N-D geometric algebras are essentially the algebras of N anticommuting scalar-squared generators. That a multivector is a linear combination of our particular generators choice is what makes it a 1-vector. Grades are "geometry dependant" and arise from a particular choice of N anticommuting generators deemed to be 1-vectors. This choice determines the restricted products and conjugations and also the signatures and associated metric.
Let (e_{1},e_{2},..,e_{N}) be an orthogonal basis for geometric algebra
Â_{p,q,r}
for p³1 with e_{1}^{2}=1.
Then {e_{1},e_{12},e_{13},..,e_{1N}} provides an anticommuting <1;2>-grade generating basis
for Â_{N} having all but the first signature negated so
Â_{p,q,r} @ Â_{q+1,p-1,r} for p³1 .
For even N this has the disconcerting effect of making
former (N-1)-blade e_{234..N} the new "N-blade" pseudoscalar i', having opposite signature to
the previous pseudoscalar i but (anti)commuting with the new "1-vector" basis in like manner to
i=e_{12..N} 's (anti)commutation with the original e_{j}.
For any N, e_{1}'ºe_{1} and e_{2}'ºe_{12} must now both be regarded as orthogonal 1-blades with their geometric product
e_{12}'=e_{2} now regarded as a 2-blade arising from a new outter product Ù' with e_{1}'e_{2}'=e_{1}(e_{12})=e_{1}Ù'e_{12}=-e_{12}Ù'e_{1}=e_{2}.
Former 1-blade e_{2} thus takes the role of a 2-blade e_{12}' while 2-blade e_{12} assumes the role of a 1-blade e_{2}'. The grade of a blade is thus
ultimately a matter of interpretation. e_{1}+e_{12} is mixed grade in Â_{p,q,r} but a pure 1-vector in
Â_{q+1,p-1,r}.
As well as new restricted products Ù', ¿' etc. we aquire new conjugations ^{§}', ^{#}' etc. with e_{12}^{§}'=e_{2}'^{§}' = e_{2}'=e_{12}
and so forth, but since ^{§}^{#} acts identically on 1 and 2 blades we have (^{§}^{#})'=^{§}^{#}
and can regard ^{§}^{#} as geometrically invarient
under the Â_{p,q,r} @ Â_{q+1,p-1,r} equivalence.
Adoption of such alternate geometric basies can be considered as differing "geometric interpretations" of the underlying multivector algebra.
Odd N
When N=p+q and ½N(N-1)+q are both odd (eg. Â_{4k+3}
or Â_{4k,1}) so that the pseudoscalar is central and negative squared
then we can replace any even number of 1-vector generators whose signature we wish to change with
their even (N-1)-vector duals. These have opposite square
and anticommutes with every other e_{i} and e_{i}^{*}.
[ If we dual an odd number we no longer have a generating basis (eg. { e_{23},e_{2},e_{3}} cannot generate e_{1})
]
For odd N>3, when an odd number of e_{1},e_{2}, and e_{3} have negative signature so that e_{123}^{2}=1 then the anticommuting
<1;4> generaters
{ e_{1},e_{2},e_{3},e_{1234},e_{1235},..,e_{123N}} negates the signatures all but the first three elements
with the same full product for odd N, giving
Â_{p,q} @ Â_{q+1,p-1} when p³2, q³1
and Â_{p,q} @ Â_{q-3,p+3} when q³3 , with ^{§} invariant.
When N=4k+1 then starting from Â_{4k+1} we can judiciously replace pairs of 1-vectors with
4k-vectors to obtain any Â_{N-2m,2m}
with ^{§} geometrically invarient.
When N=4k+3 we can change 1-vectors to (4k+2)-vectors in pairs and it is Clifford conjugation ^{#}^{§} that is invarient.
When p³4 then {e_{1}e_{1234},e_{2}e_{1234},e_{3}e_{1234},e_{4}e_{1234},e_{5},e_{6},..,e_{N}} provides anticommuting <1;3> generators with the first four positive signatures negated so Â_{p,q,r} @ Â_{p-4,q+4,r} for p³4. This works for odd and even N and keeps ^{#}^{§} invariant.
The distinct odd dimensioned geometric algebras of nonnull pseudoscalar for N£7 are
as follows, with positive signature pseudoscalared listed first and geometrically invarient conjugations indicated in []..
Â_{1} ;; Â_{0,1}
@ C_{1×1}
.
Â_{2,1} ; Â_{0,3}
@ C_{0,2}
;;
Â_{3} @ Â_{1,2}
@ C_{2}
@ C_{2×2}
[^{§}^{#}]
Â_{5} @ Â_{1,4} ;
Â_{3,2}
;;
Â_{4,1} @ Â_{2,3} @ Â_{0,5}
@ C_{4}
@ C_{4×4}
[^{§}] ;
Â_{0,7} @ Â_{4,3} ;
Â_{6,1} @ Â_{2,5}
@ C_{0,8}
;;
Â_{7} @ Â_{1,6} @ Â_{5,2} @ Â_{3,4}
@ C_{8}
@ C_{8×8}
[^{§}^{#}]
Even N
When N and ½N(N-1)+q are both even (eg Â_{4k-2l,2l}) so that i has positive signature and anticommutes with 1-vectors
we cannot selectivly dual the basis 1-vectors but we can dual them all to obtain
Â_{4k-2l,2l} @
Â_{2l,4k-2l}
and in particular Â_{4k} @ Â_{0,4k}.
Hence the distinct even dimensioned geometric algebras of nonnull pseudoscalar for N£8 are
as follows, with invarient conjugations indicated in [].
These do not divide according to pseudoscalar signature, which for even N is not geometrically invariant.
Â_{0,2}
@ Q_{1×1}
; Â_{2} @ Â_{1,1} @ Â_{2×2} [^{§}^{#}]
Â_{4} @ Â_{0,4} @ Â_{1,3} @ Q_{2×2} ;
Â_{3,1} @ Â_{2,2} @ Â_{4×4} [^{§}^{#}].
Â_{6} @ Â_{5,1} @ Â_{2,4}
@ Â_{1,5} @ Q_{4×4}
;
Â_{4,2} @ Â_{3,3} @ Â_{0,6} @ Â_{8×8}[^{§}^{#}].
Â_{8} @ Â_{0,8}
@ Â_{4,4}
@ Â_{1,7} @ Â_{5,3}
@ Â_{16×16}
;
Â_{7,1} @ Â_{3,5}
@ Â_{2,6} @ Â_{6,2} .
Thus we have Minksowski spacetime Â_{3,1} and timespace Â_{1,3} fundamentally distinct .
Lounesto [16.4] shows that
Â_{p+8,q} @
Â_{p,q+8} @
(Â_{p,q})_{16×16} , ie. the space of 16×16 matrices of Â_{p,q} multivectors.
Algebraic Product Formulations
By considering generating sets some of which commute we can reduce Â_{p,q,r} into
a "product algebra" of two smaller geometric algebras.
Suppose geometric algebra U_{N} has a basis e_{1},e_{2},..e_{N} for N > 2 with e_{N} and e_{N-1} nonnull.
We can express any
a in U_{N} not only as a real weighted sum of products of the e_{i} (the e_{i} "generate" U_{N}) but
alternatively as a real-wieghted sum of products of the N-2 3-blades
e_{N-1}e_{N}e_{1} , e_{N-1}e_{N}e_{2} , ... , e_{N-1}e_{N}e_{N-2} and the two 1-vectors e_{N-1} and e_{N} which
commute with all the 3-blade generators but anticommute with eachother. The 3-blade
generators all anticommute with eachother so we have U_{N} @ U_{N-2} × U_{2} where U_{2} is a 2-D geometric algebra
having signatures e_{N-1}^{2} and e_{N-2}^{2}
and U_{N-2} is an (N-2)-D geometric algebra with signatures
(e_{N-1}e_{N}e_{i})^{2} = -e_{N-1}^{2}e_{N}^{2}e_{i}^{2} .
Symbol × here indicates the "product" algebra obtained by allowing the generators of seperate algebras
to multiply commutatively.
If N³ 4 we can repeat the trick,
"picking out" a pair of the nonnull 3-blades to produce an alternate
set of N generators for U_{N} consitisting of two 1-vectors, two 3-blades,
and N-4 5-blades; all generators commuting with those of different grade and anticommuting with those of the same grade.
If N³6 , we can replace all but two of the 5-blades with 7-blades, and so forth.
By such means, we can express any N-D multivector in terms of N odd-graded blade generators that
each commute with all but at most one other generator (the unique generator sharing the same grade).
The signatures of these "multigrade generators" depend on the
signatures of the original 1-vector basis. We can express these results most succinctly as
Â_{p+2,q+0,r}
@ Â_{2,0} × Â_{q,p,r}
@ Â_{2×2} × Â_{q,p,r}
Â_{p+1,q+1,r}
@ Â_{1,1} × Â_{p,q,r}
@ Â_{2×2} × Â_{p,q,r}
Â_{p+0,q+2,r}
@ Â_{0,2} × Â_{q,p,r}
@ Q × Â_{q,p,r}
For p³q we thus have Â_{p,q} @ (Â_{1,1})^{q} × Â_{p-q} @ (Â_{1,1})^{q} × (Â_{2})^{2l} × Â_{m,0,r} where p-q = 4l+m with mÎ {0,1,2,3} .
"Factoring" geometric algebras in this way does not make for compacter storage since we still have
N elements generating 2^{N} distinct products amd so requiring 2^{N} real coordinates
for the genreral real-weighted sum. We merely obtain more commutative multiplication tables
for the "multigrade basis" than for the 1-vectors one.
Complexification
For odd N, if the central pseudoscalar i has negative square (ie. if ½(N-1)N+q is odd)
we can express
any mutlivector a as b+ic for even b and c giving
Â_{p,q} @ (C_{p,q})_{+}
@ ((Â_{0,1})_{p,q})_{+}
, associating i with i.
The complex conjugation b-ic is provided by the main involuation ^{#}.
If ½(N-1)N+q is even
we have
Â_{p,q} @ ((Â_{1,0})_{p,q})_{+} .
Minimal Geometric Algebras
Our geometric vocabulary is somewhat broad with various product symbols ¿,.,Ù,_{°}...; operators
+,/,^{↑},...; conjugations ^{§},^{#},... ; and so forth. Which symbols are fundamental
in that they cannot be semantically defined using other fundamental symbols?
Let us first suppose just + and ¨ ; precedence indication brackets ( and ) ; orthonormal N-frame symbols {e_{1},e_{2},...e_{N}} ;
and some multivector "variable" symbols a, b, ... together with an assigment statement = .
Unity symbols 1 and (-1) (and hence the left and right negation conjugation operators - and ^{-} ) are available from {e_{1}e_{1} , e_{2}e_{2}, e_{1}e_{2}e_{1}e_{2}} depending on the signatures
and we obtain, semantically at least, all the integer-coordinated multivectors in the given frame.
Exponmentiation ^{↑} can be defined using + and ¨ given some form of infinitite summation
å_{} symbolism and
if we then suppose logarithm ^{↓} as a particular inverse of ^{↑} we obtain a^{-1} as (-(a^{↓}))^{↑}
which we can write as ^{-1} = ^{↓}^{-}^{↑} whence a/b º
a(b^{-1}) = a(b^{↓}^{-}^{↑}) and we attain the general real-coordinate multivector space.
[under construction]
Bivectors
Left multiplication by a bivector ( a ® b_{2}a ) casts scalars into b_{2}. If b_{2} is a 2-blade It rotates directions within b_{2} (by b_{2}),
while casting directions perpendicular to b_{2} into trivectors. For N=3, it casts bivectors in b_{2} to scalars
while bivectors normal to b_{2} are rotated by b_{2} and the pseudoscalar is cast to 1-vector b_{2}^{*}.
Right multiplication by a bivector has similar effects. Indeed a 2-blade commutes with all multivectors
in its dual space , which is not the case for a 1-vector. Consequently, it is occasionally advantageous
to represent 3D 1-vectors by their bivector duals.
Let b be multivector having only bivector and scalar components.
a¿b = a.b while
b¿a = b.a + b_{0}a
b_{2}×a sends ^(a,b_{2}) to 0 and rotates ¯(a,b_{2}) in b_{2} by ½p, scaling it by |b_{2}|.
The operation a ® (ba).a for nonnull b is interesting, casting ¯(a,b) to 0 and
^(a,b) into
|^(a,b)|^{2}b
= |aÙb|^{2}(b^{-2})b
= |aÙb^{~}|^{2}b
.
Expressed as sum of commuting 2-blades
Any Euclidean N-D 2-vector b_{2} can be represented as the scalar-weighted sum of at most ½N
orthogonal (by which we mean geometrically commuting) unit 2-blades. Such a decomposition is unique except in cases where
two or more wieghts are identical.
"Decomposing" a given bivector b_{2} in this way is nontrivial but a computational algorithm exists.
[
We seek 2-blades a_{1},A_{2}..,a_{m} with a_{i}a_{j}=a_{i}Ùa_{j}=a_{j}a_{i}
and b_{2}=a_{1}+A_{2}+..+a_{m} .
Now (b_{2}^{k})_{<2k>} = b_{2}Ù...Ùb_{2}
= k!S_{r<s<..<v}a_{r}Ùa_{s}..Ùa_{v}
= k!S_{r<s<..<v}
a_{r}a_{s}..a_{v}
where there are k terms in each product and k suffices r,s,..v .
Hence (b_{2}^{k-1})_{<2k-2>} ¿
(b_{2}^{k})_{<2k>} =
k!(k-1)! å_{i=1}^{m} a_{i} (
S_{r<s<..<u ¹ i}
a_{r}^{2}a_{s}^{2}..a_{u}^{2} )
for 1£k<m
which, if the scalar a_{i}^{2} are known and distinct, provides m linear ^{N}C_{2}-dimensional equations , solvable for a_{i}
by conventional numerical methods.
The scalar a_{i}^{2} can shown to be the m roots of the m^{th} order scalar polynomial
å_{k=0}^{m} (-l)^{m-k}((b_{2}^{k})_{<2k>} ¿
(b_{2}^{k})_{<2k>})=0 .
See Hestenes & Sobczyk (3-4) for a fuller treatment.
]
Spinors
Recall that for puresquare a, a^{↑} º e^{a} = | cos(|a|) + a^{~} sin(|a|) | if a^{2} < 0 ; |
cosh(|a|) + a^{~} sinh(|a|) | if a^{2} > 0 ; | |
1+a | if a^{2} = 0. |
Authors differ in precise meaning of the terms "spinor" and "rotor".
[
Hestenes & Sobczyk define a spinor as a multivector a satisfying
(aba^{§})_{<1>}=
aba^{§}
" 1-vector b, which is equivalent to a being an even versor
except when N=p+q divides 4, in which case we have
a=(a+bi)v where v is an even versor and a and b are scalars
]
Here we use k-rotor
to indicate a nonnull 2k-versor and pure/impure k-spinor to indicate an exponentiated unit or null pure/impure k-vector.
e^{bkf} for scalar f. Postive scalars are thus pure 0-spinors.
If b is a nonnull 2-versor, then so is e^{b}
[ cos(f)+ sin(f)e_{12} = e_{1}( cos(f)e_{1}+ sin(f)e_{2}) ]
. Since any 2-blade is a 2-versor,
any 2-spinor is a 1-rotor.
Under the geometric product, the pure spinors generate Â_{N}^{+}
which for N=3 is isomorphic to
quaternion space
(the 2-spinor (inf)^{↑} corresponding to normalised quaternion
{ cos(f/2), sin(f/2)n} ).
Spinor Adjustment Rule
Whenever i squares to -1 and commutes with a
we have the spinor adjustment rule
(1+a)e^{qi} = (1+a)e^{qai}
= (1+a)e^{-qa*} (when i=i)
provided a^{2}=1.
(1-a)e^{qi} = (1-a)e^{-qai}
= (1+a)e^{qa*}
provided a^{2}=-1 .
[ Proof :
(1+a)(e^{qi} - e^{qia})
= (1+a) sinq(i - ia)
= (1+a)(1-a)i sinq
= 0 for a^{2}=1
.]
In Â_{3} we have
(1+a)e^{qi}=(1+a)e^{qai}
which means that under a (1+a) mutiplier we can replace ai exponentiations
by i exponentiations, which tend to be more commutative.
Alternate representations of spinors
Physicists tend to represent spinors in convoluted and confusing ways that obsfucate more than they reveal.
In Quantum Mechanics , a "spinor" is typically defined via two complex numbers
a_{0}+b_{0}i = r_{0}e^{q0i} and
a_{1}+b_{1}i
= r_{1}e^{q1i}
= re^{fi} r_{0}e^{q1i} when r_{0}¹0
as the singular (zero determinant) 2×2 complex matrix
b= | æ | (a_{0}+ib_{0})(a_{1}+ib_{1}) | -(a_{0}+ib_{0})^{2} | ö | = | æ | r_{0}r_{1}e^{(q0+q1)i} | -r_{0}^{2} e^{2q0i} | ö | = | r_{0}^{2}e^{2q0i} | æ | re^{fi} | -1 | ö |
è | (a_{1}+ib_{1})^{2} | -(a_{0}+ib_{0})(a_{1}+ib_{1}) | ø | è | r_{1}^{2}e^{2q1i} | -r_{0}r_{1} e^{(q0+q1)i} | ø | è | r^{2}e^{2fi} | -r e^{fi} | ø |
b(b^{§})= | æ | (a_{0}+ib_{0})(a_{0}-ib_{0}) | (a_{0}+ib_{0})(a_{1}-ib_{1}) | ö | is equal to ½. |
è | (a_{0}-ib_{0})(a_{1}+ib_{1}) | (a_{1}+ib_{1})(a_{1}-ib_{1}) | ø |
Such 2×2 complex matrices are often represented via the s_{i} and 1 "Pauli" matrix representors for 1,e_{1},e_{2},e_{3} defined above using
æ | a | b | ö | = ½(a+d)1 + ½(b+c) s_{1} + ½i(b-c) s_{2} + ½(a-d) s_{3} |
è | c | d | ø |
We can associate i with i=e_{123} and
regard b as the null (b^{2}=0) Â_{3} multivector
b=
½r_{0}^{2}e^{2q0i}
(e_{1}(r^{2}e^{2fi}-1)
-e_{2}i(r^{2}e^{2fi}+1)
+ e_{3}2re^{fi} )
=
½r_{0}^{2}e^{2q0i}(r^{2}+1)(u+vi)
where
u = (e_{1}(r^{2} cos2f-1) + e_{2}r^{2} sin2f + e_{3}2r cosf)(r^{2}+1)^{-1} ;
v = (e_{1}r^{2} sin2f - e_{2}(r^{2} cos2f+1) +e_{3}2r sinf)(r^{2}+1)^{-1}
are orthonormal 1-vectors (u^{2}=v^{2}=1) ;
uv =
(r^{2}+1)^{-1}(e_{23} 2r cosf
+ e_{31} 2r sinf
+ e_{12} ( 1-r^{2}) ) .
Letting w=-u^{-1}v i= e_{1} 2r cosf
+ e_{2} 2r sinf
+ e_{3} (1-r^{2})
= (r^{2}+1)R_{iem}(re^{fi})
we have
b=
½r_{0}^{2}e^{2q0i}(r^{2}+1)u(1-w)
where w^{2}=1 .
Riemann Sphere Representation
In Riemann sphere representation of C + ¥
we map complex "point" z=a+bi=re^{fi} to the real 3D unit 1-vector
corresponding to the intersection of the line
joining Â^{3} point ae_{1}+be_{2} to the "south pole" -e_{3} with the unit sphere centred
at 0. This point is given in spherical polar coordinates as
R_{iem}(z) º [2 tan^{-1}(r),f,1]
= e^{fe12}) e^{-2( tan-1(r))e31} e_{3}
= (1+r^{2})^{-1}( e_{3}(1-r^{2}) + 2r( cosfe_{1} + sinfe_{2}) ) .
We further define
R_{iem}(¥)=-e_{3} .
We have R_{iem}(z^{-1}) = [ p-2 tan^{-1}(r),-f,1] ;
R_{iem}(-z) = [2 tan^{-1}(r),f+p,1] ;
R_{iem}(e^{fi}) = e_{1}e^{fe12} .
[ If we site the argand plane tangent at the "north pole" e_{3} rather that at the equator
we
have
R_{iem}(z)º[ 2 tan^{-1}(½r),f,1] ]
[ Proof :
u^{2} = (r^{2} cos2f-1)^{2} + r^{4} sin2f^{2} + 4r^{2} cosf^{2}
= r^{4} + 1 - 2r^{2} cos2f+ 4r^{2} cosf^{2}
= r^{4} - 1 - 2r^{2}( cosf^{2}- sinf^{2})+ 4r^{2} cosf^{2}
= r^{4} - 1 + 2r^{2}( cosf^{2}+ sinf^{2})
= (r^{2} + 1)^{2}
v ^{2} = r^{4} sin2f^{2} + (r^{2} cos2f+1)^{2} + 4r^{2} sinf^{2}
= r^{4} + 2r^{2} cos2f +1 + 4r^{2} sinf^{2}
= 1+r^{4} + 2r^{2}
u¿v=
(r^{2} cos2f-1)r^{2} sin2f
-r^{2} sin2f(r^{2} cos2f+1)
+2r^{2} cosf sinf
=
(r^{2} cos2f-1) sin2f
- sin2f(r^{2} cos2f+1)
+ sin2f
uv =
(e_{1}(r^{2} cos2f-1) + e_{2}r^{2} sin2f + e_{3}2r cosf)
(e_{1}r^{2} sin2f - e_{2}(r^{2} cos2f+1) +e_{3}2r sinf)
= e_{23} (2r_{3} sin2f sinf+2(r^{2} cos2f+1)r cosf)
+ e_{31} (2r_{3} sin2f cosf-2(r^{2} cos2f-1)r sinf)
+ e_{12} ( -(r^{2} cos2f-1)(r^{2} cos2f+1) - r^{4} sin2f^{2})
= e_{23} 2(r_{3}( sin2f sinf+ cos2f cosf)+r cosf)
+ e_{31} (2r_{3}( sin2f cosf- cos2f sinf)+r sinf)
+ e_{12} ( 1-r^{4})
= e_{23} 2r(r^{2}+1) cosf
+ e_{31} 2r(r^{2}+1) sinf
+ e_{12} ( 1-r^{4})
= (r^{2}+1)(e_{23} 2r cosf
+ e_{31} 2r sinf
+ e_{12} (1-r^{2}) )
.]
u = e_{1}(r^{2} cos2f-1) + e_{2}r^{2} sin2f + e_{3}2r cosf ;
v = e_{1}r^{2} sin2f - e_{2}(r^{2} cos2f+1) +e_{3}2r sinf
w = e_{1} 2r cosf + e_{2} 2r sinf + e_{3} (1-r^{2})
For small r, u » -e_{1} + e_{3}2r cosf ; v » - e_{2} +e_{3}2r sinf ; w » 2r(e_{1} cosf + e_{2} sinf + e_{3} (1-r^{2})
For large r,
u » r^{2}(e_{1} cos2f + e_{2} sin2f)
v » r^{2}(e_{1} sin2f - e_{2} cos2f)
w » -r^{2}e_{3}
.
Thus with r=0 we have u=e_{1}, v=-e_{2}, w=e_{3} . As r increases u aquires a cosf e_{3} component,
When r_{0}=0 we have
b= ½ (e_{1}r_{1}^{2}e^{2q1i} - e_{2}i(r_{1}^{2}e^{2q1i}) )
= ½ e^{2q1i} r_{1}^{2} (e_{1} - e_{31} )
= ½ r_{1}^{2} e^{2q1i} e_{1} (1 + e_{3} ) .
Let z = re^{fi} =r_{1}(r_{0}^{-1})e^{q1-q0} .
A' = W.A
where
W º 2R'R^{§} is a pure bivector known as the angular veclocity bivector.
[ Proof :
RR^{§}=1
Þ R'R^{§} + R(R^{§})' = 0
Þ R'R^{§} = -R(R^{§})' = -(R'R^{§})^{§}
Þ W pure.
A'
= (RA_{0}R^{§})'
= (R'A_{0}R^{§} + RA_{0}(R^{§})')
= (R'R^{§})A - A(R'R^{§})
= (R'R^{§})×A
= (2R'R^{§}).A
= W.A
.]
For N=3 letting w=W^{*} gives
A'
= W.A
= (w^{*}).A
= (wÙA)i
= w×A
so w is the conventional 3D angular velocity 1-vector.
2R'R^{§} = W Þ R solves
R' = ½WR .
Writing W_{L} = R^{§}WR
= 2R^{§}R'
for the local angular-velocity bivector
expressed in the frame of the object, we have
R' = ½RW_{L} .
These rotor equations are generally easier to solve than their matrix counterparts.
An aesthetic advantage in defining angular velocities and momenta
as bivectors rather than vectors is the avoidance of invoking a third dimension to "hold"
them in the case of otherwise planar kinematics.
Next : Conclusion