for understanding movements and shapes in higher dimensions.
Examples and applications
Geometric algebra helps us understand shapes and movements in space. For example, it can calculate the area of a parallelogram made by two vectors.
It can also help find where a line crosses a plane. Geometric algebra gives a clear way to solve this using vectors. It also helps describe spinning motion, like the turning effect of a force (called torque), without needing extra rules. This makes working with rotations easier.
Geometric calculus
Main article: Geometric calculus
Geometric calculus is a way to study shapes and their properties. It adds rules for measuring and adding up to geometric algebra. This helps us understand how things change and how to find areas and volumes in a more general way.
One important idea in geometric calculus is the vector derivative. It lets us connect changes in a function over an area to changes along the edges of that area. This is similar to some well-known math rules, making it a useful tool for solving many geometry problems.
History
Geometric algebra, also called Clifford algebra, connects geometry and algebra. People have thought about this connection since ancient times. In 1844, Hermann Grassmann created a system to describe shapes and changes in space. Later, William Kingdon Clifford built on this work. He created a new way to multiply vectors. This helped explain how things turn and move.
In the 20th century, mathematicians kept studying these ideas. They found uses in physics to describe movements and forces. Today, geometric algebra is used in computer graphics and robotics to make calculations simpler.
Related articles
This article is a child-friendly adaptation of the Wikipedia article on Geometric algebra, available under CC BY-SA 4.0.
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