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# Geometric algebra

## Point, Line, Plane and Solid

❶Hilbert's basis theorem implies that ideals in k [ A n ] are always finitely generated.

## HOMEWORK SOLVED

In classical algebraic geometry, this field was always the complex numbers C , but many of the same results are true if we assume only that k is algebraically closed. We consider the affine space of dimension n over k , denoted A n k or more simply A n , when k is clear from the context. When one fixes a coordinate system, one may identify A n k with k n.

The purpose of not working with k n is to emphasize that one "forgets" the vector space structure that k n carries. The property of a function to be polynomial or regular does not depend on the choice of a coordinate system in A n.

When a coordinate system is chosen, the regular functions on the affine n -space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the regular functions on A n is a ring, which is denoted k [ A n ]. We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k [ A n ]. The vanishing set of S or vanishing locus or zero set is the set V S of all points in A n where every polynomial in S vanishes.

A subset of A n which is V S , for some S , is called an algebraic set. The V stands for variety a specific type of algebraic set to be defined below.

Given a subset U of A n , can one recover the set of polynomials which generate it? If U is any subset of A n , define I U to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: The answer to the first question is provided by introducing the Zariski topology , a topology on A n whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of k [ A n ].

The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that I V S is the radical of the ideal generated by S. In more abstract language, there is a Galois connection , giving rise to two closure operators ; they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection.

For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k [ A n ] are always finitely generated.

An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the irreducible components of the algebraic set.

An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring. Some authors do not make a clear distinction between algebraic sets and varieties and use irreducible variety to make the distinction when needed. Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds , there is a natural class of functions on an algebraic set, called regular functions or polynomial functions.

A regular function on an algebraic set V contained in A n is the restriction to V of a regular function on A n. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even analytic. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space , where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.

Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k [ V ]. This ring is called the coordinate ring of V. Since regular functions on V come from regular functions on A n , there is a relationship between the coordinate rings.

Using regular functions from an affine variety to A 1 , we can define regular maps from one affine variety to another.

First we will define a regular map from a variety into affine space: Let V be a variety contained in A n. Choose m regular functions on V , and call them f 1 , In other words, each f i determines one coordinate of the range of f. The definition of the regular maps apply also to algebraic sets. The regular maps are also called morphisms , as they make the collection of all affine algebraic sets into a category , where the objects are the affine algebraic sets and the morphisms are the regular maps.

The affine varieties is a subcategory of the category of the algebraic sets. This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced k -algebras.

This equivalence is one of the starting points of scheme theory. In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties next section , as an affine variety and its projective completion have the same field of functions.

If V is an affine variety, its coordinate ring is an integral domain and has thus a field of fractions which is denoted k V and called the field of the rational functions on V or, shortly, the function field of V.

Its elements are the restrictions to V of the rational functions over the affine space containing V. The domain of a rational function f is not V but the complement of the subvariety a hypersurface where the denominator of f vanishes. As with regular maps, one may define a rational map from a variety V to a variety V '.

As with the regular maps, the rational maps from V to V ' may be identified to the field homomorphisms from k V ' to k V. Two affine varieties are birationally equivalent if there are two rational functions between them which are inverse one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic. An affine variety is a rational variety if it is birationally equivalent to an affine space.

This means that the variety admits a rational parameterization. The problem of resolution of singularities is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular see also smooth completion. It was solved in the affirmative in characteristic 0 by Heisuke Hironaka in and is yet unsolved in finite characteristic. Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space.

If we draw it, we get a parabola. As x goes to negative infinity, the slope of the same line goes to negative infinity. This is a cubic curve. But unlike before, as x goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. The consideration of the projective completion of the two curves, which is their prolongation "at infinity" in the projective plane , allows us to quantify this difference: Also, both curves are rational, as they are parameterized by x , and the Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular.

Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For these reasons, projective space plays a fundamental role in algebraic geometry.

In this case, one says that the polynomial vanishes at the corresponding point of P n. This allows us to define a projective algebraic set in P n as the set V f 1 , Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced homogeneous ideals which define them. The projective varieties are the projective algebraic sets whose defining ideal is prime. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties.

The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations.

On the other hand, the field of the rational functions or function field is a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring.

The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. It follows that real algebraic geometry is not only the study of the real algebraic varieties, but has been generalized to the study of the semi-algebraic sets , which are the solutions of systems of polynomial equations and polynomial inequalities. One of the challenging problems of real algebraic geometry is the unsolved Hilbert's sixteenth problem: Decide which respective positions are possible for the ovals of a nonsingular plane curve of degree 8.

Since then, most results in this area are related to one or several of these items either by using or improving one of these algorithms, or by finding algorithms whose complexity is simply exponential in the number of the variables.

A body of mathematical theory complementary to symbolic methods called numerical algebraic geometry has been developed over the last several decades. The main computational method is homotopy continuation. This supports, for example, a model of floating point computation for solving problems of algebraic geometry.

In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential.

However, this is only a worst case complexity, and the complexity bound of Lazard's algorithm of may frequently apply. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than CAD is an algorithm which was introduced in by G.

Collins to implement with an acceptable complexity the Tarski—Seidenberg theorem on quantifier elimination over the real numbers. This theorem concerns the formulas of the first-order logic whose atomic formulas are polynomial equalities or inequalities between polynomials with real coefficients. The complexity of CAD is doubly exponential in the number of variables.

This means that CAD allows, in theory, to solve every problem of real algebraic geometry which may be expressed by such a formula, that is almost every problem concerning explicitly given varieties and semi-algebraic sets.

This implies that, unless if most polynomials appearing in the input are linear, it may not solve problems with more than four variables. Since , most of the research on this subject is devoted either to improve CAD or to find alternate algorithms in special cases of general interest.

As an example of the state of art, there are efficient algorithms to find at least a point in every connected component of a semi-algebraic set, and thus to test if a semi-algebraic set is empty. On the other hand, CAD is yet, in practice, the best algorithm to count the number of connected components.

The basic general algorithms of computational geometry have a double exponential worst case complexity. During the last 20 years of 20th century, various algorithms have been introduced to solve specific subproblems with a better complexity. The main algorithms of real algebraic geometry which solve a problem solved by CAD are related to the topology of semi-algebraic sets. One may cite counting the number of connected components , testing if two points are in the same components or computing a Whitney stratification of a real algebraic set.

Therefore, these algorithms have never been implemented and this is an active research area to search for algorithms with have together a good asymptotic complexity and a good practical efficiency. The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes , ind-schemes , algebraic spaces , algebraic stacks and so on.

The need for this arises already from the useful ideas within theory of varieties, e. Most remarkably, in late s, algebraic varieties were subsumed into Alexander Grothendieck 's concept of a scheme. Dotting the " Pauli vector " a dyad:. However, a useful inner product cannot be defined in the space and so there is no geometric product either leaving only outer product and non-metric uses of duality such as meet and join.

Nevertheless, there has been investigation of 4-dimensional alternatives to the full 5-dimensional CGA for limited geometries such as rigid body movements. Other useful references are Li and Bayro-Corrochano This allows all conformal transformations to be done as rotations and reflections and is covariant , extending incidence relations of projective geometry to circles and spheres. A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.

In a postscript to that paper, reference is made to a further paper [21] that Dorst describes as resolving most of the weaknesses in this research area.

Simple reflections in a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of general rotoreflections and rotations. The result of the reflection will be. A general reflection may be expressed as the composite of any odd number of single-axis reflections. We can also show that. The descriptions for rotations and reflections, including their outermorphisms, are examples of such sandwiching.

The outermorphisms have a particularly simple algebraic form. Specifically, a mapping of vectors of the form. Since both operators and operand are versors there is potential for alternative examples such as rotating a rotor or reflecting a spinor always provided that some geometrical or physical significance can be attached to such operations.

Clifford group, although Lundholm deprecates this usage. Spinors are defined as elements of the even subalgebra of a real GA; an analysis of the GA approach to spinors is given by Francis and Kosowsky. It could be any shape, although the volume equals that of the parallelotope. The mathematical description of rotational forces such as torque and angular momentum often makes use of the cross product of vector calculus in three dimensions with a convention of orientation handedness.

The cross product can be viewed in terms of the exterior product allowing a more natural geometric interpretation of the cross product as a bivector using the dual relationship.

For example, torque is generally defined as the magnitude of the perpendicular force component times distance, or work per unit angle.

Geometric calculus extends the formalism to include differentiation and integration including differential geometry and differential forms. Essentially, the vector derivative is defined so that the GA version of Green's theorem is true,.

Also developed are the concept of vector manifold and geometric integration theory which generalizes Cartan's differential forms. Although the connection of geometry with algebra dates as far back at least to Euclid 's Elements in the third century B. In that year, Hermann Grassmann introduced the idea of a geometrical algebra in full generality as a certain calculus analogous to the propositional calculus that encoded all of the geometrical information of a space.

From his point of view, the quaternions described certain transformations which he called rotors , whereas Grassmann's algebra described certain properties or Strecken such as length, area, and volume.

Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra. Nevertheless, another revolutionary development of the 19th-century would completely overshadow the geometric algebras: Vector analysis was motivated by James Clerk Maxwell 's studies of electromagnetism , and specifically the need to express and manipulate conveniently certain differential equations.

Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adopted it as their geometrical toolkit of choice, particularly following the influential textbook Vector Analysis by Edwin Bidwell Wilson , following lectures of Gibbs.

In more detail, there have been three approaches to geometric algebra: Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work of abstract algebraists such as Hermann Weyl and Claude Chevalley. The geometrical approach to geometric algebras has seen a number of 20th-century revivals. In mathematics, Emil Artin 's Geometric Algebra [32] discusses the algebra associated with each of a number of geometries, including affine geometry , projective geometry , symplectic geometry , and orthogonal geometry.

In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism, together with more advanced topics such as quantum mechanics and gauge theory. In computer graphics and robotics, geometric algebras have been revived in order to efficiently represent rotations and other transformations.

For applications of GA in robotics screw theory, kinematics and dynamics using versors , computer vision, control and neural computing geometric learning see Bayro GA is a very application-oriented subject.

There is a reasonably steep initial learning curve associated with it, but this can be eased somewhat by the use of applicable software.

The following is a list of freely available software that does not require ownership of commercial software or purchase of any commercial products for this purpose:. Software allowing script creation and including sample visualizations, manual and GA introduction. From Wikipedia, the free encyclopedia. For other uses, see Geometric algebra disambiguation. Symmetric bilinear form and Exterior algebra.

Reversed orientation corresponds to negating the exterior product. Comparison of vector algebra and geometric algebra.

Some authors may extend the meaning of inner product to the entire algebra, but there is little consensus on this. Even in texts on geometric algebras, the term is not universally used. The Grassmann—Cayley algebra regards the meet relation as its counterpart and gives a unifying framework in which these two operations have equal footing Grassmann himself defined the meet operation as the dual of the outer product operation, but later mathematicians defined the meet operator independently of the outer product through a process called shuffle , and the meet operation is termed the shuffle product.

It is shown that this is an antisymmetric operation that satisfies associativity, defining an algebra in its own right. Thus, the Grassmann—Cayley algebra has two algebraic structures simultaneously: Cross product Triple product Seven-dimensional cross product. Geometric algebra Exterior algebra Bivector Multivector. Category Outline Portal Wikibook Wikiversity.

Wikibooks has a book on the topic of: Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space. Wikiversity has learning resources about Investigating 3D geometric algebra.

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Geometry and Algebra. Suppose that during Sally's first year of college, she decides to take algebra and geometry. Though these are both considered to be mathematics courses, the course catalog.