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| <!--This paragraph outlines what CGA is, with particular reference to points and transformations-->In [[mathematics]], with application in [[computational geometry]], '''conformal geometric algebra''' (CGA) is the [[geometric algebra]] constructed over the resultant space of a projective map from an ''n''-dimensional [[Euclidean space|Euclidean]] or [[pseudo-Euclidean space|pseudo-Euclidean]] base space ℝ<sup>''p'',''q''</sup> into ℝ<sup>''p''+1,''q''+1</sup>. This allows operations on the ''n''-dimensional space, including rotations, translations and reflections to be represented using [[Versor#Geometric algebra|versor]]s of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations.
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| <!--This paragraph extends the conformal idea to generalized "conformal geometric objects"-->The effect of the mapping is that generalized (i.e. including zero curvature) [[n-sphere|''k''-spheres]] in the base space map onto (''k''+2)-[[Blade (geometry)|blades]], and so that the effect of a translation (or ''any'' [[conformal mapping]]) of the base space corresponds to a rotation in the higher-dimensional space. In the algebra of this space, based on the [[geometric product]] of vectors, such transformations correspond to the algebra's characteristic [[Inner automorphism|sandwich operation]]s, similar to the use of [[Quaternions and spatial rotation|quaternions for spatial rotation in 3D]], which combine very efficiently. A consequence of rotors representing transformations is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly. A geometric object (a ''k''-sphere) can be synthesized as the wedge product of ''k''+2 linearly independent vectors representing points on the object; conversely, the object can be decomposed as the repeated [[wedge product]] of vectors representing ''k''+2 points in its surface. Some intersection operations also acquire a very tidy algebraic form: for example, for the Euclidean base space ℝ<sup>3</sup>, applying the [[wedge product]] to the dual of the tetravectors representing two spheres produces the dual of the trivector representation of their circle of intersection.
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| <!--This paragraph introduces applications and motivates the use of CGA-->As this algebraic structure lends itself directly to effective computation, it facilitates exploration of the classical methods of [[projective geometry]] and [[inversive geometry]] in a concrete, easy-to-manipulate setting. It has also been used as an efficient structure to represent and facilitate calculations in [[screw theory]]. CGA has particularly been applied in connection with the projective mapping of the everyday Euclidean space ℝ<sup>3</sup> into a five-dimensional space ℝ<sup>4,1</sup>, which has been investigated for applications in robotics and computer vision. It can be applied generally to any Euclidean or pseudo-Euclidean space, and the mapping of [[Minkowski space]] ℝ<sup>3,1</sup> to the space ℝ<sup>4,2</sup> is being investigated for applications to relativistic physics.
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| {{cleanup-remainder|date=February 2012}}
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| ==Construction of CGA==
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| === Notation and terminology===
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| The Euclidean space containing the objects being modelled is referred to here as the ''base space'', and the algebraic space used to projectively model these objects is referred to here as the ''representation space''. A ''homogeneous subspace'' refers to a linear subspace of the algebraic space.
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| <!--Some thought is need for these terms.
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| * Hestenes uses "inhomogeneous" and "homogeneous" as adjectives (but this clashes with "homogeneous subspace")
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| * Perwass refers to "Euclidean space" and "conformal space" respectively.
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| * Dorst uses "base space" and "representation space"
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| Dorst's appears to me the best terminology – Quondum -->
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| The terms for objects: ''point'', ''line'', ''circle'', ''sphere'' etc. are used to mean either the geometric object in the base space, or the homogeneous subspace of the representation space that represents that object, with the latter generally being intended unless indicated otherwise. Algebraically, any element of the homogeneous subspace will be used, with one element being referred to as ''normalized'' by some criterion.
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| Boldface lowercase Latin letters are used to represent position vectors from the origin to a point in the base space. Italic symbols are used for other elements of the representation space.
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| === Base and representation spaces ===
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| The base space {{math|ℝ<sup>''p'',''q''</sup>}} is extended by adding two basis vectors {{math|1=''e''<sub>−</sub>}} and {{math|1=''e''<sub>+</sub>}} orthogonal to the base space and to each other, with {{math|1=''e''<sub>−</sub><sup>2</sup> = −1}} and {{math|1=''e''<sub>+</sub><sup>2</sup> = +1}}, creating the representation space {{math|ℝ<sup>''p''+1,''q''+1</sup>}}.
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| It is convenient to use two null vectors {{math|''n''<sub>o</sub>}} and {{math|''n''<sub>∞</sub>}} as basis vectors in place of {{math|1=''e''<sub>+</sub>}} and {{math|1=''e''<sub>−</sub>}}, where {{math|1=''n''<sub>o</sub> = (''e''<sub>−</sub> − ''e''<sub>+</sub>)/2}}, and {{math|1=''n''<sub>∞</sub> = ''e''<sub>−</sub> + ''e''<sub>+</sub>}}.
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| It can be verified, where {{math|'''x'''}} is in the base space, that:
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| :<math>\begin{align}
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| {n_o}^2 & = 0 \qquad & n_\text{o} \cdot n_\infty & = -1 \qquad & n_\text{o} \cdot \mathbf{x} & = 0 \\
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| {n_\infty}^2 & = 0 \qquad & n_\text{o} \wedge n_\infty & = e_{-}e_{+} \qquad & n_\infty \cdot \mathbf{x} & = 0
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| \end{align}</math>
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| These properties lead to formulas which may seem slightly counter-intuitive for the {{math|''n''<sub>o</sub>}} and {{math|''n''<sub>∞</sub>}} coefficients of a general vector {{math|''r''}} in the representation space:
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| :The coefficient of {{math|''n''<sub>o</sub>}} for {{math|''r''}} is {{math|−''n''<sub>∞</sub> ⋅ ''r''}}
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| :The coefficient of {{math|''n''<sub>∞</sub>}} for {{math|''r''}} is {{math|−''n''<sub>o</sub> ⋅ ''r''}}
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| === Mapping between the base space and the representation space ===
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| The mapping from a vector in the base space (being from the origin to a point in the affine space represented) is given by the formula:{{refn|group="upper-alpha"|The mapping can also be written {{math|''F'' : '''x''' → −('''x''' − ''e''<sub>+</sub>) ''n''<sub>∞</sub> ('''x''' − ''e''<sub>+</sub>)}}, as given in [[David Hestenes|Hestenes]] and Sobczak (1984), p.303.<ref>Hestenes, D. and Sobczak, G. (1984), ''Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics''. Dordrecht: Reidel; pp. 302–303.</ref> The equivalence of the two forms is noted in Lasenby and Lasenby (2000).<ref>Lasenby, AN and Lasenby, J (2000), [http://www.mrao.cam.ac.uk/~clifford/publications/ps/ll_surface.ps.gz Surface evolution and representation using geometric algebra]; in ''The Mathematics of Surfaces IX: the 9th IMA Conference, Cambridge, 4–7 September 2000'', pp. 144–168</ref>}}
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| :<math> F : \mathbf{x} \mapsto n_\text{o} + \mathbf{x} + \tfrac{1}{2} \mathbf{x}^2 n_\infty </math>
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| Points and other objects that differ only by a nonzero scalar factor all map to the same object in the base space. When normalisation is desired, as for generating a simple reverse map of a point from the representation space to the base space or determining distances, the condition {{math|1=''F''('''x''') ⋅ ''n''<sub>∞</sub> = −1}} may be used.
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| [[Image:Conformal Embedding.svg|right|300px|thumb|Change of normalisation: mapping the null cone from the hyperplane {{math|1=''r'' ⋅ (''n''<sub>∞</sub> − ''n''<sub>o</sub>) = 1}} to the hyperplane {{math|1=''r'' ⋅ ''n''<sub>∞</sub> = −1}}.]]
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| The forward mapping is equivalent to:
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| * first conformally projecting {{math|'''x'''}} from {{math|''e''<sub>123</sub>}} onto a unit 3-sphere in the space {{math|''e''<sub>+</sub> ∧ ''e''<sub>123</sub>}} (in 5-D this is in the subspace {{math|1=''r'' ⋅ (−''n''<sub>o</sub> − {{sfrac|1|2}}''n''<sub>∞</sub>) = 0}});
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| * then lift this into a projective space, by adjoining {{math|1=''e''<sub>–</sub> = 1}}, and identifying all points on the same ray from the origin (in 5-D this is in the subspace {{math|1=''r'' ⋅ (−''n''<sub>o</sub> − {{sfrac|1|2}}''n''<sub>∞</sub>) = 1}});
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| * then change the normalisation, so the plane for the homogenous projection is given by the {{math|''n''<sub>o</sub>}} co-ordinate having a value {{math|1}}, i.e. {{math|1=''r'' ⋅ ''n''<sub>∞</sub> = −1}}.
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| === Inverse mapping ===
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| An inverse mapping for {{math|''X''}} on the null cone is given (Perwass eqn 4.37) by
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| :<math>X \mapsto \mathcal{P}^\perp_{n_\infty \wedge n_o}\left( \frac{X}{- X \cdot n_\infty}\right)</math>
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| This first gives a stereographic projection from the light-cone onto the plane {{math|1=''r'' ⋅ ''n''<sub>∞</sub> = −1}}, and then throws away the {{math|''n''<sub>o</sub>}} and {{math|''n''<sub>∞</sub>}} parts, so that the overall result is to map all of the equivalent points {{math|1=''αX'' = ''α''(''n''<sub>o</sub> + '''x''' + {{sfrac|2}}'''x'''<sup>2</sup>''n''<sub>∞</sub>)}} to {{math|'''x'''}}. | |
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| === Origin and point at infinity ===
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| The point {{math|1='''x''' = 0}} in {{math|ℝ<sup>''p'',''q''</sup>}} maps to {{math|''n''<sub>o</sub>}} in {{math|ℝ<sup>''p''+1,''q''+1</sup>}}, so {{math|''n''<sub>o</sub>}} is identified as the (representation) vector of the point at the origin.
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| A vector in {{math|ℝ<sup>''p''+1,''q''+1</sup>}} with a nonzero {{math|''n''<sub>∞</sub>}} coefficient, but a zero {{math|''n''<sub>o</sub>}} coefficient, must (considering the inverse map) be the image of an ''infinite'' vector in {{math|ℝ<sup>''p'',''q''</sup>}}. The direction {{math|''n''<sub>∞</sub>}} therefore represents the (conformal) [[point at infinity]]. This motivates the subscripts {{math|<sub>o</sub>}} and {{math|<sub>∞</sub>}} for identifying the null basis vectors.
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| The choice of the origin is arbitrary: any other point may be chosen, as the representation is of an [[affine space]]. The origin merely represents a reference point, and is algebraically equivalent to any other point. Changing the origin corresponds to a rotation in the representation space.{{clarify|date=February 2012}}
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| == Geometrical objects ==
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| === As the solution of a pair of equations ===
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| Given any nonzero blade {{math|''A''}} of the representing space, the set of vectors that are solutions to a pair of homogeneous equations of the form<ref>[http://arxiv.org/abs/cs.CG/0310017 Chris Doran (2003), ''Circle and sphere blending with conformal geometric algebra'']</ref>
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| :<math>X^2 = 0</math>
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| :<math>X \wedge A = 0</math>
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| are homogeneous 1-d subspaces of null vectors, and are thus representations of sets of points in the base space. This leads to the choice of a blade {{math|''A''}} as being a useful way to represent a class of geometric object. Specific cases for the blade {{math|''A''}} (independent of the number of dimensions of the space) when the base space is Euclidean space are: | |
| * a scalar: the empty set
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| * a vector: a single point
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| * a bivector: a pair of points
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| * a trivector: a generalized circle
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| * a 4-vector: a generalized sphere
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| * etc.
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| These each may split into three cases according to whether {{math|''A''<sup>2</sup>}} is positive, zero or negative, corresponding (in reversed order in some cases) to the object as listed, a degenerate case of a single point, or no points (where the nonzero solutions of {{math|1=''X'' ∧ ''A''}} exclude null vectors).
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| The listed geometric objects are replaced by the corresponding shapes of constant magnitude from a center when the base space is pseudo-Euclidean.
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| Flat objects my be identified by the point at infinity being included in the solutions. Thus, if {{math|1=''n''<sub>∞</sub> ∧ ''A'' = 0}}, and the blade {{math|''A''}} is of grade 3 or higher, the object will be a line, plane, etc.
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| === As derived from points of the object ===
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| A blade {{math|''A''}} representing of one of this class of object may be found as the outer product of linearly independent vectors representing points on the object. In the base space, this linear independence manifests as each point lying outside the object defined by the other points. So, for example, a fourth point lying on the generalized circle defined by three distinct points cannot be used as a fourth point to define a sphere.
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| === odds ===
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| :Points in '''e'''<sub>123</sub> map onto the null cone—the null ''parabola'' if we set ''r'' . ''n''<sub>∞</sub> = -1.
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| :We can consider the locus of points in '''e'''<sub>123</sub> s.t. in conformal space ''g''('''x''') . A = 0, for various types of geometrical object A.
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| :We start by observing that <math>g(\mathbf{a}) . g(\mathbf{b}) = -\frac{1}{2} \| \mathbf{a} - \mathbf{b} \|^2 </math>
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| compare:
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| * x. a = 0 => x perp a; x.(a∧b) = 0 => x perp a ''and'' x perp b
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| * x∧a = 0 => x parallel to a; x∧(a∧b) = 0 => x parallel to a ''or'' to b (or to some linear combination)
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| the inner product and outer product representations are related by dualisation
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| :x∧A = 0 <=> x . A* = 0 (''check''—works if x is 1-dim, A is n-1 dim)
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| ==== g(x) . A = 0 ====
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| :* A ''point'': the locus of ''x'' in '''R'''<sup>3</sup> is a ''point'' if A in '''R'''<sup>4,1</sup> is a vector on the null cone.
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| :::(N.B. that because it's a homogeneous projective space, vectors of any length on a ray through the origin are equivalent, so g(x).A =0 is equivalent to g(x).g(a) = 0).
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| ::: *** ''warning'': apparently wrong codimension—go to the sphere as the general case, then restrict to a sphere of size zero. Is the dual of the equation affected by being on the null cone?
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| :* A ''sphere'': the locus of '''x''' is a ''sphere'' if A = S, a vector off the null cone.
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| :::If
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| ::::<math>\mathbf{S} = g(\mathbf{a}) - \frac{1}{2} \rho^2 \mathbf{e}_\infty</math>
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| :::then '''S'''.'''X''' = 0 => <math> -\frac{1}{2} (\mathbf{a}-\mathbf{x})^2 + \frac{1}{2} \rho^2 = 0 </math>
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| :::these are the points corresponding to a sphere
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| :::::''make pic to show hyperbolic orthogonality'' --> for a vector S off the null-cone, which directions are hyperbolically orthogonal? (cf Lorentz transformation pix)
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| ::::::in 2+1 D, if S is (1,a,b), (using co-ords e-, {e+, e<sub>i</sub>}), the points hyperbolically orthogonal to S are those euclideanly orthogonal to (-1,a,b)—i.e., a plane; or in ''n'' dimensions, a hyperplane through the origin. This would cut another plane not through the origin in a line (a hypersurface in an ''n''-2 surface), and then the cone in two points (resp. some sort of ''n''-3 conic surface). So it's going to probably look like some kind of conic. This is the surface that is the image of a sphere under ''g''.
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| :*A ''plane'': the locus of '''x''' is a ''plane'' if ''A'' = ''P'', a vector with a zero ''n''<sub>o</sub> component. In a homogeneous projective space such a vector ''P'' represents a vector on the plane ''n''<sub>o</sub>=1 that would be infinitely far from the origin (ie infinitely far outside the null cone) , so g(x).P =0 corresponds to ''x'' on a sphere of infinite radius, a plane.
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| ::In particular:
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| ::* <math>\mathbf{P} = \hat{\mathbf{a}} + \alpha \mathbf{e}_\infty </math> corresponds to ''x'' on a plane with normal <math>\hat{\mathbf{a}}</math> an orthogonal distance α from the origin.
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| ::* <math>\mathbf{P} = g(\mathbf{a}) - g(\mathbf{b})</math> corresponds to a plane half way between '''a''' and '''b''', with normal '''a''' - '''b'''
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| :*''circles''
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| :*''tangent planes''
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| :*''lines''
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| :*''lines at infinity''
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| :*''point pairs''
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| == Transformations ==
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| :* ''reflections''
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| :: It can be verified that forming '''P''' g('''x''') '''P''' gives a new direction on the null-cone, g('''x' '''), where '''x' ''' corresponds to a reflection in the plane of points '''p''' in '''R'''<sup>3</sup> that satisfy g('''p''') . '''P''' = 0.
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| :: g('''x''') . A = 0 => '''P''' g('''x''') . A '''P''' = 0 => '''P''' g('''x''') '''P''' . '''P''' A '''P''' (and similarly for the wedge product), so the effect of applying '''P''' sandwich-fashion to any the quantities A in the section above is similarly to reflect the corresponding locus of points '''x''', so the corresponding circles, spheres, lines and planes corresponding to particular types of A are reflected in exactly the same way that applying '''P''' to g('''x''') reflects a point '''x'''.
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| This reflection operation can be used to build up general translations and rotations:
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| :* ''translations''
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| :: Reflection in two parallel planes gives a translation,
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| :: <math>g(\mathbf{x}^\prime) = \mathbf{P}_\beta \mathbf{P}_\alpha \; g(\mathbf{x}) \; \mathbf{P}_\alpha \mathbf{P}_\beta</math>
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| :: If <math>\mathbf{P}_\alpha = \hat{\mathbf{a}} +\alpha \mathbf{e}_\infty</math> and <math>\mathbf{P}_\beta = \hat{\mathbf{a}} +\beta \mathbf{e}_\infty</math> then <math>\mathbf{x}^\prime = \mathbf{x} + 2 (\beta-\alpha) \hat{\mathbf{a}}</math>
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| :* ''rotations''
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| :: <math>g(\mathbf{x}^\prime) = \hat{\mathbf{b}}\hat{\mathbf{a}} \; g(\mathbf{x}) \; \hat{\mathbf{a}}\hat{\mathbf{b}}</math> corresponds to an '''x' ''' that is rotated about the origin by an angle 2 θ where θ is the angle between '''a''' and '''b''' -- the same effect that this rotor would have if applied directly to '''x'''.
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| :* ''general rotations''
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| :: rotations about a general point can be achieved by first translating the point to the origin, then rotating around the origin, then translating the point back to its original position, i.e. a sandwiching by the operator <math>\mathbf{TR{\tilde{T}}}</math> so
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| :: <math>g (\mathcal{G}x) = \mathbf{TR{\tilde{T}}} \; g(\mathbf{x}) \; \mathbf{T\tilde{R}\tilde{T}}</math>
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| :* ''screws''
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| :: the effect a ''[[screw theory|screw]]'', or ''motor'', (a rotation about a general point, followed by a translation parallel to the axis of rotation) can be achieved by sandwiching g('''x''') by the operator <math>\mathbf{M} = \mathbf{T_2T_1R{\tilde{T_1}}}</math>.
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| :: '''M''' can also be parametrised <math>\mathbf{M} = \mathbf{T^\prime R^\prime}</math> ([[Chasles' theorem]])
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| :* ''inversions''
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| :: an [[Inversion transformation|inversion]] is a reflection in a sphere – various operations which can be achieved using such inversions are discussed at [[inversive geometry]]. In particular, the combination of inversion together with the [[Euclidean transformation]]s translation and rotation is sufficient to express ''any'' [[conformal map]]ping – i.e. any mapping which universally preserves angles. ([[Liouville's theorem (conformal mappings)|Liouville's theorem]]).
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| :* ''dilations''
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| :: two inversions with the same centre produce a [[Dilation (metric space)|dilation]].
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| == Notes ==
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| {{reflist|group=upper-alpha}}
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| == References ==
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| {{reflist}}
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| == Bibliography ==
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| {{refbegin}}
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| === Books ===
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| * Hestenes ''et al'' (2000), in G. Sommer (ed.), ''Geometric Computing with Clifford Algebra''. Springer Verlag. ISBN 3-540-41198-4 ([http://books.google.com/books/about/Geometric_computing_with_Clifford_algebr.html?id=DD5QcR6t3h8C Google books]) (http://geocalc.clas.asu.edu/html/UAFCG.html Hestenes website)
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| ** Ch. 1: [http://geocalc.clas.asu.edu/pdf/CompGeom-ch1.pdf New algebraic tools for classical geometry]
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| ** Ch. 2: [http://geocalc.clas.asu.edu/pdf/CompGeom-ch2.pdf Generalized Homogeneous Coordinates for Computational Geometry]
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| ** Ch. 3: [http://geocalc.clas.asu.edu/pdf/CompGeom-ch3.pdf Spherical Conformal Geometry with Geometric Algebra]
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| ** Ch. 4: [http://geocalc.clas.asu.edu/pdf/CompGeom-ch4.pdf A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces]
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| * Hestenes (2001), in E. Bayro-Corrochano & G. Sobczyk (eds.), ''Advances in Geometric Algebra with Applications in Science and Engineering'', Springer Verlag. ISBN 0-8176-4199-8 [http://books.google.co.uk/books?id=GVqz9-_fiLEC Google books]
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| ** [http://geocalc.clas.asu.edu/pdf/OldWine.pdf Old Wine in New Bottles] (pp. 1–14)
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| * Hestenes (2010), in E. Bayro-Corrochano and G. Scheuermann (2010), ''Geometric Algebra Computing in Engineering and Computer Science''. Springer Verlag. ISBN 1-84996-107-7 ([http://books.google.co.uk/books?id=A_f3ADC-h2wC Google books]).
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| ** [http://geocalc.clas.asu.edu/pdf/New_Tools_for_Comp_Geom.pdf New Tools for Computational Geometry and rejuvenation of Screw Theory]
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| * Doran, C. and Lasenby, A. (2003), ''Geometric algebra for physicists'', Cambridge University Press. ISBN 0-521-48022-1 §10.2; p. 351 et seq
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| * Dorst, L. ''et al'' (2007), ''Geometric Algebra for Computer Science'', Morgan-Kaufmann. ISBN 0-12-374942-5 Chapter 13; p. 355 et seq
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| * Vince, J. (2008), ''Geometric Algebra for Computer Graphics'', Springer Verlag. ISBN 1-84628-996-3 Chapter 11; p. 199 et seq
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| * Perwass, C. (2009), ''Geometric Algebra with Applications in Engineering'', Springer Verlag. ISBN 3-540-89067-X §4.3: p. 145 et seq
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| * Bayro-Corrochano, E. and Scheuermann G. (2010, eds.), ''Geometric Algebra Computing in Engineering and Computer Science''. Springer Verlag. ISBN 1-84996-107-7 pp. 3–90
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| * Bayro-Corrochano (2010), ''Geometric Computing for Wavelet Transforms, Robot Vision, Learning, Control and Action''. Springer Verlag. ISBN 1-84882-928-0 Chapter 6; pp. 149–183
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| * Dorst, L. and Lasenby, J. (2011, eds.), ''Guide to Geometric Algebra in Practice''. Springer Verlag, pp. 3–252. ISBN 978-0-85729-810-2.
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| === Online resources ===
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| * Wareham, R. (2006), ''[http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pdf Computer Graphics using Conformal Geometric Algebra]'', PhD thesis, University of Cambridge, pp. 14–26, 31—67
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| * Bromborsky, A. (2008), [http://www.montgomerycollege.edu/Departments/planet/planet/Numerical_Relativity/GA-SIG/CFgeom5-15-2008.pdf Conformal Geometry via Geometric Algebra] (Online slides)
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| * Dell’Acqua, A. ''et al'' (2008), [http://home.dei.polimi.it/tubaro/Journals/Journal_2008_DA.pdf 3D Motion from structures of points, lines and planes], ''Image and Vision Computing'', '''26''' 529–549
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| * Dorst, L. (2010), [http://www.springer.com/cda/content/document/cda_downloaddocument/9781849961073-c2.pdf?SGWID=0-0-45-938847-0 Tutorial: Structure-Preserving Representation of Euclidean Motions through Conformal Geometric Algebra], in E. Bayro-Corrochano, G. Scheuermann (eds.), ''Geometric Algebra Computing'', Springer Verlag.
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| * Colapinto, P. (2011), [http://www.wolftype.com/versor/colapinto_masters_final_02.pdf VERSOR Spatial Computing with Conformal Geometric Algebra], MSc thesis, University of California Santa Barbara
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| * Macdonald, A. (2013), [http://faculty.luther.edu/~macdonal/GA&GC.pdf A Survey of Geometric Algebra and Geometric Calculus]. (Online notes) §4.2: p. 26 et seq.
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| * on the motor algebra over ℝ<sup>n+1</sup>:
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| ** Eduardo Bayro Corrochano (2001), ''Geometric computing for perception action systems: Concepts, algorithms and scientific applications''. ([http://books.google.co.uk/books?id=oDrCziWyGo0C Google books])
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| {{refend}}
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| [[Category:Geometric algebra]]
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| [[Category:Conformal geometry]]
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| [[Category:Inversive geometry]]
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| [[Category:Computational geometry]]
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