The zero blade
From this perspective, the zero blade 0=0 represents U^{N}, as does the pseudoscalar i.
This is unsatisfactory partly because 0 should not imply a particular dimension N. Bouma suggests
interpreting 0 instead as an indeterminate subspace. This is appropriate in a programming context,
since an absolute magnitude measure of a blade then becomes a measure of its "ambiguity",
with large values indicating robust exact representations, small values indicating possibly
inexact results, and zero indicating a total absence of subspace identification
rather than a particular subspace.
Lines and Planes
We now generalise the concepts of lines and planes.
Programmers traditionally represent 3D planes with a normal 1-vector
and either a scalar distance (aka. directance)
from a given origin point, or a particular "base" point in the plane. This approach fails to extend to higher
dimensions where there is no unique normal 1-vector; rather a normal (dual) (N-k)-blade.
A k-plane (aka a displaced subspace or a flat) is a set of the form
{ r : (r-a)Ùb_{k} = 0 }
= { r : rÙb_{k} = aÙb_{k} } where b_{k} is a k-blade.
In particular, a lies in the k-plane.
An (N-1)-plane is known as a hyperplane.
If b_{k} is invertible (ie. is nondegenerate), we can represent the k-plane by the
mixed-grade multivector
b_{k} + aÙb_{k}
= (1+a)Ùb_{k}
A 0-plane a is the singular pointset { a} .
A 1-plane b + aÙb is the line { r: r = a + lb}.
A 2-plane is the traditional plane.
We call b_{k} the tangent of the k-plane, and aÙb_{k} its moment.
We say the k-plane is parallel to b_{k}.
The dual of the k-plane equation is
(rÙb_{k})i^{-1} = (aÙb_{k})i^{-1}
Û r.(b_{k}i^{-1}) = a.(b_{k}i^{-1}).
Hence for k=2, N=3 the dual of the tangent is the conventional normal vector n while
the dual of the moment is the scalar directance a.n of
the conventional 2-plane representation { r : r.n = a.n }.
For N=3, we can accordingly encode lines by a vector and a pseudovector
instead of the usual two vectors; and planes by a pseudovector and a pseudoscalar
instead of the usual vector and scalar.
The k-plane (1+a)Ùb_{k} can be expressed as (1 + ^_{bk}(a))b_{k} ,
the 1-vector ^_{bk}(a) being the vector directance of the k-plane.
Since the representations of k-planes not containing the origin point 0 are not pure blades, the operators
¯, ^ ,È, Ç are undefined with respect to them. The higher dimensional embeddings discussed below
provide a solution to this.
Simplexes
Given a frame of k+1 vectors (a_{0},a_{1},a_{2},...a_{k}) where k<N,
their k-simplex is the convex hull defined by the points.
The 1-simplex for
(a_{0},a_{1}) , for example, is the line segment connecting a_{0} to a_{1}.
The 2-simplex for (a_{0},a_{1},a_{2}) is the flat triangular "surface element"
defined by them.
Writing a_{k} º (a_{1}-a_{0})Ù(a_{2}-a_{0})Ù...Ù(a_{k}-a_{0}) ;
where a_{0} is known as the base point, we note that the simplex
is contained within the k-plane
a_{k} + a_{0}Ùa_{k}
which we will refer to as the extended k-simplex.
a_{k} + a_{0}Ùa_{k}
has
moment a_{0}Ùa_{1}Ù...Ùa_{k} = a_{0}Ùa_{k}
which is nonzero only if all k+1 vectors are linearly independant.
The scalar k!^{-1} |a_{k}| is known as the content of
the k-simplex; where |a_{k}| º |a_{k}^{2}|^{½}
is welldefined for any pureblade a_{k}. For k=1 this is the length |a_{1}-a_{0}|; for k=2 it is the area ½|(a_{2}-a_{0})Ù(a_{1}-a_{0})|.
Simplexes provide the basic construction element for the multidimensional integration operations of Geoemetric calculus.
One of the fundamental limitations of multivectors is that while they can naturally be used to
concisely represent and manipulate extended k-simplexes (the plane containing a given three points for example)
and, as we will see below, spherical surfaces , with simple blades, they do not provide so ready a means of directly representing and manuipulating bounded k-simplexes
such as as the triangular "facet" defined by a particular three points or the line "segment" defined by two.
From a programming perspective, this is matter of some frustation.
Frames
The obvious way to represent a k-frame (a_{1},..,a_{k}) is as k 1-vectors, typically as the columns of a
N×k real matrix.
However alternate seldom discussed possibilities are the mixed multivectors
a_{k} = a_{1} + a_{1}Ùa_{2} + ... + a_{1}Ùa_{2}Ù..Ùa_{k}
or a_{1} + a_{1}a_{2} + ... + a_{1}a_{2}..a_{k} .
[ If we allow the operation _{<1>} (ie. taking only the 1-vector component) then we can clearly recover
a_{1}=a_{k}_{<1>} and thence a^{1}=a_{1}/(a_{1}^{2}) . a_{2} is then available as a^{1}¿a_{k},
a^{3} as a^{2}¿(a^{1}¿a_{k}), and so forth
]
Suppose we transform the frame as a_{i}'ºBa_{i}B^{#}^{-1} where
B^{#}^{-1}B = b .
Ba_{k}B^{#}^{-1} is not the correct representor for the transformed frame,
(Ba_{k}B^{#}^{-1})_{<i>} requiring scaling by b^{1-i} . However,
provided b>0 we can unambiguously renormalise the a_{i} while reconstructing them
from Ba_{k}B^{#}^{-1} so it is a representor for the transformed frame.
Higher Dimensional Embeddings
We can think of k-blades as representing k-dimensional subspaces in
Â^{N}. For N=3, these correspond to lines and planes through the
origin. We would like to be able to use the operators
¯, ^ ,È, Ç to manipulate general displaced subspaces of Â^{N}
(ie. ones not containing the origin).
One way of doing this is to "embedd" Â^{N} points in a higher dimensional
space via a function ¦ : Â^{N} ®
Â^{N+p,q,r} and work with the multivectors of that space.
With each point x in Â^{N} we associate a point x = ¦(x) in the higher space.
With each multivector in Â_{N} we associate the "same" multivector with regard to an
extended basis.
Multivectors in the higher space are associated with particular structures in Â^{N}
and we can investigate these structures by working with the multivectors.
We look here at three popular embeddings.
Homogeneous coordinates
We move from Â_{N} to Â_{N+1} by the incorportaion of a new basis vector
e_{0} with Sig(e_{0})=1.
The embedding is trivial: ¦(x) = x + e_{0} .
Some authors favour x ® e_{0}x, mapping 1-vectors to 2-blades (when e_{0}¿x=0), but we do not
take this approach here.
For N=3, our multivectors now require 16 coordinates.
Homogeneous coordinates are so called because they "desingularise" the origin point 0.
In a Euclidean space, all points save 0 have a geometric inverse. By displacing all points by 1-vector e_{0}
outside Â_{N} we ensure that all points are invertible.
k-planes
Consider the Â_{N+1} (k+1)-blade (a_{0}+e_{0})Ùa_{k} where a_{k} is tangent to the Â_{N} k-simplex for frame { a_{0},a_{1},..a_{k}}.
(x+e_{0}) Î (a_{0}+e_{0})Ùa_{k} Û (x+e_{0})Ù(a_{0}+e_{0})Ùa_{k} = 0
Û xÙa_{0}Ùa_{k} = e_{0}(x-a_{0})Ùa_{k}
Û (x-a_{0})Ùa_{k} = 0 which is the condition for membership in the k-plane
containing the k-simplex.
Hence the k-simplex of any k+1 points in Â^{N} corresponds to a
(k+1)-blade in Â_{N+1}. The simplex frame is not uniquely recoverable from the blade, only
the extended simplex and measure of the content.
In particular, the Â_{N} k-plane (1+x)Ùa_{k} is represented by Â_{N+1} (k+1)-blade (e_{0}+x)Ùa_{k} .
For example:
We can accordingly use joins and meets in Â_{N+1} to construct and intersect displaced subspaces in Â_{N}. The advantage of using joins rather than outer products to construct simplex representatives is that we will obtain appropriate results for degenerate simplexes. If p=q, for example, then (e_{0}+p)È(e_{0}+q)=e_{0}+p. If a,b and c are colinear, then (e_{0}+a)È((e_{0}+b)È(e_{0}+c)) represents the line containing them.
If a given Â_{N} k- and m-planes intersect then the meet of their Â_{N+1} representatives represents that intersection, and we can recover it as follows. We have an c Î Â_{N+1} which we wish to express as (e_{0}+c_{0})ÙC_{n} where c_{0} and C_{n} are in Â_{N}. If we have c expressed via coordinates with respect to the extended basis for { e_{0},e_{1},...,e_{N} } this is a trivial computation ( C_{n}_{[ij..m]} = c_{[0ij..m]} ).
Suppose the Â_{N} k- and m-planes with k+m=N-1 containing given simplexes do not intersect.
The corresponding Â_{N+1} blades ((a_{0}+e_{0})Ùa_{k}) and ((b_{0}+e_{0})Ùb_{l})
are then complimentary and have a scalar meet equal to
the directed perpendicular Euclidean distance between the Â_{N} planes.
[ Proof :
((a_{0}+e_{0})Ùa_{k})Ù((b_{0}+e_{0})Ùb_{l})
= a_{0}Ùa_{k}Ùb_{0}Ùb_{l}
+e_{0}Ù(a_{k}Ùb_{0}Ùb_{l}-a_{k}Ùa_{0}Ùb_{l})
= 0+e_{0}Ù(a_{k}Ùb_{0}Ùb_{l}-a_{k}Ùa_{0}Ùb_{l})
= e_{0}Ùa_{k}Ù(b_{0}-a_{0})Ùb_{l} .
.]
Let us review with regard to N=3. By the adoption of homogeneous coordinates, we can represent 3D points, lines, and planes by blades of Â_{4} which are implementable as 2^{4}=16 dimensional vectors. The use of a sixteen-real multivector to represent a 3D point may seem extravagent, but of course such a multivector has sparsity 13. Similarly, the multivector representations of lines and planes have sparsities 10 and 12 respectively. In compensation for this wastage we have the following advantages:
ab in e_{¥0} | |||||||
1 | e_{0} | e_{¥} | e_{¥0} | e_{+} | e_{-} | ||
1 | 1 | e_{0} | e_{¥} | e_{¥0} | e_{+} | e_{-} | |
b | e_{0} | e_{0} | 0 | -1+e_{¥0} | e_{0} | ½(-1+e_{¥0}) | ½(-1+e_{¥0}) |
e_{¥} | e_{¥} | -1-e_{¥0} | 0 | -e_{¥} | 1+e_{¥0} | -1-e_{¥0} | |
e_{¥0} | e_{¥0} | -e_{0} | e_{¥} | 1 | e_{-} | e_{+} | |
e_{+} | e_{+} | ½(-1-e_{¥0}) | 1-e_{¥0} | -e_{-} | 1 | -e_{¥0} | |
e_{-} | e_{-} | ½(-1-e_{¥0}) | -1+e_{¥0} | -e_{+} | e_{¥0} | -1 |
We will denote the "extended" Â_{N+1,1} dual in ie_{¥0} by a^{*} º ai^{-1}e_{¥0}
and the "unextended" Â_{N+1} dual in i by a^{*} º ai^{-1}.
e_{0}^{*} = e_{0}(e_{¥0}i)^{-1} = (e_{0}e_{¥0})i^{-1} = -e_{0}i^{-1} ;
e_{¥}^{*} = e_{¥}(e_{¥0}i)^{-1} = -e_{¥}i^{-1} so the extended and unextended duals of the nullextendors
differ only in sign. This runs counter to our experience of nonnull basies, which might lead us to expect
e_{0}^{*} to exclude e_{0} and include e_{¥}.
Geometric Interpretation Overview
We shall see below that for N=3, by the adoption of GHC we
can represent traditional 3D points, lines, planes, bipoints, circles, and spheres by pure blades in
Â_{4,1} which are implementable as 2^{5}=32 dimensional vectors. More generally we can represent N-D
k-spheres and k-planes by particular (k+2)-blades in U_{N}^{%} implimentable as 2^{N+2} dimensional 1-vectors.
We will associate
U^{N}^{%} point s=l(c+e_{0}+½e_{¥}(c^{2}±r^{2}) ) via its dual with
the U^{N} hyper(anti)sphere of squared radius s^{2}=±r^{2} and centre c
and can think of e_{¥} as providing a "squared radius coordinate".
Thus 1-curves in
U^{N}^{%} represent paths through the space of U^{N} hyperspheres and we can represent the trajectory of a hypersphere with
constant radius r as a U^{N}^{%} 1-curve confined to x^{2}=r^{2}.
If points a,b are distinct
then a¿b = (aÙb)^{2} = -½(a-b)^{2} and
aÙb represents the bipoint {a,b}
while e_{¥}ÙaÙb represents the extended line through a,b
; with (e_{¥}ÙaÙb)^{2} = (a-b)^{2} . Here a is the horoemebdding of
U^{N} point a into U^{N}^{%} described below.
If points a,b,c are noncolinear aÙbÙc represents the
1-sphere (ie. the circle) through a,b,c
with (aÙbÙc)^{2} = r^{2} 4 Area(a,b,c)^{2}
where r is the circle radius;
while
e_{¥}ÙaÙbÙc represents the 2-plane (ie. the plane) containing {a,b,c}
; with
(e_{¥}ÙaÙbÙc)^{2} = -2! Area(a,b,c)^{2} .
If points a,b,c,d are noncoplanar
aÙbÙcÙd represents the
2-sphere (ie. 3D sphere) through a,b,c,d, with
(aÙbÙcÙd)^{2}
= -r^{2} 3! Volume(a,b,c,d) ; while
e_{¥}ÙaÙbÙcÙd represents the 3-plane containing them
with
(e_{¥}ÙaÙbÙcÙd)^{2} = -3! Volume(a,b,c,d) .
These results generalise for higher N with the k-blade outter product of k£N+1 embedded points representing the (k-2)-sphere through those points and having magnitude r (k-1)! V where r is the radius of the (k-2)-sphere and V is the "volume" content of the (k-1)-simplex formed by the points. And the (k+1)-blade formed by wedging this with e_{¥} represents the (k+1)-plane containing the points, with magntitude (k-1)! V. What could be more useful?
We can associate more general U_{N}^{%} k-blade s_{1}Ùs_{2}..Ùs_{k} via its (N+2-k)-blade dual (s_{1}Ùs_{2}..Ùs_{k})^{*} with the space of all planes and spheres that include the (N-k)-sphere s_{1}^{*}Çs_{2}^{*}...Çs_{k}^{*} .
A Â_{N+1,1} (k+1)-blade a_{k+1} = a_{0}Ùa_{1}Ù...Ùa_{k} derived from Â^{N} points a_{0},a_{1},..,a_{k} expands as a_{0}Ùa_{1}Ù...Ùa_{k} = a_{0}Ùa_{1}Ù...a_{k} + e_{0}Ù(a_{1}-a_{0})Ù(a_{2}-a_{0})Ù..Ù(a_{k}-a_{0}) + _{O}(terms in e_{¥} and e_{¥0})
This all constitutes a magnificent "payback" for investing in two additional dimensions and GHC blades provide an excellent way to represent extended lines and planes and circles and k-spherical surfaces, but they are less accomodating when it comes to representing finite line segments and the interiors of k-spheres. Informally, they embody boundaries rather than the regions bound. Not all Â_{N+1,1} blades are of this type, others represent not k-planes or k-spheres but more general "families" of hyperspheres.
We can form an m-blade meet a_{k}Çb_{l} of (k-2)-sphere a_{k} and (l-2)-sphere b_{l} in the usual way, imposing Euclidean signatures if desired. Any null 1-vector x within their meet corresponds to either a U^{N} point in their geometric intersction, or to ¥x when e^{0}¿x=0 . A nonnull 1-vector will be dual to a (possibly anti) hypersphere whose centre will have geometric relevance to the nature of their nonintersection.
Suppose for example that we have two 3-horoblades in Â^{4,1} each representing either a circle or an infinite lines in Â^{3}. Two 3-blades must intersect in a 5D space so the meet of the two blades will be a proper blade, ie. have grade ³ 1. If the meet is a null 1-vector, then the two objects intersect in Â^{3} at a single point. If the meet is a nonnull 1-vector m then the circles do not intersect and the meet is dual to a hypersphere that intersects with both, the sign of m^{2} indicating whether the line passes through the inside (negative) or outside (postive) of the circle as it crosses its tangent plane.
Hence the meet may not be scalar when the Â^{N} structures do not intersect, representing "imaginary" hyper(anti)spherical solutions. The nature of an intersection in Â^{N} is indicated by the grade of the delta product and the sign of the square of the meet, which is often most directly computaable via the square of the delta product. When the meet is scalar valued, it provides a measure of the seperation such as when providing the squared radius of a hypersphere tangent to both. When the meet is proper-blade valued, it either geometrically represents the Â^{N} intersection or an "imaginary intersection" embodying information pertaining to the geometry of the Â^{N} nonintersection, eg. providing a seperating 2-plane. The meet Ç acts precisely as programmers might wish it to, essentially providing the biggest possible seperator between a given two nonintersecting pointsets.
We will now substantiate and justify the associations summarised above, a process which will of necessity be somewhat mathematically intensive.
The reader should remember that when we speak of associating a particular Â_{N}^{%}
or U_{N}^{%} (k+2)-blade a_{k+2} with a particular
Â_{N} or U^{N} pointset such as a k-sphere, we should more correctly speak
of associating the intersection of a_{k+2} with the horosphere
since a_{k+2} actually represents a set of hyperspheres in Â^{N} or U^{N}.
The horosphere point embedding
We are interested in the point embedding taking Â^{N} to the subspace of Â^{N+1,1} defined by
H^{N}_{e¥} = { r Î Â^{N+1,1} : r^{2} = 0 ; e^{0}¿r = 1 }.
This is the horosphere, the intersection of a hyperplane normal to e_{¥} containing e_{0},
and the null cone { x Î Â^{N+1,1} : x^{2} = 0 }.
x = x + ae_{0} + be_{¥} is null
Û x^{2} = 2ab so
H^{N}_{e¥} = { x + ½x^{2}e_{¥} + e_{0} : x Î Â^{N} } .
Our embeding is accordingly
x = f_{0}(x)
º f_{0}_{e¥}(x) º x + e_{0} + ½x^{2}e_{¥}
=
x + ½e_{-}(x^{2}+1) + ½e_{+}(x^{2}-1)
though some authors favour
lx + l^{2}e_{0} + ½x^{2}e_{¥}
= lx + ½e_{-}(x^{2}+l^{2}) + ½e_{+}(x^{2}-l^{2})
with l a unit length in order to homogenise the "dimension" or "units" of x .
We will refer to such null x of unit e_{0} coordinate (-e_{¥}¿x=1)
as the horosphere embedding or horoembedding of
x = ¯_{(e¥0*)}(x).
In particular,
e_{0} corresponds to the Â^{N} origin 0 while e_{¥} represents a hypothetical Â^{N} point at infinity which we will denote ¥.
When x^{2}=1 this coincides with the spherical conformal embedding x = x+e_{0}+½e_{¥} = x+e_{-} .
Some authors consider an embedding Â^{N} ® Â_{N+1,1}
defined by x ® e_{¥0}x, mapping 1-vectors to 3-blades. We do not take this
approach here, although similar correspondances arise as consequences of the
point embedding since we associate the horosphere intersection of e_{¥0}x=e_{¥}Ùe_{0}Ùx with the 1-plane (line)
through 0 and x. The 3-blade e_{¥0}x also spans U^{N}^{%} points of the form
e_{0}+l(x-e_{0})
= (1-l)((1-l)^{-1}x + e_{0})
= (1-l)(((1-l)^{-1}x)' - ½(1-l)^{-2}x^{2}e_{¥})
dual to hyperspheres of centre (1-l)^{-1}x and radius (1-l)^{-1}|x|,
ie. hyperspheres that contain (pass through) 0 with centre on the line through 0 and x.
In Homogenous coordinates, we "de-specified" the origin by replacing 0 with 0^{2}=0 by e_{0} with e_{0}^{2}=1, however (when extending a Euclidean space) the embedded origin remains a unique point in that it is to sole point minimally attaining the inequality x^{2}³1 so one might more properly refer to "quasi-homogenised coordinates". With GHC the embedded origin e_{0} is truly equivalent to every other embedded point in that all nontrivially satisfy x^{2}=0, and in this sence GHC can be regarded as a "true" homogenisation.
By embedding into a nullcone we arrange that x is noninvertible even when x^{-1} exists. Although "inverting a point" seldom has a physical interpretation, we will see that self-inverse 2-blade flat point e_{¥}Ù(x+e_{0})=e_{¥}Ùx also represents the Â^{N} point x but in a slightly different way.
Inverse Point Embedding
The simplest inverse mapping
f_{0}^{-1}(x) º (e^{0}¿x)^{-1} ^_{}(x, e_{¥0})
= -( e_{¥}¿x)^{-1}(xÙe_{¥0})e_{¥0}
= (e_{¥}¿x)^{-1}(e_{¥0}Ùx)e_{¥0} taking U^{N}^{%} into U^{N}
is unsatisfactory because f_{0}^{-1}(e_{¥}) is undefined and we wish to define f_{0}^{-1}
fully over U^{N}^{%} rather than merely over H^{N}_{e¥}.
Thus we seek a
f_{0}^{-1} : U^{N}^{%} ® U^{N} È ¥ with f_{0}^{-1}(e_{¥})=¥.
Some authors favour f_{0}^{-1}(x) =
e_{+}¿(xÙe_{¥})
= -e_{-}¿(xÙe_{¥})
= e_{0}¿(xÙe_{¥})
.e_{+}
= (xe_{¥} - e_{¥0}).e_{+} = x + e_{-} but we will take
f_{0}^{-1} : U^{N}^{%} ® U^{N}^{%} defined by f_{0}^{-1}(x) º e^{¥}¿(e_{¥}Ù((e^{0}¿x)^{-1}x) - e_{¥0})
which returns the horosphere to the U^{N}^{%} subspace { x : x¿e_{¥0} = 0 } .
Clearly f_{0}^{-1}f_{0} is an identity mapping 1 : U^{N}®U^{N}
but f_{0}f_{0}^{-1} : U^{N}^{%} ® H^{N}_{e¥} is a many-to-one "projection" into the horosphere
dependant on our choice of f_{0}^{-1}.
All f_{0}^{-1} coincide over H^{N}_{e¥}, however, and so given any g U^{N} ® H^{N}_{e¥} we can unambigusously define
f_{0}^{-1}g U^{N}^{%} ® U^{N} .
Embedded Products
Note that (x+e_{0}+le_{¥})^{2} = x^{2} - 2l.
Note also the scaled idempotent 2-versors
(e_{0}x)^{2} = (-x^{2})e_{0}x
and
(xe_{0})^{2} = (-x^{2})xe_{0} with similar reslts for e_{¥}x and xe_{¥}.
Further,
e_{0}xe_{¥}x
= -e_{0}(x^{2}e_{¥} + 2x) has
(e_{0}xe_{¥}x)^{2}
= e_{0}(x^{2}e_{¥} + 2x)e_{0}(x^{2}e_{¥} + 2x)
= e_{0}(x^{2}e_{¥})e_{0}(x^{2}e_{¥} + 2x)
= -2x^{2} (e_{0}xe_{¥}x)
while
(e_{0}xe_{¥}x)^{§}
(e_{0}xe_{¥}x) =
(e_{0}xe_{¥}x)
(e_{0}xe_{¥}x)^{§} = 0.
Let a,b be the null Â_{N+1,1} 1-vectors associated with Â^{N} points a and b.
ab =
-½(a-b)^{2} + aÙb +½(a^{2}b-b^{2}a)e_{¥}
+ (a-b)e_{0} - ½(a^{2}-b^{2})e_{¥0} .
[ Proof :
(a + ½a^{2}e_{¥} + e_{0})(b + ½b^{2}e_{¥} + e_{0})
= ab - ½a^{2}be_{¥} - be_{0}
+ ½ab^{2}e_{¥} + ½b^{2}e_{0}e_{¥}
+ ae_{0} + ½a^{2}e_{¥}e_{0}
=
a¿b + aÙb - be_{0}
+ ½(ab^{2}-a^{2}b)e_{¥} + ½b^{2}(-1-e_{¥0})
+ ae_{0} + ½a^{2}(-1+e_{¥0})
=
-½(a-b)^{2} + aÙb +½(a^{2}b-b^{2}a)e_{¥}
+ (a-b)e_{0} - ½(a^{2}-b^{2})e_{¥0} .
.]
Hence
a¿b =a.b = -½(a-b)^{2} = -½(a-b)^{2}
and so for nullseperated points a and b ab=-ba=aÙb is a null 2-blade.
Also (aÙb)^{2} = (a¿b)^{2} = 4^{-1}(a-b)^{4}
.
The 1-vector linear combination l_{1}a+l_{2}b represents a point
(ie. is null) only if (a-b)^{2}=0 .
aÙb =
aÙb +½(a^{2}b-b^{2}a)e_{¥}
+ (a-b)e_{0} - ½(a^{2}-b^{2})e_{¥0}.
For N=3, our multivectors now require 2^{5}=32 coordinates but are frequently sparse.
An embedded point has just five nonzero coordinates, for example.
Dropping to the horosphere
We can express a general 1-vector y Î Â^{N+1,1} as
x + ae_{0} + be_{¥} where x Î Â^{N} but this appears less useful than
the
decomposition a_{e¥}x_{e¥} + b_{e¥}e_{¥} where x_{e¥} lies in the horosphere (ie. x_{e¥}^{2}=0 and e^{0}¿x_{e¥}=1) ;
achieved by a_{e¥} = e^{0}¿y = -e_{¥}¿y ;
b_{e¥} = ½(e_{¥}¿y)^{-1}y^{2} ; and
x_{e¥} = - ½(e^{0}¿y)^{-2} ye_{¥}y
corresponding to reflecting e_{¥} in y and rescaling.
More generally, for a given unit or null 1-vector e in U^{N}^{%} we can define
null horodrop along e or e-horodrop of y denoted D_{e}(y)
by removing just enough e from y to make it null.
If this has nonzero e_{0} coordinate then we can rescale to give the normalised e-horodrop
D_{e}^{~}(y).
with unit e_{0} coordinate.
For e=e_{¥} we can think of moving from scaled hyper(anti)sphere dual
y=l(c+e_{0}+½(c^{2}±r^{2})e_{¥})
to the embedded centre point c as a natural definition of f_{0}^{-1} f_{0}.
This fails for y=x+be_{¥}
where x Î U^{N} and so e^{0}¿y=0, which we can intrepret as representing a particular "directed infinity"
¥x .
Thus we define the centre of U_{N}^{%} 1-blade s by
Ú(s) = s_{[e0]} + ½s_{[e0]}^{2}e_{¥} where
s_{[e0]} º (e^{0}¿s)^{-1} s when
(e^{0}¿s) ¹ 0 , and e_{¥} else is s reccaled for unit e_{0} coordinate when possible.
Unlike the disembedded centre ¯_{I}(s_{[e0]})
For general null e
then provided e¿y ¹ 0 we can
remove ½(e¿y)^{-1}y^{2} e
and rescale for unit e_{0} coordinate equivalent to yey rescaled.
For general unit e with (e¿y)^{2} ³ e^{2}y^{2}
we can remove
e^{2}(-e¿y ±
((e¿y)^{2}-e^{2}y^{2})^{½}) e
and reach H^{N}_{e¥} at two alternative nullvectors having opposite e coordinate and typically
corresponding
to x^{4} greater and less than 1.
If we obtain unrescalable l(x+be_{¥}) we interpret it as ¥x .
Clearly D_{e¥}(y)
= (e^{0}¿y) f_{0}((e^{0}¿y)^{-1} ^_{e¥0}(y))
= (e^{0}¿y) f_{0}f_{0}^{-1}(y)
if y has nonzero e_{0} coordinate. f_{0}^{-1} D_{e¥} for y with nonzero e_{0} coordinate thus
trivially consists of dividing the U^{N} component by the e_{0} coordinate, independant of the e_{¥} coordinate.
The question arises of how much
e must be added to a null x to make it unit, ie. we
seek b so that the e-horolift L_{e}^{±}(x) = x + be has
L_{e}^{±}(x) ^{2} = ±1.
For e=e_{¥} we solve (x+be_{¥})^{2} = ±1 with
b = -/+½(e^{0}¿x)^{-1} corresponding
(for "normalised" null x with e^{0}¿x=1) to obtain
L_{e¥}^{±}(x) =
x + e_{0} + ½(x^{2} -/+ 1)e_{¥}
= x + ½(e_{-}-e_{+}) -/+ ½(e_{-}+e_{+}) + ½x^{2}e_{¥}
= x -/+ e_{±} + ½x^{2}e_{¥} .
which we can think of as moving from scaled null point embedding
lx = lf_{0}(x)
to the (dual of) the unit-radius (anti)hypersphere with centre x.
Hyperspheres
The non degenerate N-D sphere (aka. hypersphere) { x : (x-c)^{2} = r^{2} }
where r>0
corresponds to the Â_{N+1,1} equation
x¿c = -½ r^{2}.
Writing
s = c-½r^{2}e_{¥}
= c + e_{0} +½(c^{2}-r^{2})e_{¥}
= c + ½(1+c^{2}-r^{2})e_{-}
+ ½(-1+c^{2}-r^{2})e_{+}
we have
s^{2} = r^{2} ;
e^{0}¿s=1
, and x¿s=0 Û x lies on the hypersphere.
[ Proof : x¿s = x¿(c-½r^{2}e_{¥})
= -½ r^{2} - ½r^{2}x¿e_{¥}
= 0.
.]
So for any plussquare s Î Â^{N+1,1} satisfying e^{0}¿s = 1
the solution set
{ x Î H^{N}_{e¥} : x¿s = 0 } =
{ x Î H^{N}_{e¥} : xÙ(s^{*}) = 0 }
corresponds to a hypersphere (ie. a spherical surface) in Â^{N} having centre
c = ^_{e¥0}(s) and radius r=(s^{2})^{½} ;
and we can associate any hyperblade
s^{*}=se_{¥0}i^{-1}
with an N-D hypersphere provided e^{0}¿s¹1
and s^{2}>0.
In particular, (e_{0}-½e_{¥})^{*}=-e_{+}^{*} represents the unit sphere at 0.
Conversely, any nondegenerate hypersphere in Â^{N} corrseponds to a solution set
{ x Î H^{N}_{e¥} : x¿s = 0 } where
s=c+e_{0} +½(c^{2}-r^{2})e_{¥} satisfies
s^{2}=r^{2} ; e_{¥}¿s = -1.
Further, s = a¿(cÙe_{¥})
where a is any point on the hypersphere.
Whence s^{*} = aÙ(cÙe_{¥})^{*} .
[ Proof :
Setting a=c+rb^{~} for arbitary unit b^{~} we have
a¿(cÙe_{¥}) = (c+rb^{~} + e_{0}
+ ½(c^{2}+2rc¿b^{~} + r^{2})e_{¥})
¿(ce_{¥}-e_{¥0})
= c^{2}e_{¥} + r(b^{~}¿c)e_{¥}+c
+ e_{0}
- ½(c^{2}+2rc¿b^{~} + r^{2})e_{¥}
= c+e_{0}+½(c^{2}-r^{2})e_{¥}
.]
The U^{N} centre c can be recovered from s as ¯_{I}((e^{0}¿_ve2(s))^{-1}s) = (e^{0}¿_ve2(s))^{-1} (s¿I)I^{-1} corresponding to recaling for unit e_{0} coordinate and then discarding the e_{0} and e_{¥} (or e_{+} and e_{-}) coordinates.
To what does the horosphere solution set for x¿s = 0 correspond if
s^{2}>0 but e^{0}¿s=0 rather than 1?
s = ^_{e¥0}(s) + ¯_{e¥0}(s)
= s+(s¿e^{¥})e_{¥} when e^{0}¿s=0 so
s = |s|(n-(s¿e_{0}/|s|)e_{¥}) where n is a unit vector
in Â^{N} and |s|=(s^{2})^{½}=(s^{2})^{½}.
Thus x¿s=0
Û x¿(n-(s¿e_{0}/|s|)e_{¥}) = 0
Û x¿n - e_{0}¿(s¿e_{0}/|s|)e_{¥} = 0
Û x¿n + (s¿e_{0}/|s|) = 0
Û x¿n + d = 0
where d = (s¿e_{0}/|s|) = s^{~}¿e_{0}.
This defines a Â^{N} hyperplane having normal s and
directance -ds.
Since e_{¥}¿s = 0, e_{¥} lies in the solution set, corrseponding to a
"point at infinity" ¥
attained by the hyperplane. We can accordingly view a hyperplane as a sphere
(of infinite radius) that passes through ¥.
We will refer to s^{2} = r^{2} , the squared radius of the U^{N} hypersphere to which s is dual,
as the squadius of plussquare U^{N}^{%} point s. For s^{2}<0 we have a negative squadius
corresponding to the squared radius of an antihypersphere ( { x : x^{2}=-r^{2} } ) in U^{N}.
The dual equation to x¿s = 0 is
xÙ(s^{*}) = 0
where s^{*} is a pseudovector in Â_{N+1,1}.
In particular, consider a nondegenerate (N+1)-blade
s^{*}=a_{0}Ùa_{1}Ù....Ùa_{N} .
We have s¿e_{¥} = 0 Û (e_{¥}Ùs^{*})^{2} = 0
so if e_{¥}Ùs^{*} ¹ 0 the blade represents a
sphere in Â^{N} with x lieing on the sphere iff
xÙs^{*}=0.
In particular, a_{i}Ùs^{*}=0 for i=0,1,..N so
a_{0},a_{1},...a_{N} lie on the sphere.
[ Proof : (s¿e_{¥})^{2} - (sÙe_{¥})^{2}
= (s¿e_{¥} + sÙe_{¥})(s¿e_{¥} - sÙe_{¥})
= (se_{¥})(e_{¥}s)
= s(e_{¥}e_{¥})s) = 0 .]
If e_{¥}Ùs^{*} = 0 the blade represents a
hyperplane in Â^{N} which can similarly be shown to contain
a_{0},a_{1},...a_{N}.
We can thus represent the hyperplane containing a_{1},a_{2},...a_{N}
by the blade e_{¥}Ùa_{1}Ù...Ùa_{N}.
The Power Distance
If s_{1}^{*} and s_{2}^{*} represent hyperspheres then s_{1}¿s_{2} = ½(r_{1}^{2}+r_{2}^{2} - (c_{1}-c_{2})^{2} ) provides a natural metric on spherespace sometimes refered to outside the GA literature (without the ½ factor) as the power distance [ Amenta ]. This apparently disappointing inner product (we might instead have wished for the closest distance between the spheres) has a definite geometric interpretation and has significant utility. If a is a point common to both hyperspheres then the cosine of the angle subtended by the "radial normals" (a-c_{1}) and (a-c_{2}) is given (via the traditional triangular cosine rule) as ½(r_{1}^{2} +r_{2}^{2}-(c_{1}-c_{2})^{2})(r_{1}r_{2})^{-1} . Thus s_{1}¿s_{2} is r_{1}r_{2} times the cosine of the angle subtended by the radial normals at any common point. This result also holds when one of s_{1} or s_{2} represents a hyperplane.
If two sepeperate real hyperspheres do not interesect with |c_{1}-c_{2}| > r_{1}+r_{2} then s_{1}¿s_{2} is -½ times the square of the length of a tangent to s_{2}^{*} from the meet of a tangent from c_{2} to s_{1}^{*} with s_{1}^{*}. (c_{1}-c_{2})^{2} - r_{1}^{2} - r_{2}^{2} is the average of the squared common external and internal tangent lengths (c_{1}-c_{2})^{2} - (r_{1}-r_{2})^{2} and (c_{1}-c_{2})^{2} - (r_{1}+r_{2})^{2} .
(N-1)-spheres s_{1}^{*} and s_{2}^{*} are considered orthogonal if s_{1}¿|s_{2} = 0 , ie. Û
r_{1}^{2}+r_{2}^{2} = (c_{1}-c_{2})^{2} .
For real spheres in a Euclidean space with postive r_{1}^{2} and r_{2}^{2}, this coincides with the spheres intersecting at right angles;
When r_{1}^{2} < 0, for othogonality we require r_{2}^{2} ³ |r_{1}^{2}| and
c_{1} placed inside s_{2}^{*} such that the intersection
of real sphere s_{1}'^{*} centered at c_{1} with radius |r_{1}^{2}|^{½} is a great circle of s_{1}'^{*} .
When r_{2}^{2} < |r_{1}^{2}| and r_{1}^{2} < 0 then hypersheres s_{1}^{*} and s_{2}^{*} in Â^{N} cannot be orthogonal.
s_{1}Ùs_{2}
= c_{1}Ùc_{2} + e_{0}d + ½e_{¥}c + ½e_{¥0}( c_{1}^{2}-c_{2}^{2}+r_{2}^{2}-r_{1}^{2})
where c º (c_{1}^{2}-r_{1}^{2})c_{2} - (c_{2}^{2}-r_{2}^{2})c_{1}
and d º c_{2}-c_{1}.
(s_{1}Ùs_{2})^{2}
= (s_{1}¿s_{2})^{2} - r_{1}^{2}r_{2}^{2}
= ¼((r_{1}+r_{2})^{2}-(c_{1}-c_{2})^{2})((r_{1}-r_{2})^{2}-(c_{1}-c_{2})^{2})
= ¼( (r_{1}^{2}-r_{2}^{2})^{2} - 2(c_{1}-c_{2})^{2}(r_{1}^{2}+r_{2}^{2})
+ (c_{1}-c_{2})^{4} )
.
[ Proof :
(s_{1}Ùs_{2})^{2}
= (c_{1}Ùc_{2})^{2} + c¿d + ¼(c_{1}^{2}-c_{2}^{2}+r_{2}^{2}-r_{1}^{2})^{2}
Set c_{1}=0
so that c=-r_{1}^{2}c_{2} and d=c_{2} .
We then have
(s_{1}Ùs_{2})^{2} =
-r_{1}^{2}c_{2}^{2} + ¼(c_{2}^{2}-r_{2}^{2}+r_{1}^{2})^{2}
=
¼c_{2}^{4} + ¼(r_{1}^{2}-r_{2}^{2})^{2} +
-½c_{2}^{2}(r_{1}^{2}+r_{2}^{2}) .
Replacing c_{2} with c_{2}-c_{1} gives
¼( (r_{1}^{2}-r_{2}^{2})^{2} - 2(c_{1}-c_{2})^{2}(r_{1}^{2}+r_{2}^{2})
+ (c_{1}-c_{2})^{4} )
which rearranges as
¼((r_{1}^{2}+r_{2}^{2})^{2} - 2(c_{1}-c_{2})^{2}(r_{1}^{2}+r_{2}^{2})
+ (c_{1}-c_{2})^{4}
- 4r_{1}^{2}r_{2}^{2})
= (½(r_{1}^{2}+r_{2}^{2} - (c_{1}-c_{2})^{2} ))^{2} - r_{1}^{2}r_{2}^{2}
.]
When c_{1}=a+b , c_{2}=a-b, this is (a+b)Ù(a-b) - 2e_{0}b + ½e_{¥}
( ((a+b)^{2}-r_{1}^{2})(a-b) - ((a-b)^{2}-r_{2}^{2})(a+b) )
+ ½e_{¥0}( (a+b)^{2}-(a-b)^{2}+r_{2}^{2}-r_{1}^{2})
and when r_{1} = r_{2} = 0 we have the 0-sphere
2(bÙa - e_{0}b + ½e_{¥}(aba -b^{3}) + e_{¥0}(a¿b) )
representing the bipoint a±b .
[ Proof :
(a+b)Ù(a-b) + 2e_{0}b + ½e_{¥}
( (a+b)^{2}(a-b) - (a+b)(a-b)^{2} )
+ ½e_{¥0}( (a+b)^{2}-(a-b)^{2})
=
2bÙa + 2e_{0}b + ½e_{¥}(
( (a+b)(a+b-(a-b))(a-b)
+ ½e_{¥0}( 4 a¿b)
=
2(bÙa - e_{0}b + ½e_{¥}(a+b)b(a-b) + e_{¥0}(a¿b) )
= 2(bÙa - e_{0}b + ½e_{¥}(aba -b^{3}) + e_{¥0}(a¿b) )
.]
When s_{1}=x and r_{1}=0 we have
x¿s_{2} = ½(r_{2}^{2}-(c_{2}-x)^{2}) which is positive only when x lies outside
s_{2}^{*} and zero only (for Euclidean U^{N}) when it lies on it. Also (xÙs_{2})^{2}=
¼(r_{2}^{2}-(c_{2}-x)^{2})^{2} is zero if x lies in hypersphere
s_{2}^{*}
.
More generally, the sign of (s_{1}Ùs_{2})^{2} indicates whether s_{1}^{*} intersects s_{2}^{*}.
Negative implies an intersection while zero implies tangential contact.
s+le_{¥} is dual to a hypersphere with centre c and squadius r^{2}-2l.
s+le_{0} = (1+l)( (1+l)^{-1}c + e_{0}
+ ½e_{¥}(1+l)^{-1}(c^{2}-r^{2})) is dual to
a hypersphere with centre (1+l)^{-1}c and squadius
(1+l)^{-2}(r^{2}-lc^{2}).
ls_{1} + ms_{2} is dual to a sphere of center (_lamba+m)^{-1}(ls_{1}+ms_{2}) and squadius
(l+m)^{-2}( ls_{1} + ms_{2})^{2} =
(l+m)^{-2}(l^{2}r_{1}^{2} + m^{2}r_{2}^{2}
+lm(r_{1}^{2}+r_{2}^{2} - (c_{1}-c_{2})^{2} ) )
..
In particular ½(p_{1}+p_{2}) has square -¼(c_{1}-c_{2})^{2} and so is dual to a sphere passing through the two points centered on their midpoint.
.
We saw that the meet Ç can be evaluated with forced Eucliden signatures and when interscting blades in
Â_{p,q}^{%} we can stay in
Â_{p,q}^{%} or "lift" into
Â_{p+q,0}^{%} or Â_{p+q+2,0}
according to choice.
The point x=X+x^{N}e_{N} is represented as x=x+e_{0}+½(X^{2}-(x^{N})^{2}) in
Â_{N,1}^{%}
which represents the dual of a hypersphere centre x and radius 2^{½}x^{N} .
If x^{2}<0 this sphere encloses 0, if x^{2}=0 it includes 0.
The Â^{N,1} squaredlength x^{2} represents the Â^{N+1,0} squeperation
of 0 from the spherical surface.
k-planes
Let a_{0},a_{1},..a_{k} be k+1 1-vector horopoints
in Â_{N+1,1} corresponding to points a_{0},a_{1},s..,a_{k} in Â^{N}
and set Â^{N} k-blade
a_{k} º (a_{1}-a_{0})Ù(a_{2}-a_{0})Ù...Ù(a_{k}-a_{0}) .
We have
e_{¥}Ùa_{0}Ùa_{1}Ù...Ùa_{k}
= e_{¥}Ù(a_{0} + e_{0})Ùa_{k}
.
[ Proof :
e_{¥}Ù(a_{0}+½a_{0}^{2}e_{¥}+e_{0})Ùa_{1}Ù...Ùa_{k}
= (e_{¥}Ùa_{0}+e_{¥0})Ùa_{1}Ù...Ùa_{k}
= (e_{¥}a_{0}+e_{¥0})Ù(a_{1}+½a_{1}^{2}e_{¥}+e_{0})Ùa_{3}Ù...Ùa_{k}
= (e_{¥}a_{0}+e_{¥0})Ù(a_{1}+e_{0})Ùa_{3}Ù...Ùa_{k}
= ((e_{¥}a_{0})Ùa_{1}+e_{¥0}Ùa_{1}+e_{¥}Ùa_{1}Ùe_{0})Ùa_{3}Ù...Ùa_{k}
= (e_{¥}a_{0}Ùa_{1}+e_{¥0}Ù(a_{1}-a_{0}))Ùa_{3}Ù...Ùa_{k}
= ...
= (e_{¥}Ùa_{0}Ù...Ùa_{k} + e_{¥0}Ù(a_{1}-a_{0})Ù...Ù(a_{k}-a_{0}))
= e_{¥}Ù(a_{0}Ù(a_{1}-a_{0})Ù...Ù(a_{k}-a_{0}) + e_{0}Ù(a_{1}-a_{0})Ù...Ù(a_{k}-a_{0}))
.]
Also, (e_{¥}Ùa_{0}Ùa_{1}Ù...Ùa_{k})^{2}
= a_{k}^{2}
= (-1)^{½k(k-1)}(k!V_{k})^{2}
where V_{k} is the volume (content) of the k-simplex.
[ Proof :
For a_{0}=0 we trivially have
(e_{¥}Ùe_{0}Ùa_{k})^{2} = (-e_{¥0}a_{k})^{2} = a_{k}^{2}.
More generally
(e_{¥}Ù(e_{0}+a_{0})Ùa_{k})^{2} = (½e_{¥}(a_{0}Ùa_{k})-e_{¥0}a_{k})^{2}
= (-e_{¥0}(½e_{¥}(a_{0}Ùa_{k})+a_{k})^{2}
= (½e_{¥}(a_{0}Ùa_{k})+a_{k})^{2}
= (½(e_{¥}(a_{0}Ùa_{k})a_{k} + a_{k}e_{¥}(a_{0}Ùa_{k}))+a_{k}^{2})
= (½e_{¥}((a_{0}Ùa_{k})a_{k} + a_{k}^{#}(a_{0}Ùa_{k}))+a_{k}^{2})
= a_{k}^{2} since we know result is scalar
.]
Now, x + e_{0} + ½ x^{2}e_{¥} Î
e_{¥}Ù(a_{0} + e_{0})Ùa_{k}
Û (x-a_{0})Ùa_{k} = 0 . Thus
e_{¥}Ùa_{0}Ùa_{1}Ù...Ùa_{k}
= e_{¥}Ùa_{0}Ùa_{k}
= -i^{2} (a_{0}¿(e_{¥}a_{k}^{*}))^{*}
represents the Â^{N} k-plane containing a_{0},a_{1},...a_{k}.
.
[ Proof :
(x + e_{0} + ½ x^{2}e_{¥})Ùe_{¥}Ù(a_{0} + e_{0})Ùa_{k} = 0
Û e_{¥}Ù(x + e_{0})Ù(a_{0} + e_{0})Ùa_{k} = 0
Û e_{¥}Ù(xÙa_{0}Ùa_{k} - e_{0}(x-a_{0})Ùa_{k}) = 0 .
Also, e_{¥}Ùa_{0}Ùa_{k}
= -(a_{0}¿((e_{¥}Ùa_{k})^{*}))^{-*}
= -(a_{0}¿((e_{¥}Ak)^{*}))^{-*}
= -(a_{0}¿(e_{¥}Ake_{¥0}i^{-1}))^{-*}
= -(a_{0}¿(e_{¥}a_{k}i^{-1}))^{-*}
.]
In particular
e_{¥0}Ùb_{k}
= e_{¥0}b_{k} represents the k-plane through 0 with tangent k-blade b_{k}.
If a_{0}¿a_{k}=0 we have
e_{¥}Ù(a_{0} + e_{0})Ùa_{k}
= (e_{¥}(a_{0} + e_{0})+1)a_{k} and it is sometimes convenient to express the (k+2)-blade in product form
as (e_{¥}(e_{0}+dn)+1)a_{k} where a_{k} is the tangent blade
and dn is the directance 1-vector normal to a_{k}, ie. the Â^{N}
point in the k-plane closest to the origin.
For example, the 3-blade e_{¥}Ù(a+e_{0})Ùd represents the 1-plane (extended line) through a and a+d If a is the closest point of the line to the origin so that a¿d=0 we have e_{¥}Ù(a+e_{0})Ùd = (e_{¥}(a+e_{0})+1)Ùd = (e_{¥}(a+e_{0})+1)d represents the 1-plane (line) through a and a+ld
The unit plusquare 2-blade e_{¥}Ùa = e_{¥}Ù(a+e_{0}) = e_{¥}(a+e_{0}) + 1 = e_{¥0}+e_{¥}a represents the 0-plane (point) {a} and so represents the same thing as does the null 1-blade a=a+e_{0}+½a^{2}e_{¥} . It is this invertible 0-plane representation that is "output" by the meet and join operations, eg. the meet of lines e_{¥}Ù(a+e_{0})Ùu and e_{¥}Ù(a+e_{0})Ùv is (e_{¥}Ù(a+e_{0})Ùu) Ç (e_{¥}Ù(a+e_{0})Ùv) = e_{¥}Ù(a+e_{0}) .
We can recover the tangent a_{k} from A_{k+2} = e_{¥}Ù(a_{0} + e_{0})Ùa_{k}
via a_{k} = e_{¥0}¿A_{k+2}
= e^{0}¿(e^{¥}¿A_{k+2}). To retrieve an a_{0}
we form
s = a_{k}¿(e^{¥}¿A_{k+2}) parallel to
¯_{e¥¿Ak+2}(e^{0})
and rescale so that
e^{0}¿s=1. We then have null a_{0} = s + ½s^{2}e_{¥} with
a_{0}ÙA_{k+2}=0.
We can also represent a_{k+2} as a translated k-plane through 0 [ See next chapter ]
as
a_{k+2} = (1+½e_{¥}a_{0})(e_{¥}Ùe_{0}Ùa_{k})(1-½e_{¥}a_{0})
= (1+½e_{¥}a_{0})e_{¥0}a_{k}(1-½e_{¥}a_{0})
= (1+½e_{¥}(a_{0}+e_{0}))e_{¥0}a_{k}(1-½e_{¥}(a_{0}+e_{0})) .
Since the U^{N} hyperplane through point a with normal n is represented by
U^{N}^{%} hyperblade
e_{¥}Ù(a+e_{0})Ù(ni^{-1}) , the hyperplane of points equidistant
from points a and b
is represented by U_{N}^{%} hyperblade (a-b)^{*} where ^{*} denotes the GHC extended dual.
[ Proof :
e_{¥}Ù(½(a+b)+e_{0})Ù((a-b)i^{-1})
= e_{¥}Ù((½(a+b)¿(a-b))i^{-1} + e_{0}(a-b)i^{-1})
= ½(a^{2}-b^{2})e_{¥}i^{-1} + e_{¥0}(a-b)i^{-1}
= (½(a^{2}-b^{2})e_{¥}e_{¥0}i^{-1} + (a-b)e_{¥0}i^{-1}
= (½(a^{2}-b^{2})e_{¥} + a-b)(e_{¥0}i)^{-1}
= (a-b)(e_{¥0}i)^{-1} .
Verification: xÙ((a-b)^{*})=0 Û x¿a=x¿b Û (x-a)^{2}=(x-b)^{2}
.]
k-spheres
An N-D k-sphere is the intersection of a hypersphere and a (k+1)-plane in N-D.
[
A strong case can be made for refering to this as a (k+1)-sphere rather than a k-sphere, as does much topology literature,
but the term 2-sphere is so uniformly used in the geometry literature to represent a spherical surface in 3D
that we retain the terminology here.
We will use the terms "sphere" and "hypersphere" to mean an (N-1)-sphere in the N-D space
currently under discussion, favouring "hypersphere" except when N=3.
]
When we speak of a point being "within" a k-sphere, we mean that it lies in the "surface". A circle divides its plane into "inside" and "outside" regions containing c and e_{¥} respectively. More generally a k-sphere divides its (k+1) dimensional tangent space into two regions, or four in the case of a tsphere in a Minkowski space.
We will denote the "solid interior" of a k-sphere a by a^{·}, although we do not yet have a multivector representation of it.
We will refer to the outter product of k U_{N}^{%} embedded point 1-vectors as a k-pointblade.
When we say a particular U_{N}^{%} (k+2)-blade a_{k+2} corresponds to a pointset A such as a k-sphere in U^{N} what we actually mean is that the meet of a_{k+2} and the U^{N}^{%} horosphere { x : x^{2}=0 } maps 1-1 (apart from scale) to the embedded set ¦(A). Furthermore, a U_{N}^{%} (k+2)-blade actually represents a set of hypersphere duals, each U^{N}^{%} point l(c+e_{0}+½(c^{2}±r^{2})e_{¥}) within a_{k+2} corresponding dually to a U^{N} hyper(anti)sphere of squadius ±r^{2}. The horosphere ponts are dual to zero radius hyperspheres, ie. to points for Euclidean U^{N}, nullcones for Minkowski.
The (k+2)-blade representation a_{k+2} of a k-sphere of centre c, radius r and (k+1)-plane through c parallel to tangent (k+1)-blade a_{k+1}=(a_{1}-a_{0})Ù(a_{2}-a_{0})Ù...(a_{k+1}-a_{0}) has four alternatively covenient representational forms:
a_{k+2} splits naturally into unextended 1, e_{0}, e_{¥}, and e_{¥0} "factors". Observe that the scaled tangent(k+1)-blade is available as the e_{0} factor and having obtained a unit tangent blade, it can be contracted with the (k+2)-blade 1 factor of a_{k+2} to recover the (same scaled) a_{0}.
To recover the enclosing hypersphere (aka. surround) s^{*}
(and hence r^{2} and c) and
the tangency a_{k+1} (independant of c) of a given (k+2)-horoblade
a_{k+2}=la_{0}Ùa_{1}Ù...Ùa_{k+1} , we can
recover s
as the dual of a_{k+2} in e_{¥}Ùa_{k+2} ie. s = -a_{k+2}¿((e_{¥}Ùa_{k+2})^{-1}) = ^_{ak+2}(e_{¥}))^{-1}
( ie. the dual to a_{k+2} in e_{¥}Ùa_{k+2}) .
We can neglect the inversion and calculate
s =
a_{k+2}¿(e_{¥}Ùa_{k+2})
= a_{k+2}(e_{¥}Ùa_{k+2})
provided we then rescale so that e^{0}¿s=1 .
The squared radius r^{2} is then available as s^{2}; or we can compute it directly as
r^{2}=(-1)^{k}a_{k+2}^{2}(e_{¥}Ùa_{k+2})^{-2} .
Note that s = a_{k+2}(e_{¥}Ùa_{k+2})^{-1}
= (-1)^{(k+1)(N+1)}(¯_{(ak+2*)}(e_{¥})^{-1})^{*}
[ Proof :
a_{k+2}(e_{¥}Ùa_{k+2})^{-1}
= ((e_{¥}Ùa_{k+2})a_{k+2}^{-1})^{-1}
= ((e_{¥}¿a_{k+2}^{*}))^{-*}a_{k+2}^{-1})^{-1}
= (-1)^{k(N+1)}((e_{¥}¿(a_{k+2}^{*}))a_{k+2}^{-1}^{-*})^{-1}
= (-1)^{k(N+1)}(¯_{(ak+2*)}(e_{¥})^{-*})^{-1}
= (-1)^{k(N+1)}i^{-1}¯_{(ak+2*)}(e_{¥})^{-1}
= (-1)^{(k+1)(N+1)}¯_{(ak+2*)}(e_{¥})^{-1}^{*}
.]
Thus we can define the centre of a (k+2)-blade as the centre of the 1-blade dual surround
Ú(a_{k+2}) º Ú(_Akp¿(e_{¥}Ùa_{k+2})), or e_{¥} if e_{¥}Ùa_{k+2}=0.
If s^{2}=r^{2}<0 for Euclidean U^{N} then we interpret the blade as
representing an "imaginary" U^{N} pointset.
Once we have s then provided r¹0 we can form
a_{k+1} =e_{¥0}¿(sa_{k+2})
with a_{k+2}=2r^{-2} s¿(e_{¥}ÙsÙa_{k+1})
because
sa_{k+2} = i^{2} s^{2} (e_{¥}ÙsÙa_{k+1})
= i^{2} r^{2} e_{¥}Ù(c+e_{0})Ùa_{k+1} .
Note that a_{k+2}e_{¥}a_{k+2} ¹ a_{k+2}¿(e_{¥}Ùa_{k+2}) but is rather a multiple of c , residing entirely within Â_{N+1}.
If a_{k+2} is a k-plane rather than a k-sphere we will have e_{¥}Ùa_{k+2}=0 and can recover the tangent k-blade and point a_{0} withing the k-plane from a_{k+2} as previously described.
If a_{k+2} is not a pointblade, then s = a_{k+2}¿(e_{¥}Ùa_{k+2}) may have e^{0}¿s=0 and so not represent a hypersphere dual. If r^{2}=0 (such as for a nullcone) then b_{k+1} cannot be recovered in this way.
When s=-e_{+} for the origin-centred unit hypersphere we have
a_{k+2}
= -i^{2} e_{+}e_{¥0}a_{k+1} = -i^{2} e_{-}a_{k+1} = -i^{2} e_{-}Ùa_{k+1} .
Contents of k-spheres
It can be inductively shown that a (2k-1)-sphere has interior "volume" content
|S_{2k-1,R}|_{·}
= R^{2k} p^{k} (k!)^{-1}
= R^{2k} p^{k} (G(k+1))^{-1}
while a 2k-sphere has
content
|S_{2k,R}|_{·} =
(2R)^{2k+1}p^{k} k! ( (2k+1)!)^{-1}
= R^{2k+1}p^{k+½} (G(k+1+½))^{-1} .
The "surface area" boundary content of a k-sphere, is (k+1)/R times the
"volume" interior content.
Thus a 5-sphere (existing in 6 or more dimensions) has content R^{6} p^{3}/6 and boundary R^{5} p^{3} ;
a 6-sphere has content (2R)^{7} p^{3} 3!/7! =
R^{7} 16p^{3}/105
and boundary
R^{6} 16p^{3}/15 ;
a 7-sphere has content
R^{8} p^{4} / 4!
and boundary
R^{7} p^{4} / 3 ; and so forth.
We can unify the content expressions for odd and even dimensions by recourse to the conventional
scalar gamma function ;
a k-sphere having interior content
|S_{k,R}|_{·} = R^{k+1}p^{½(k+1)} G(½(k+3))^{-1}
= (R^{2}p)^{½(k+1)} G(½(k+3))^{-1} ;
and "surface" content
|S_{k,R}| =
(K+1) (R^{2}p)^{½(k-1)}p G(½(k+3))^{-1}
= 2 (R^{2}p)^{½(k-1)}p G(½(k+1))^{-1} .
This prompts us to define real scalar constant
o_{k} º
|S_{k-1,1}| = 2p^{½k} (G(½k))^{-1}
as the surface area of a unit (k-1)-sphere _{[ HS 7-4.12 ]}.
o_{N} is thus the boundary content of the N-D hypersphere and we have
o_{k} ^{-1} = ½p^{-½k} G(½k) .
In particular
o_{1}^{-1} = ½ ;
o_{2}^{-1} = ½p^{-1} ;
o_{3}^{-1} = ¼p^{-1} ;
o_{4}^{-1} = ½ p^{-2} ;
o_{5}^{-1} = 3/8 p^{-2} ;
o_{6}^{-1} = p^{-3} ;
o_{7}^{-1} = 15/16 p^{-3} .
k-antispheres
If the j-2 and (k-2)-spheres represented by Â_{N+1,1} j and k-blades a_{j} and b_{k}
where j£k intersect
it will be in a l-sphere for some l£j represented by the meet a_{j}Çb_{k} . If they do not inertsect then the meet may not vanish, but instead represent a l-antisphere, ie.
a l-sphere of imaginary radius and so negative real squared-radius.
So, for example, in N=3 the meet of the 4-blades representing hyperspheres of radius ½
centred at points e_{1} and e_{2} is a 4-blade representing a 2-antisphere of squared radius ¼ centred at ½(e_{1}+e_{2}) .
This 4-blade is dual to a 1-blade rpresenting the 0-sphere bipoint consisting of the two closest points
on the hyperspheres.
If we replace the e_{2} centred hypersphere with the 0-plane e_{¥}Ù(e_{0}+e_{1}) through 0 and e_{1}
then the meet is the bipoint { ½e_{1}, 1½e_{1}} as expected but if we replace it with
e_{¥}Ù(e_{0}+e_{2}) through 0 and e_{2} that does not intersect O_{e1,½}then the meet is
the 0-antisphere centre 0 tangent e_{2} of radius ½3^{½} which can be regharded as the bipoint
0± ½3^{½}ie_{2} both of which are squared distance ¼ from e_{1}.
This is dual to the 1-sphere centre 0 of radius ½3^{½} , axis vector e_{2}, and tangent e_{13} .
e_{-}-negation
If we flip the e_{-} coordinate of
s =
c-½r^{2}e_{¥}
= c + e_{0} +½(c^{2}-r^{2})e_{¥}
= c + ½(1+c^{2}-r^{2})e_{-} + ½(-1+c^{2}-r^{2})e_{+}
we obtain
s^{[e-]}
= -(e_{-}^{*})s(e_{-}^{*})^{-1}
= c - ½(1+c^{2}-r^{2})e_{-}
+ ½(-1+c^{2}-r^{2})e_{+}
= c - ½(1+c^{2}-r^{2})(e_{0}+½e_{¥})
+ ½(-1+c^{2}-r^{2})(-e_{0}+½e_{¥})
= c - (c^{2}-r^{2})e_{0}
- ½e_{¥}
= - (c^{2}-r^{2})^{-1} (-(c^{2}-r^{2})^{-1}c + e_{0}
+ ½e_{¥}(c^{2}-r^{2})^{-1})
-(c^{2}-r^{2})^{-1}c + e_{0} + ½e_{¥}(c^{2}-r^{2}) with square r^{2}(c^{2}-r^{2})^{2} represents a sphere of centre c'=-(c^{2}-r^{2})^{-1}c (so c'^{2} = (c^{2}-r^{2})^{-2}c^{2} ) and squared radius r^{2}(c^{2}-r^{2})^{2} ; when r=0 we have -c^{-2} times the horoembedding of -c^{-2}c which is reflection (aka. inversion) in the unit hypersphere combined with negation (reflection in e_{0}).
Note that e_{0}^{[e-]} = -½e_{¥} and e_{¥}^{[e-]}=-2e_{0}.
s^{[e-]})s = c^{2} + (½(1+c^{2}-r^{2}))^{2} + (½(-1+c^{2}-r^{2}))^{2} + (c + ½(-1+c^{2}-r^{2})e_{+})Ùe_{-} with positive scalar part ³½.
s^{[+-]}s
= (c-e_{0}-½(c^{2}-r^{2})) (c+e_{0}+½(c^{2}-r^{2}))
= c^{2} - (e_{0}+½(c^{2}-r^{2})e_{¥})^{2}
= 2c^{2} - r^{2}
vanishes only for s=e_{0} or
e_{+}-negation
Flipping the e_{+} coordinate of s
we obtain
s^{[e+]}
= -(e_{+}^{*})s(e_{+}^{*})^{-1}
= c + ½(1+c^{2}-r^{2})e_{-}
- ½(-1+c^{2}-r^{2})e_{+}
= c + ½(1+c^{2}-r^{2})(e_{0}+½e_{¥})
- ½(-1+c^{2}-r^{2})(-e_{0}+½e_{¥})
= c + (c^{2}-r^{2})e_{0} + ½e_{¥}
= (c^{2}-r^{2})((c^{2}-r^{2})^{-1}c + e_{0} + ½(c^{2}-r^{2})^{-1}e_{¥} )
with square r^{2} corresponding dually to a (c^{2}-r^{2}) weighted
hypershere centre (c^{2}-r^{2})^{-1}c and squared radius
r^{2}(c^{2}-r^{2})^{-2} .
When r=0 this c^{-2} times the horoembedding of c^{-2}c corresponding to reflection (ie. inversion) in the unit hypersphere.
s^{[e+]}¿s
= (c + ½(1+c^{2}-r^{2})e_{-})^{2} - (½(-1+c^{2}-r^{2})e_{+})^{2}
= c^{2} - ½(1+(c^{2}-r^{2})^{2})
= -½ + c^{2} - ½(c^{2}-r^{2})^{2} which vanishes only for hyperspheres of radius ½^{½} containing 0.
s^{[e+]}Ùs = 2
(c + ½(1+c^{2}-r^{2})e_{-})Ù(½(-1+c^{2}-r^{2})e_{+}
= (-1+c^{2}-r^{2})cÙe_{+} - ½(1+c^{2}-r^{2})(-1+c^{2}-r^{2})e_{¥0}
= (-1+c^{2}-r^{2})cÙe_{+} - ½((c^{2}-r^{2})^{2}-1)e_{¥0} .
Hyperspheres containing 0 have
s^{[e+]}s
= -½ + c^{2} - cÙe_{+} .
If we instead eliminate the e_{+} component we obtain ¯_{e+*}(s) = c + ½(1+c^{2}-r^{2})e_{-} = c + ½(1+c^{2}-r^{2})(e_{0}+½e_{¥} ) = ½(1+c^{2}-r^{2})( 2(1+c^{2}-r^{2})^{-1}c+e_{0}+½e_{¥}) with square c^{2} - ¼(1+c^{2}-r^{2})^{2} which is a ½(1+c^{2}-r^{2}) weighted hypersphere of centre 2(1+c^{2}-r^{2})^{-1}c and squared radius (c^{2} - ¼(1+c^{2}-r^{2})^{2}) (½(1+c^{2}-r^{2}))^{-2} = 4c^{2}(1+c^{2}-r^{2}))^{-2} - 1 .
e_{0}^{[e+]} = ½e_{¥}; e_{¥}^{[e+]} = 2e_{0}
Example Geometric Manipulations
Let us define the squeperation (short for squared seperation) of a point from a
hypersphere to be the square of the shortest distance from the point to the sphere surface. We can think of this as the "squared height"
of points "outside" the hypershere, zero for points "within" (ie. "on") the sphere, and the "squared depth" of points
"inside" it.
Leting e_{i} º e_{i}+e_{0}+½e_{¥} = e_{i}+e_{-} for i=1,2,..N denote N of the 2^{N} embedded corners of a particular unit U^{N} hypercube we have
Now consider le_{1}+me_{2}
= (l+m)( (l+m)^{-1}(le_{1}+me_{2}) + e_{0} + ½e_{¥})
= (l+m)( ((l+m)^{-1}(le_{1}+me_{2}))' + e_{0} + ½e_{¥}(1-
(l+m)^{-2}(le_{1}+me_{2})^{2}))
which for l+m¹0 is dual to the (N-1)-sphere of centre
(l+m)^{-1}(le_{1}+me_{2})
and radius (l+m)^{-1} (l^{2}+m^{2})^{½} .
Now (e_{1}-(l+m)^{-1}(le_{1}+me_{2}))^{2}
= (l+m)^{-2} ( m(e_{1}-e_{2}))^{2}
= 2 m^{2}(l+m)^{-2} so |e_{1}-c|=2^{½}|m(l+m)^{-1}|
which differs from the radius by
(l+m)^{-1}((l^{2}+m^{2})^{½}-m
and we recognise le_{1}+me_{2} as dual to the hypersphere
having centre (l+m)^{-1}(le_{1}+me_{2}) and radius chosen so that the sum of the
squared distances of e_{1} and e_{2} from the hyperspherical surface is (e_{1}-e_{2})^{2} . We can regard this as
the hypersphere moving and expanding to keep the "combined squeperation" from points e_{1} and e_{2} constant.
Pencils
Consider a Â_{N+1,1} 2-blade s_{2} = s_{1}Ùs_{2} where
s_{1} and s_{2} represent hyperspheres in Â^{N}.
The hyperspheres intersect in an (N-2)-sphere s_{2}^{*} = s_{1}^{*} Ç s_{2}^{*}
Û s_{2}^{2} < 0.
Since e_{¥}Ùs_{2} ¹ 0 the 1-vector
s = s_{2}(e_{¥}Ùs_{2})^{-1}
= ¯_{s2}(e_{¥})^{-1} represents a sphere
having the same centre and radius as the intersection.
s_{2}^{*} represents an intersective pencil of all
spheres containing s_{1}^{*} Ç s_{2}^{*} as well as the hyperplane containing it.
Further, if a is any point in the intersective (N-2)-sphere
s_{1}*s_{2} = (a-c_{1})*(a-c_{2})
= ½(r_{1}r_{2})^{-1}(r_{1}^{2}+r_{2}^{2}-|c_{1}-c_{2}|^{2})
[ where a*b º (a¿b) (|a||b|)^{-1}
is the inversive product ]
.
The geometric product of the 1-vectors representing two intersecting hyperspheres is
thus a 2-blade representing all hyperspheres sharing that intersection and a scalar having
value the product of the radii and the scalar cosine of the angle subtended by the hypersphere centres
from any point on the intersection.
s=l_{1}s_{1}+l_{2}s_{2} = (l_{1}+l_{2})
(l_{1}(l_{1}+l_{2})^{-1}c_{1}
+(l_{2}(l_{1}+l_{2})^{-1}c_{2} + e_{0} +½e_{¥}(...)
has
s^{2} =
l_{1}^{2}r_{1}^{2} +l_{2}^{2}r_{2}^{2} + l_{1}l_{2}(r_{1}^{2}+r_{2}^{2}-(c_{1}-c_{2})^{2})
= (l_{1}+l_{2})(l_{1}r_{1}^{2} +l_{2}r_{2}^{2}) - l_{1}l_{2}(c_{1}-c_{2})^{2}
so the weighted sum of two spheres is another sphere.
When l_{1}+l_{2}=1 for colinearity with s_{1} and s_{2} we have
s =
(l_{1}c_{1}+l_{2}c_{2} + e_{0} +½e_{¥}(...) with s^{2}
= (r_{2}^{2} +l_{1}(r_{1}^{2} -r_{2}^{2}) - l_{1}(1-l_{1})(c_{1}-c_{2})^{2} .
Setting r_{1}=r_{2}=0 we can now interpret the wieghted sum of non null-seperated points as dual to a
hypersphere of centre l_{1}c_{1}+l_{2}c_{2} and squared radius -l_{1}l_{2}(c_{1}-c_{2})^{2} . Thus a-b is dual to a hypersphere of
centre a-b and radius (a-b)^{2} for spacelike seperated a,b.
Spheres s_{1} and s_{2} are nonintersecting iff s_{2}^{2} > 0 in which case s_{2}
contains two noncolinear null vectors which represent two particular Â^{N}
points. s_{2}^{*} represents
a Poncelet pencil of all spheres and hyperplames with respect to which these two points are inversive.
The spheres are tangent iff s_{2}^{2} = 0. The contact point is then represented by
^_{s1}(s_{2}) and s_{2}^{*} represents
a tangent pencil of all spheres tangent to this point.
Thus if c_{1}¹c_{2} then (s_{1}Ùs_{2})^{*} represents all hyperspheres that intersect identically with s_{1} as does s_{2}, while if c_{1}=c_{2}=c but r_{1}¹r_{2} then s_{1}Ùs_{2} = ½(r_{2}^{2}-r_{1}^{2})e_{¥}Ù(c+e_{0}) and (s_{1}Ùs_{2})^{*} represents all hyperspheres with centre c.
Suppose now that s_{2} represents a hyperplane n_{2} + d_{2}e_{¥}.
Sphere s_{1} intersects the hyperplane iff s_{2}^{2} < 0 as above. The interesction
has the same centre and radius as sphere ^_{s2}(s_{1})^{*}.
s_{2}^{*} = (s_{1}Ùs_{2})^{*} represents a concurrent pencil of spheres as above.
Further, if a is any point in the intersective (N-2)-sphere
s_{1}*s_{2} = -(a-c_{1})*n_{2} = (c_{1}.n_{2}-d_{2})/r_{1} .
The sphere and hyperplane are seperate iff s_{2}^{2} > 0 in which case
s_{2}^{*} represents a Poncelet pencil as above.
The sphere and hyperplane are tangent (at ^(s_{1},s_{2})) iff s_{2}^{2} = 0 in which case
s_{2}^{*} represents a tangent pencil as above.
If both s_{1} and s_{2} are hyperplanes then once again they intersect iff
s_{2}^{2} < 0.
If s_{1} and s_{2} both contain 0 then their intersection is the (N-2)-space
s_{2}i, otherwise it is (e_{0}¿s_{2})^{*} reprenting
an (N-2)-plane (a line if N=3) having the same normal and distance from 0 as
^(e_{0},s_{2})^{*}.
s_{2}^{*} represents a concurrent pencil of hyperplanes
containing the (N-2)-plane.
Further, s_{1}*s_{2} = n_{1}*n_{2} .
Otherwise s_{2}^{2} = 0 and s_{1},s_{2} are parallel a distance
|e_{0}¿^(s_{1}~,s_{2})| apart.
s_{2}^{*} represents a parallel pencil of hyperplanes
normal to ^(s_{1},s_{2}).
We can generalise pencils as 2-bunches where a k-bunch is represented
by (the dual of) a k-blade s_{1}Ùs_{2}Ù...Ùs_{k} where
s_{1},s_{2},...,s_{k} are representors of hyperspheres or hyperplanes.
Those interested should consult
Li et al. A key result is that if the k hyperspheres s_{1},s_{2},..s_{k} intersect in an
(N-k)-sphere s_{1}^{*}Çs_{2}^{*}Ç...s_{k}^{*} then (s_{1}Ùs_{2}Ù..s_{k})^{*} is an intersecting k-bunch representing
all m-spheres and m-planes containing s_{1}^{*}Çs_{2}^{*}Ç...s_{k}^{*} where m³N-k.
In particular a point a satisfies aÙ(s^{*})=0 for any hypersphere s^{*} containing
a in its surface, so a can be regarded as a 1-bunch m-representing
the set of (N-m)-spheres and (N-m)-planes containing a.
Geometric Interpretation of GHC Blades
We now summarise the GHC geometric interpretations of particular Â_{N+1,1} or U_{N}^{%} pure blades.
Let b_{k}
be a pure k-blade in Â_{N} ;
null 1-vectors a_{0},a_{1},... be embedded points ; and 1-vectors s_{1},s_{2},... represent (duals of) hyperspheres.
Blade | Grade | Square | Horosphere intersection represents |
e_{0} | 1 | 0 | Origin point 0 (embedded) |
e_{¥} | 1 | 0 | Infinity point ¥ |
e_{¥0} | 2 | 1 | Origin point {0} (0-plane) |
a+e_{0}+½a^{2}e_{¥} | 1 | 0 | Point a (embedded) |
e_{¥}Ù(a+e_{0}) | 2 | 1 | Point {a} (0-plane) |
e_{¥}Ùa | 2 | 0 | Direction a |
e_{+}^{*} | N+1 | (-1)^{N+1}i^{2} | Unit hypersphere (squared radius=1) centre 0 |
e_{-}^{*} | N+1 | (-1)^{N}i^{2} | Unit antihypersphere (squared radius=-1) centre 0 |
e_{0}^{*} | N+1 | 0 | Null (ie. zero radius) hypersphere centre 0 |
e_{¥}^{*} | N+1 | 0 | No horosphere intersection |
e_{¥0}b_{k} | k+2 | b_{k}^{2} | k-plane through 0 with tangent b_{k} |
(c+e_{0}+½(c^{2}-r^{2})e_{¥})^{*} | N+1 | (-1)^{N+1}i^{2}r^{2} | Hypersphere centre c squared radius r^{2} |
a_{0}Ùa_{1}...Ùa_{k} | k+1 | (k!^{-1} r Volume(a_{0},a_{1},...,a_{k}))^{2} | (k-1)-sphere through a_{0},a_{1},...and a_{k} |
(e_{0}+a_{0})Ùb_{k} | k+1 | (-1)^{k}(^_{bk}(a_{0}))^{2}b_{k}^{2} | (k-1)-sphere through a_{0} with tangent b_{k} |
e_{-}Ùb_{k} | k+1 | (-1)^{k}b_{k}^{2} | Unit (k-1)-sphere with centre 0 and tangent b_{k} |
e_{¥}Ù(e_{0}+a_{0})Ùb_{k} | k+2 | b_{k}^{2} | k-plane through a_{0} with tangent b_{k} |
(a_{1}-a_{0})^{*} | N+1 | (a_{1}-a_{0})^{2} | Bisecting hyperplane between a_{0} and a_{1} |
(s_{1}Ùs_{2}Ù...Ùs_{k})^{*} | N+2-k | ? | k-bunch of spheres and planes containing s_{1}^{*}Çs_{2}^{*}...Çs_{k}^{*}. |
Each such blade b has a variety of "duals". We have the unextended dual ^{*} in i which sends e_{-} and e_{+} to (N+1)-vectors and the e_{i} to (N-1)-vectors; and the extended dual ^{*} in i = ie_{¥0} . For example consider 3-blade b= (e_{0}+8e_{¥})e_{12} = ½(8e_{-}+7e_{+})e_{12} representing the 1-sphere of radius 4, centre 0 and tangent 2-plane e_{12} . When N=5, its unextended dual is (e_{0}+8e_{¥})e_{345} representing the 2-sphere of radius 4, centre 0, and tangent 3-plane e_{345} whereas the extended dual is b^{*} = (e_{0}-8e_{¥})e_{345} representing a 2-antisphere of radius -4, centre 0 and tangent volume e_{345}. In Euclidean Â_{5} this is an empty pointset but if e_{4}^{2}=-1 (for example) then b does contain points in the GHC horosphere such as the embedding of Â^{4,1} point e_{3} + 15^{½}e_{4}.
In an GHC extended Euclidean space, the extended dual of
a k-sphere can be regarded as an "imaginary (N-k)-sphere". Imaginary in that it has negative squared radius and
so does not contain any representors of points in Â^{N}.
We can then more safely consider the extended dual b^{*} as representing the same geometric construct as b,
considering the multiplication by imaginary i=e_{¥0}i as being an "irrelevant scaling" of b much as we
consider multiplication of b by a scalar value a to be irrelevant. Encountering a nonnull 1-vector s, for example,
we might consider it to represent the (N-1)-sphere more properly represented by (N+1)-blade s^{*}.
If N is odd so that i is central, we can "normalise away" a complex rather than a real
"scale" for a nonnull blade b. All that is required is a reversing conjugation ^{^} that preserves
k-blade b while negating (N+2)-blade i and we have
(ab)^{~} º |(ab)^{^}(ab)|^{-½} ab.
However in a nonEuclidean space, the dual may represent a distinct "actual" or "real" geometric pointset.
Point Versors
Conisder the point k-versor p_{1}p_{2}p_{3}...p_{k} where the p_{i} is the GHC embeddings p_{i} = p_{i} + e_{0} + ½p_{i}^{2}e_{¥} .
For k=2 we have p_{1}¿p_{2} + p_{1}Ùp_{2} which comprises the squared seperation distance and the 0-sphere bipoint { p_{1},p_{2} } .
For k=3 we have p_{1}(p_{2} ¿ p_{3}) + p_{1}¿(p_{2}Ùp_{3}) + p_{1}Ùp_{2}Ùp_{3}
comprising the 3-blade 1-sphere through the points; and the 1-vector
S = -½(a^{2}p_{1} - b^{2}p_{2} + c^{2}p_{3} ) with
S^{2} = ¼a^{2}b^{2}c^{2} , where
a^{2} º (p_{2}-p_{3})^{2} ; b^{2} = (p_{1}-p_{3})^{2}; c^{2}=(p_{1}-p_{2})^{2} .,
The e_{0} coordinate -½(a^{2}-b^{2}+c^{2}) of S is zero when the p_{i} form a right angled triangle with p_{2} opposing the hypotenuse;
but is otherwise positive in a Euclidean space with s = -2(a^{2}-b^{2}+c^{2})^{-1}S
= (a^{2}-b^{2}+c^{2})^{-1}(a^{2}p_{1}-b^{2}p_{2}+c^{2}p_{3})
then dual to a hypersphere with center
(a^{2}-b^{2}+c^{2})^{-1}(a^{2}p_{1}-b^{2}p_{2}+c^{2}p_{3}) and squadius
a^{2}b^{2}c^{2}(a^{2}-b^{2}+c^{2})^{-2} ;
passing through the endpoints p_{1} and p_{3} and a squared distance ???
from p_{2}..
.
[ Proof : S
= p_{1}(p_{2} ¿ p_{3}) + p_{1}¿(p_{2}Ùp_{3})
= p_{1}(p_{2} ¿ p_{3}) + (p_{1}¿p_{2})p_{3} - p_{2}(p_{1}¿p_{3})
= -½(p_{1}a^{2} - p_{2}b^{2} + p_{3}c^{2})
º -½(a^{2}-b^{2}+c^{2})s
S^{2} =
(p_{1}(p_{2} ¿ p_{3}) - p_{2}(p_{1}¿p_{3}) + p_{3}(p_{1}¿p_{2}) )^{2}
= 2(
-(p_{1}¿p_{2})(p_{2} ¿ p_{3})(p_{1}¿p_{3})
+(p_{1}¿p_{3})(p_{2} ¿ p_{3})(p_{1}¿p_{2})
- (p_{2}¿p_{3})(p_{1}¿p_{3})(p_{1}¿p_{2}) )
= -2(p_{1}¿p_{2})(p_{2} ¿ p_{3})(p_{1}¿p_{3})
= ¼a^{2}b^{2}c^{2}
Þ s^{2} = (a^{2}-b^{2}+c^{2})^{-2}a^{2}b^{2}c^{2} .
.
Now p_{1}¿S
= p_{3}¿S = 0 while
p_{2}¿S = 2(p_{1}¿p_{2})(p_{2} ¿ p_{3}) = ½a^{2}c^{2}
hence p_{2}¿s = -(a^{2}-b^{2}+c^{2})^{-1}a^{2}c^{2}
so
(p_{2}-c)^{2} = r^{2} - 2(p_{2}¿s)
= r^{2} + 2(a^{2}-b^{2}+c^{2})^{-1}a^{2}c^{2}
= a^{2}c^{2}(a^{2}-b^{2}+c^{2})^{-2}(b^{2}+2(a^{2}-b^{2}+c^{2}))
= a^{2}c^{2}(a^{2}-b^{2}+c^{2})^{-2}(2a^{2}-b^{2}+2c^{2}))
from which we deduce that s is dual to a hyperplane or hypersphere that passes through p_{1} and p_{3} and
within squared seperation a^{2}c^{2} (a^{2}-b^{2}+c^{2})^{-1} of p_{2} .
.]
For example p_{1} = 2e_{1} + e_{0} - 2e_{¥} ; p_{2}=e_{0} ; p_{3} = e_{2}+e_{0}-½e_{¥} with
a^{2}=4; b^{2}=5; c^{2} = 1;
and p_{2}Ùp_{3} = _e02 + ½ e_{¥0}
genretes the circle centred at e_{1}+½e_{2} with squadius 2^{-2}5 and tangent e_{12} .
Convergent Point Projection
The point x =
e_{0} + xe_{1} + ze_{3} + ½x^{2}e_{¥}
= xe_{1} + ze_{3} + ½(x^{2}+1)e_{-} + ½(x^{2}-1)e_{+} where x=xe_{1}+ze_{3}
inverts in 3-blade e_{-12} representing the unit radius circle centered at e_{0} with tangent e_{12}
to e_{-12}xe_{-12}^{-1} =
xe_{1} - ze_{3} + ½(x^{2}+1)e_{-} - ½(x^{2}-1)e_{+}
and projects to e_{-12} as
¯_{e-12}(x) =
((xe_{1} + xe_{3} + ½(x^{2}+1)e_{-} + ½(x^{2}-1)e_{+})¿e_{-12})e_{-12}^{-1}
= ((-xe_{-2} -½(x^{2}+1)e_{12} )(-e_{-12})
= ((xe_{-2} + ½(x^{2}+1)e_{12} )e_{-12}
= -xe_{1} - ½(x^{2}+1)e_{-}
= -xe_{1} - ½(x^{2}+1)(e_{0}+½e_{¥})
= -½(x^{2}+1)( (½(x^{2}+1))^{-1}xe_{1} + e_{0}+½e_{¥} )
, dual to an antihypersphere centre
(½(x^{2}+z^{2}+1))^{-1}xe_{1}
with
squared radius (½(x^{2}+1))^{-2} x^{2} - 1 , which is
-(x^{2}+1)^{-2} (x^{2} - 1)^{2} when z=0.
Note that this projection is also given by ½(x + e_{-12}xe_{-12}^{-1}),
The dualed contraction projection provides a more efficiemt and robust grade limited computation,
but the averaged inversion can be more useful intuitively.
If z=0 and x=1+d with |d| < 1 then d' = x'-1 has |d'| < d^{2}
so the repeatedly projected centre will converge reasonably rapidly to e_{1}
[ Proof : (½(x^{2}+1))^{-1}x -1
= (½(x^{2}+1))^{-1} (x-½(x^{2}+1))
= (½((1+d)^{2}+1))^{-1} (1+d - ½((1+d)^{2}+1))
= -(½((1+d)^{2}+1))^{-1} ½d^{2}
= -((1+d)^{2}+1)^{-1} d^{2} .]
We accordingly define the convergent point projection of x into (k+2)-blade a_{k+2}
¯_{Ú}(x)_{ak+2}(x) as the
limit of x_{j+1} = ¯_{Ú}(x_{j}, a_{k+2}),
ie. we have ¯_{Ú} º (¯Ú)^{¥} .
Nonflat Embeddings
Our approach for intersecting planes and spheres in U^{N} is to embed k+2 nondegenerate points
in a k-sphere into the U^{N}^{%} horosphere H^{N}_{e¥} with
f_{0}(x) º x+e_{0}+½x^{2}e_{¥} and then form their
(k+2)-blade outterproduct. We intersect such blades using the meet, and we can do this forcing any signatures we like.
The actual signatures (or metric) of U^{N} contribute only to the e_{¥} coordinate of f_{0}(x), and then arise
again when considering where and whether the m-blade meet intersects H^{N}_{e¥}.
Suppose instead we embedd with ¦: U^{N} ® U^{N}^{%} defined by
¦(x) = h(f_{0}(x)) where
h : U^{N}^{%} ® U^{N}^{%}
is an arbitary function.
We can extend linear h to an outtermorphism mapping U_{N}^{%} ® U_{N}^{%} and we are particularly interested in h()
with h(e_{¥})=le_{¥} for nonzero scalar l so that
¦(¥) = le_{¥} and k-planes map to k-planes.
If (k+2)-horoblade a_{k+2} represents a k-sphere through U^{N} points a_{0},a_{1},..,a_{k+1} then h(a_{k+2}) is a more general (k+2)-blade containing nonhoro h(a_{0}), h(a_{1}),..h(a_{k+1}) whose intersection with H^{N}_{e¥} correspnds to a k-curve in U^{N} of a type dependant on h(). More interestingly, if isomorphic h^{-1} exists we can form h(h^{-1}(e_{¥}Ùb_{l+1}) Ç a_{k+2}) propotionate to h((e_{¥}Ùh^{-1}(b_{l+1})) Ç a_{k+2}) to obtain a (nonhoro) m-blade whose intersection with H^{N}_{e¥} represents the intersection of l-plane e_{¥}Ùb_{l+1} with the particular k-curve of type dictated by h() .
We can also compose h with a horodrop to obtain
D_{e}h : U^{N}^{%}®H^{N}_{e¥} inducing a U^{N} transformation
h_{e} º f_{0}^{-1} D_{e}hf_{0} : U^{N} ® U^{N} È {¥}.
k-conics
Conics are naturally present in GA as the medial axies of Dupin Cycides. For example:
3-blade e_{-12} represents the unit radius circle centered at e_{0} with tangent e_{12}.
Rotating s = xe_{1} + ½(1+x^{2}-r^{2})e_{-} + ½(-1+x^{2}-r^{2})e_{+}
= xe_{1} + ½(1+D)e_{-} + ½(-1+D)e_{+}, where D = x^{2}-r^{2},
in 3D dual e_{-12}^{*} = e_{+3} we have
(½qe_{+3})^{↑}_{§}(s) =
xe_{1} + ½(1+D)e_{-} + ½(-1+D) cos(q)e_{+}
- ½(-1+D) sin(q)e_{3} .
A hypershere dual is "normalised" with unit e_{0} coordinate if the e_{-} coordinate minus the e_{+} coordinate is unity, so the
centre of the rotated hypersphere is
c(q) = X(q)e_{1} +Z(q)e_{3} =
(½(1+D) - ½(-1+D) cos(q))^{-1}
(xe_{1} - ½(-1+D) sin(q)e_{3})
which for x^{2} > r^{2} is an ellipse centre
½xD^{-1}(1+D)e_{1} eccentricity x^{-1}D^{½}
passing through
(½xD^{-1}(1+D) ± ½xD^{-1}(-1+D))e_{1} at q=0,p and
½xD^{-1}(1+D)e_{1} ± ½D^{-½} (-1+D)e_{3} at
q = ± cos^{-1}((1+D)^{-1}(-1+D)) .
The conic transform y =
h(x) = S_{a,e0}(x)
º x + ½(a¿x)Ùe_{0} for U_{N} 1-vector a
with inverse
h^{-1}(y) = S_{-a,e0} = y - ½(a¿y)e_{0}
has y^{2} = (a¿x)x¿e_{0}
= -½(a¿x)x^{2} .
S_{a,e0}(x)
= x + ½(a¿x)e_{0}
is dual to the hypersphere of radius
(-½a¿x)^{½}|x| centred at (1+½a¿x)^{-1} x .
h(x)=x for all x perpendicular to a (including e_{¥} and e_{0}) and is
linear in x , so only h(la) is nontrivial.
y = h(f_{0}(la))
= la + e_{0}(1+½la^{2}) + ½l^{2}a^{2}e_{¥}
has
y^{2}
= -½l^{3}a^{4}
and e_{¥}-horodrops to
D_{e¥}(h(f_{0}(la))) =
(1+½la^{2}) f_{0}(la(1+½la^{2})^{-1}) .
For l<0, y thus represents a hypersphere with c=la(1+½la^{2})^{-1}
and radius 2^{-½}l^{3/2}a .
For nonnull a we have
f_{0}^{-1} D_{e¥}hf_{0}(la) =
la^{~}(1±½al)^{-1}
according as a^{2}=±a^{2}.
h(aÙb) = h(a)Ùh(b) so h maps k-blades to k-blades but while h(e_{¥}Ùa_{0}Ùa_{1}...Ùa_{k+1}) = e_{¥}Ùh(a_{0})Ùh(a_{1})...Ùh(a_{k+1})) is proportionate to e_{¥}Ùf_{0}h_{e¥}(a_{0}) Ùf_{0}h_{e¥}(a_{1}) Ù...Ùf_{0}h_{e¥}(a_{k}) so that h maps k-planes representors to k-planes repersentors, it maps k-sphere representors to more general k-curve representors.
U^{N} transformation h_{e¥}(x) = (1+½a¿x)^{-1}x
with inverse
h_{e¥}^{-1}(y) = (1-½a¿y)^{-1}y
acts as a sort of "directed dilation" with the 0-centred hypersphere of radius r < 2a^{-2} mapping under h_{e¥} to an ellipsoid
that passes through ±r(1±½ar)^{-1} a and intersects the orginal hypersphere in a (N-2)-sphere
of radius r and (N-2)-plane through 0 perpendicular to a. Taking a=ae_{1} and
N=3 we have an ellipsoid having 0 as its rightmost focus and
lefthand focus at -r^{2}a(1-¼a^{2}r^{2})^{-1}e_{1} with
eccentricity e=½ra, passing through
-r(1-½ar)^{-1}e_{1} ;
r(1+½ar)^{-1}e_{1} ; ± re_{2}; and ± re_{3}
with "semi-major axis" r(1-¼a^{2}r^{2})^{-1}
and b = r(1-¼a^{2}r^{2})^{-½} .
Points with a¿x=-2 map to ¥ provoiding hyperbolids and parabaloids.
To intersect such a 0-focussed k-conic with a ray e_{¥}Ùf_{0}(a_{0})Ùf_{0}(a_{1}) through points a_{0} and a_{1} it thus suffices to intersect the k-sphere with the ray e_{¥}Ùf_{0}h_{e¥}^{-1}(a_{0})Ùf_{0}h_{e¥}^{-1}(a_{1}) through points h_{e¥}^{-1}(a_{0}) and h_{e¥}^{-1}(a_{1}) and apply h_{e¥} to the resultant two points.
More generally, given any k+2 points a_{0},a_{1},..a_{k+1} then for arbitary a we have a k-conic
passing through a_{0},a_{1},..,a_{k} corresponding to the image under S_{a,e0} of the k-sphere through
the k+2 points S_{-a,e0}(a_{i}) .
a=0 gives the k-sphere through a_{0},a_{1},..,a_{k+1} and while
the k-sphere through the h_{e¥}^{-1}(a_{i}) has radius < 2 we will have an k-ellipsoid.
PolarGrid(f_{0}^{-1} D_{e}(x+½a(x¿a)b)) where D_{e} is dropping to the horosphere in direction e | |||
a=e_{1} ;b=e_{0} ; e=e_{¥} | |||
8×8 cells r=½ a=0 | 8×8 cells r=½ a=½ | 8×8 cells r=½ a=1 | 8×8 cells r=½ a=8 |
a=e_{1} ; b=e_{0} ; e=e_{+} ; - in root | |||
4×4 cells r=½ a=½ | 4×4 cells r=½ a=1 | 4×4 cells r=½ a=2 | |
b=e_{0} ; e=e_{+} ; + in root | |||
2×2 cells r=½ a=¼ | 2×2 cells r=½ a=½ | 2×2 cells r=½ a=1 | |
s _{1} ¿ s _{2} =
s_{1} ¿ s_{2} + ½(r_{1}^{2}+r_{2}^{2})(e_{0}¿e_{¥})
= ½(r_{1}^{2}+r_{2}^{2}-(c_{1}-c_{2})^{2}) - ½(r_{1}^{2}+r_{2}^{2})
= -½(c_{1}-c_{2})^{2}
and hence
( s _{1}Ù s _{2})^{2} = ¼(c_{1}-c_{2})^{4}
independant of r_{1} and r_{2} .
.
[ Proof : s_{1}¿s_{2} + (e_{0} + ½r_{1}^{2}e_{¥})¿(e_{0} + ½r_{2}^{2}e_{¥})
= ½(r_{1}^{2}+r^{2}-(c_{1}-c_{2})^{2}) + ½r_{1}^{2}e_{¥}¿e_{0}+r_{2}^{2}e_{0}¿e_{¥})
.]
s _{e1,½} Ù s _{e2,½} Ù s _{-e1,½} represents a 1-torus
formed by sweeping a sphere of radius ½ along the 1-sphere circular path
having centre e_{0} and tangent plane e_{12}. However if we intersect this with another
torus we can obtain only spheres in which they intersect (if any) rather than the more desirable
set of shared U_{N} points .
If, for example, we intersect
s _{1}Ù s _{2}Ù s _{3}
with the cone s _{0,0} Ù s _{2} Ù e_{¥}
we retrieve null 1-vector
s _{2}
but if instead we use
s _{0,0} Ù s _{c2,d} Ù e_{¥} we obtain a minussquared 1-vector
having positive e_{0},e_{¥},e_{0}, and e_{¥} coordinates represnting a 4-antisphere
of radius 0.6i in U^{N}^{%} whose "centre" is a U^{N} 2-antisphere centred at 0.8e_{1}
of radius .331i .
Line Segments
If we add a scaled line representation to the representation of a point on the line
g = e_{¥}ÙaÙd + a we obtain a <1;3> grade pointed line
(aka. flag) satisfying gg^{§} = -d^{2} .
pg is a <0;2;4> multivector with zero 4-vector component when p lies on the line
and scalar component p¿a = -½(p-a)^{2}
Consider the noninvertible 3-versor e_{¥}(a+e_{0})d . Its 3-vector component
e_{¥}Ù(a+e_{0})Ùd embodies the extended line through a and a+d and we can recover d from this
(as the "coefficient" of e_{¥0}) but a is ambiguous since any point on the line will give the same 3-blade.
The 1-vector component e_{¥}(a¿d) - d resolves the ambiguity in a by specifying a¿d.
However this is not a very useful form because
e_{¥}(a+e_{0})de_{¥}(b+e_{0}) = -de_{¥}(b+e_{0}) independant of a.
The "One Up" Embeddings
Moving off the horosphere
We saw that adding -/+ ½(e^{0}¿x)^{-1} e_{¥} to x gives a 1-vector
with square ±1 corresponding to moving
from a scaled embedded point lx to the (dual of) the unit (anti)hypersphere centre x,
but we can also move off the horosphere in directions e_{+} and e_{-} and this is the basis for the one-up embeddings
proposed by Lasenby.
Recalling that x = x + e_{0} + ½x^{2}e_{¥}
=
x + ½e_{-}(x^{2}+1) + ½e_{+}(x^{2}-1)
we define
x_{±} º ¦_{±}(x) º
(^_{e-/+}(x))^{~} =
-2(x^{2}±1)^{-1}x + e_{-/+}
= -2(x^{2}±1)^{-1} (x + ½(x^{2} -/+ 1)e_{±})
with
x_{±}¿e_{-/+}=0 and x_{±}^{2}=±1 ;
but with the caveat that we have null x_{±} º x when x^{2}=-/+1 , ie. when
x¿e_{-/+} = 0. Otherwise, we can recover x from x_{±} via
x = -½(x^{2}±1)(x_{±} - e_{-/+})
= -(e_{±}¿x_{±} ±1)^{-1}(x_{±} - e_{-/+})
.
[ Proof :
e_{±}¿x_{±} = -/+ (x^{2}±1)^{-1}(x^{2} -/+ 1)
Û
(e_{±}¿x_{±})(x^{2}±1) = -/+ (x^{2} -/+ 1)
Û x^{2}(e_{±}¿x_{±} ± 1) = 1 -/+ e_{±}¿x_{±}
Þ x^{2} = (1 -/+ e_{±}¿x_{±})(e_{±}¿x_{±} ± 1)^{-1}
Þ x^{2} ± 1 = (e_{±}¿x_{±} ± 1)^{-1}
((1 -/+ e_{±}¿x_{±}) ± (e_{±}¿x_{±} ± 1))
= 2(e_{±}¿x_{±} ± 1)^{-1}
.]
Lasenby suggests regarding x with x^{2}=1 as boundary points for the x_{-} embedding. Since e_{0 ±} = e_{±} while e_{¥ ±} = -e_{±} we lack a distinct e_{¥ ±} .
a_{±}¿b_{±}
= ±1 - 2(a^{2}±1)^{-1}(b^{2}±1)^{-1}(a-b)^{2}
provides our inner product.
[ Proof :
(-2(a^{2}±1)^{-1}a + e_{-/+})¿
(-2(b^{2}±1)^{-1}b + e_{-/+})
=
4(a^{2}±1)^{-1}(b^{2}±1)^{-1}a¿b - e_{-/+}^{2}
.]
k-planes and k-spheres
Let U_{N}^{%} (k+2)-blade a_{k+2} = a_{0}Ùa_{1}Ù..Ùa_{k+1}
represent a k-sphere in U^{N} in the usual way. Provided no a_{j}^{2}=1 we have
a_{k+2} = l(a_{0 -}-e_{+})Ù(a_{1 -}-e_{+})Ù...Ù(a_{k+1 -}-e_{+})
where l =
(-½)^{k+2 }(a_{0}^{2}-1)
(a_{1}^{2}-1)...
(a_{k+1}^{2}-1)
and we can expand this as
a_{k+2} = l(a_{k+2 -} + e_{+}Ùb_{k+1 -})
where (k+2)-blade
a_{k+2 -} º a_{0 -}Ùa_{1 -}Ù...Ùa_{k+1 -}
lies within e_{+}^{*}
and (k+1)-vector
b_{k+1 -} = å_{i=0}^{k+1} a_{i -} lies within a_{k+2 -} and so
either commutes or anticommutes with it according as k is odd or even.
Here (k+1)-blade a_{i -} º
(-1)^{i}
a_{0 -}Ùa_{1 -}Ù...^{[i]}...Ùa_{k+1 -}
with the ^{[i]} denoting ommission of a_{i -} .
a_{k+2 -} and b_{k+1 -} thus provide a (N+1)-D representation of the k-sphere a_{k+2}
with
the ^{k+2}C_{N+1} coordinate condition x_{-}Ùb_{k+1 -}=a_{k+2 -}
replacing ^{k+3}C_{N+2} coordinate condition xÙa_{k+2}=0
as the criteria for x lieing in the k-sphere.
Note that e_{-}Ùa_{k+2 -} = l^{-1} e_{-}Ùa_{0}Ùa_{1}Ù..Ùa_{k+1} and e_{-}Ùb_{k+1 -} = e_{-}Ùb_{k+1} where b_{k+1} º å_{i=0}^{k+1} (P_{j¹i}(-2(a_{j}^{2}-1)^{-1})a_{0}Ùa_{1}Ù...^{[i]}..Ùa_{k+1} .
e_{¥}Ù(a_{0 -}-e_{+})Ù(a_{1 -}-e_{+})Ù...Ù(a_{k -}-e_{+})
expands proportionate to
e_{¥}Ù(a_{k+1 -} + e_{+}Ùb_{k})
=
e_{¥}Ùa_{k+1 -} - e_{¥0}Ùb_{k}
so we can also represent k-planes with (N+1)-D vectors rather than (N+2)-D (k+2)-blades.
k-Planes and k-Spheres
la_{±} + (1-l)b_{±} =
-2(l(a^{2}±1)^{-1}(a+½(a^{2} -/+ 1)e_{±})
+(1-l)(b^{2}±1)^{-1}(b+½(b^{2} -/+ 1)e_{±})
is a scaled multiple of the unit representor for U^{N} point
l(a^{2}±1)^{-1}a + (1-l)(b^{2}±1)^{-1}b
which does not in general lie on the line through a and b.
Thus a_{±}Ùb_{±} actually represents a particular 1-curve through a, b rather than
the 1-plane through them.
Lasenby asserts that the tangent at x_{-} for x_{-}Ùa_{-}Ùb_{-} = 0
has direction x_{-}¿(a_{-}Ùb_{-}).
For want of better terms, we will capitalise, and refer to the k-curve
{ x : x_{-}Ùa_{0 -}Ù..a_{k -} = 0 } as a k-Plane and the (N-1)-curve
{ x : x_{-}¿d_{-} = -/+½m^{2} } as a hyper(anti)Sphere of squadius m^{2}
and centre d. A k-Sphere is of course the intersection of a k-Plane with a hyperSphere.
We have a_{k+2}e_{+}a_{k+2} =
2a_{k+2 -}b_{k+1 -} + (b_{k+1}^{2}+(-1)^{k}a_{k+2 -}^{2})e_{+} .
[ Proof :
(a_{k+2 -} + e_{+}b_{k+1 -})e_{+}(a_{k+2 -} + e_{+}b_{k+1 -})
= (-1)^{k}a_{k+2 -}^{2}e_{+} + (-1)^{k+1}b_{k+1 -}a_{k+2 -}
+ a_{k+2 -}b_{k+1 -} + e_{+}b_{k+1 -}^{2}
.]
Numerical experimentation confirms Lasenby's assertion [6.2] that
¯_{e+*}(a_{k+2}e_{+}a_{k+2}) =
2a_{k+2 -}b_{k+1 -} is a multiple of d_{-}, the centre of the k-Sphere through
a_{0},a_{1},...,a_{k+1}.
Spherical Conformal Coordinates
As described by
Hestenes et al,
the intersection of the Â_{N+1,1} = Â_{N}^{%} plane e_{-}¿x=-1
and the horosphere { x : x^{2}=0 } represents the unit hypersphere in Euclidean space
Â^{N+1},
ie. an N-sphere. Thus, for example, the surface of a 3D globe (a 2-sphere) is represented by the e_{-}¿x=-1
plane in Â_{3,1} .
There are a variety of equivalent distance measures one can adopt between two points a and b on the unit N-sphere
(perhaps the most natural is the subtended angle) but we will favour the chord distance
(a-b)^{2} ^{½} = (2(1-a¿b))^{½}
even though Â_{N+1} 1-vector a-b does not strictly speaking exist inside S^{N} .
This ranges from 0 when a=b through 2^{½} when a¿b=0 to a maximum of 2 when a=-b corresponmding to a single point.
Our embedding ¦: Â^{N+1} ® Â^{N+1,1} is
¦(x) = e_{-} + x . Those x
mapping to the horosphere have x^{2} = 1. This coincides over Â^{N}
with our GHC embedding over the unit origin centred hypersphere -e_{+}^{*}
of xÎÂ^{N} with x^{2}=1 since then
x = x + e_{0} + ½e_{¥} = x + ½(e_{-}-e_{+}) + ½(e_{-}+e_{+}) = x + e_{-}
Within S^{N}, we can define a k-sphere having centre c (with c^{2}=1) as the set { x: x^{2}=1, (x-c)^{2}=R^{2} }
for any 0£r£2 which we can regard from our S^{N} Ì Â^{N+1} perspective as the intersection
of S^{N} and a (k-1)-plane in Â^{N+1}.
But spheres of radius greater than 2^{½} can arguably be said to have two centres c and -c
with -c the "better" one, so it is natural to
restrict attention to k-spheres of radius r £ 2^{½} and refer to S^{N} k-spheres having
radius 2^{½} < r £ 2 as a k-antisphere.
There are no k-planes within S^{N} but analagous to k-planes are the great k-spheres of radius 2^{½}.
The dual of 1-vector s = c + (1-½r^{2})e_{-} = c - ½r^{2}e_{-}
with s^{2} = r^{2}(1-¼r^{2})
corresponds to the (N-1)-sphere of radius 0£r<2^{½} when s^{2} > 0 ;
to the great (N-1)-sphere with r=2^{½} when s^{2} = 0 ; and to the (N-1)-antisphere
with 2^{½}<r£2 when s^{2} < 0 .
Given k+1 points
a_{0},
a_{1},...
a_{k} in S^{N} , the (k+1)-blade
a_{k+1} = a_{0}Ùa_{1}Ù...Ùa_{k}
represents the (k-1)-sphere containing the k+1 points if e_{-}Ùa_{k+1} ¹ 0 or the
great (k-1)-sphere containing them if e_{-}Ùa_{k+1} = 0.
Intersection results similar to those for the GHC emebedding follow, and conformal transformations of S^{N}
can be represented by multivectors in like manner to our following discussion of Lorentz transformations
in GHC. See Hestenes et al for a full treatment.
Tspherical Conformal Coordinates
The intersection of the Â_{N,2} = Â_{N-1,1}^{%} plane e_{+}¿x=-1
and the horosphere { x : x^{2}=0 } represents the unit hypertsphere in Â^{N,1}
corresponding to the unit hypertsphere in Â^{N-1,2} .
Our embedding ¦: Â^{N,1} ® Â^{N+1,1} is
¦(x) = e_{+} + x . Those x
mapping to the horosphere have x^{2} = -1.
This coincides over Â^{N-1,1}
with our GHC embedding over the unit origin centred hypersphere -e_{+}^{*}
of xÎÂ^{N} with x^{2}=-1 since then
x = x + e_{0} - ½e_{¥} = x + e_{+} .
Soft Geometry
We can now represent particular geometric pointsets like a (k-2)-spheres in Â^{N}
as a pure k-blade b_{k} in Â_{N+1,1} with the understanding that x is in the set iff
(k+1)-blade xÙb_{k}=0.
Consider the scalar field ¦(x)= (-(xÙb_{k})^{4})^{↑}
= (-((xÙb_{k})_{*}(xÙb_{k}))^{2})^{↑} .
The product (xÙb_{k})^{2} is scalar (being the square of a pure (k+1)-blade)
and squaring again ensures a nonnegative value. ¦(x)=1 for all x with
xÙb_{k}=0 and lies in (0,1) everywhere else, falling rapidly in magnitude
for increasing |(xÙb_{k})^{2}| .
¦ thus "peaks" at 1 over our geometric pointset and falls rapidly to positive nearzero away
from it, with the caveat that it is also 1 at x such that xÙb_{k} is null rather than zero. This
includes every null x that commutes or anticommutes with b_{k}.
We require a measure that is small everywhere "away" from b_{k} and one (frame dependant) way to ensure
this is to force Euclidean signatures
using ¦(x)= (-((xÙb_{k})¿_{+}(xÙb_{k})^{2})^{↑} .
Next : Multivectors as Transformations