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. The term plane is traditionally used for a 2-plane.
    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}.
    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_ki^-1) = a.(b_ki^-1). Û r.(b_ki^-1) = a.(b_ki^-1). Hence for k=2, N=3 the dual of the tangent is the conventional normal vector 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.

Simplexes
    Given a frame of k+1 vectors (a₀,a₁,a₂,...a_k) where k<N, their k-simplex is the convex hull defined by the points. The 1-simplex for (a₀,a₁) , for example, is the line segment connecting a₀ to a₁. The 2-simplex for (a₀,a₁,a₂) is the flat triangular "surface element" defined by them.
     Writing a_k º (a₁-a₀)Ù(a₂-a₀)Ù...Ù(a_k-a₀) ; where a₀ is known as the base point, we note that the simplex is contained within the k-plane a_k + a₀Ùa_k which we will refer to as the extended k-simplex.
      a_k + a₀Ùa_k has moment a₀Ùa₁Ù...Ùa_k = a₀Ùa_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_k²|^½ is welldefined for any pureblade a_k. For k=1 this is the length |a₁-a₀|; for k=2 it is the area ½|(a₂-a₀)Ù(a₁-a₀)|.

    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.

    The representations of general k-planes not containing the origin are not simple blades, so the operators ¯, ^ ,È, Ç are undefined with respect to them. The higher dimensional embeddings   discussed below provide a solution to this.

Frames
    The obvious way to represent a k-frame (a₁,..,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 = a₁ + a₁Ùa₂ + ... + a₁Ùa₂Ù..Ùa_k   or  a₁ + a₁a₂ + ... + a₁a₂..a_k . [  If we allow the operation _<1> (ie. taking only the 1-vector component) then we can clearly recover a₁=a_k<1> and thence a¹=a₁/(a₁²) . a₂ is then available as a¹¿a_k, a₃ as a²¿(a¹¿a_k), and so forth ]
    Suppose we transform the frame as a_i'ºBa_iB^#-1 where B^#-1B = b .
    Ba_kB^#-1 is not the correct representor for the transformed frame, (Ba_kB^#-1)_<i> requiring scaling by b^1-i . However, provided b>0 we can unambiguously renormalise the a_i while reconstructing them from Ba_kB^#-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 e₀ with Sig(e₀)=1. The embedding is trivial: ¦(x) = x + e₀ .
    Some authors favour x ® e₀x, mapping 1-vectors to 2-blades (when e₀¿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 e₀ outside Â_N we ensure that all points are invertible.

k-planes

    Consider the Â_N+1 (k+1)-blade (a₀+e₀)Ùa_k where a_k is tangent to the Â_N k-simplex for frame { a₀,a₁,..a_k}.
    (x+e₀) Î (a₀+e₀)Ùa_k Û (x+e₀)Ù(a₀+e₀)Ùa_k = 0 Û xÙa₀Ùa_k = e₀(x-a₀)Ùa_k
    Û (x-a₀)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 (e₀+x)Ùa_k .
    For example:

• The Â^N point (0-plane) p corresponds to the Â_N+1 vector e₀+p.
• The Â^N line (1-plane) through 2 distinct points p,q corresponds to the Â_N+1 2-blade (e₀+p)Ù(q-p) = (e₀+p)Ù(e₀+q) = È(e₀+p , e₀+q). Any segment of the same length of the extended line through p and q will generate this same Â_N+1 bivector,
• The Â^N 2-plane containing 3 noncolinear points a,b, and c corresponds to the Â_N+1 3-blade
      (e₀+a)Ù(b-a)Ù(c-a) = (e₀+a)Ù(e₀+b)Ù(e₀+c) = (e₀+a)È((e₀+b)È(e₀+c)) .
      Any three points in the a,b,c plane defining a triangle of the the same area will generate this same trivector.

    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 (e₀+p)È(e₀+q)=e₀+p. If a,b and c are colinear, then (e₀+a)È((e₀+b)È(e₀+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 (e₀+c₀)ÙC_n where c₀ and C_n are in Â_N. If we  have c expressed via coordinates with respect to the extended basis for { e₀,e₁,...,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 ((a₀+e₀)Ùa_k) and  ((b₀+e₀)Ùb_l) are then complimentary and have a scalar meet equal to the directed perpendicular Euclidean distance between the Â_N planes.
[ Proof :   ((a₀+e₀)Ùa_k)Ù((b₀+e₀)Ùb_l)   =   a₀Ùa_kÙb₀Ùb_l +e₀Ù(a_kÙb₀Ùb_l-a_kÙa₀Ùb_l)
    = 0+e₀Ù(a_kÙb₀Ùb_l-a_kÙa₀Ùb_l)   =   e₀Ùa_kÙ(b₀-a₀)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 Â₄ which are implementable as 2⁴=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:

• Uniformity of representation.
• Generally applicable operators for geometric projection, rejection, and intersection/seperation replacing the conventional unholy suite of point-line, line-line, line-plane, point-plane, line-plane, plane-plane geometric primitves.
• Affine invarient representation. Given a linear or affine map g:³ ® ³, we have (g(r)-g(a))Ùg(b_k)=g(r-a)Ùg(b_k)=g((r-a)Ùb_k) so we can represent the image under g of a k-plane by the image under g of its "pseudo representors". We cannot do this for the "normal representation" since g(a.n) ¹ g(a).g(n).
• Ease of extension into higher dimensions. The conventional vector cross product so pivotal in conventional 3D geometry, for example, only exists for N=3 or N=7 (when it is a product of three vectors, see Lounesto Ch 23.4)

    e₀^* = e₀(e_¥0i)^-1 = (e₀e_¥0)i^-1 = -e₀i^-1 ;

    e_¥^* = e_¥(e_¥0i)^-1 = -e_¥i^-1 .

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 2⁵=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+e₀+½e_¥(c²±r²) ) via its dual with the U^N hyper(anti)sphere of squared radius s²=±r² 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 x²=r².

    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 = r²   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 = -r² 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 will associate more general U_N^% k-blade s₁Ùs₂..Ùs_k via its (N+2-k)-blade dual (s₁Ùs₂..Ùs_k)^* with the space of all planes and spheres that include the (N-k)-sphere meet s₁^*Çs₂^*...Çs_k^* .

    This all constitutes a magnificent "payback" for investing in two additional dimensions. A Â_N+1,1 (k+1)-blade   a_k+1 = a₀Ùa₁Ù...Ùa_k derived from Â^N points a₀,a₁,..,a_k expands as a₀Ùa₁Ù...Ùa_k = a₀Ùa₁Ù...a_k + e₀Ù(a₁-a₀)Ù(a₂-a₀)Ù..Ù(a_k-a₀) + _O(terms in e_¥ and e_¥0)

• If   (a₁-a₀)Ù(a₂-a₀)Ù..Ù(a_k-a₀) ¹ 0 , then a_k+1 represents the (k-1)-sphere containing the k+1 points.
• Otherwise if a₀Ùa₁Ù...a_k ¹ 0 it represents the extended k-simplex {a₀,..a_k} .
• In the "doubly degenerate" case of both (a₁-a₀)Ù(a₂-a₀)Ù..Ù(a_k-a₀) and a₀Ùa₁Ù...a_k vanishing, a_k represents a (k-1)-plane through the origin.

    Generalised homogenised blades thus provide an excellent way to represent extended lines and planes and circles and spherical surfaces, but they are less accomodating when it comes to representing finite line segments and the interiors of k-spheres. Informally, they capture boundaries rather than the regions bound.
    But 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_kÇb_l of (k-2)-sphere a_k and (l-2)-sphere b_l in the usual way, imposing Eucluidean 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 e⁰¿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-pointblades in Â^4,1 each representing either a circle or an infinite lines in 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 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 m² indicates 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 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 provideing the squared radius of a hypersphere tangent to both. When it 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 nonintesrsecting pointsets.

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 : r² = 0 ; e⁰¿r = 1 }. This is the horosphere, the intersection of a hyperplane normal to e_¥ containing e₀, and the null cone   { x Î Â^N+1,1 : x² = 0 }.
    x = x + ae₀ + be_¥ is null Û x² = ab so H^N_e¥ = { x + ½x²e_¥ + e₀    :     x Î Â^N }   .
    Our embeding is accordingly
    x   =   f₀(x) º f_0e¥(x) º x + e₀ + ½x²e_¥   =   x + ½e_-(x²+1) + ½e₊(x²-1)   though some authors favour lx + l²e₀ + ½x²e_¥   =   lx + ½e_-(x²+l²) + ½e₊(x²-l²) with l a unit length in order to homogenise the "dimension" or "units" of x .
    We will refer to such null x of unit e₀ coordinate (-e_¥¿x=1) as the horosphere embedding or horoembedding of x = ¯_(e¥0^*)(x).
    In particular, e₀ corresponds to the Â^N origin 0 while e_¥ represents a hypothetical Â^N point at infinity which we will denote ¥.
    When x²=1 this coincides with the spherical conformal embedding x = x+e₀+½e_¥ = x+e_- .
    Some authors consider an embedding Â^N ® Â_N+1,1 defined by x ® e_¥0x, 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_¥0x=e_¥Ùe₀Ùx with the 1-plane (line) through 0 and x. The 3-blade e_¥0x also spans U^N% points of the form e₀+l(x-e₀) = (1-l)((1-l)^-1x + e₀) = (1-l)(((1-l)^-1x)' - ½(1-l)^-2x²e_¥) dual to hyperspheres of centre (1-l)^-1x 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 0²=0 by e₀ with e₀²=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 x²³1 so one might more properly refer to "quasi-homogenised coordinates". With GHC the embedded origin e₀ is truly equivalent to every other embedded point in that all nontrivially satisfy x²=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+e₀)=e_¥Ùx also represents the Â^N point x but in a slightly different way.
    The nearest we have to a 1-vector inverse for the embedded point is x' = 2(x²+1)^-2x^[+] where
    x^[+] = x - ½e_-(x²+1) + ½e₊(x²-1) = x - x²e₀ - ½e_¥ = -x²f₀(-x^-1) for nonnull x has xx^[+] = ½(x² + 1)²  + 2xÙe_- + (x²-1)e_¥0 so, provided x²¹-1, we have xx' = 1 + 4(x²+1)^-1((x²+1)^-1xÙe_- + (x²-1)e_¥0) , though this is of little practical use.

Inverse Point Embedding
    The simplest inverse mapping
    f₀^-1(x) º (e⁰¿x)^-1 ^(x, e_¥0)     =   -( e_¥¿x)^-1(xÙe_¥0)e_¥0   =   (e_¥¿x)^-1(e_¥0Ùx)e_¥0     taking U^N% into U^N is unsatisfactory because f₀^-1(e_¥) is undefined and we wish to define f₀^-1 fully over U^N% rather than merely over H^N_e¥. Thus we seek a f₀^-1 : U^N% ® U^N È ¥ with f₀^-1(e_¥)=¥.
    Some authors favour f₀^-1(x) = e₊¿(xÙe_¥) = -e_-¿(xÙe_¥) = e₀¿(xÙe_¥) .e₊ = (xe_¥ - e_¥0).e₊ = x + e_- but we will take f₀^-1 : U^N% ® U^N% defined by f₀^-1(x) º e^¥¿(e_¥Ù((e⁰¿x)^-1x) - e_¥0) which returns the horosphere to the U^N% subspace  { x : x¿e_¥0 = 0 } .
    Clearly f₀^-1f₀ is an identity mapping 1 : U^N®U^N   but f₀f₀^-1 : U^N% ® H^N_e¥ is a many-to-one "projection" into the horosphere dependant on our choice of f₀^-1.
    All f₀^-1 coincide over H^N_e¥, however, and so given any g U^N ® H^N_e¥ we can unambigusously define f₀^-1g U^N% ® U^N .

Embedded Products

    Note that (x+e₀+le_¥)² = x² - 2l.
    Note also the scaled idempotent 2-versors (e₀x)² = (-x²)e₀x and (xe₀)² = (-x²)xe₀ with similar reslts for e_¥x and xe_¥.
    Further, e₀xe_¥x   =   -e₀(x²e_¥ + 2x)     has
    (e₀xe_¥x)²   =   e₀(x²e_¥ + 2x)e₀(x²e_¥ + 2x)   =   e₀(x²e_¥)e₀(x²e_¥ + 2x)   =   -2x² (e₀xe_¥x)     while (e₀xe_¥x)^§ (e₀xe_¥x)   =   (e₀xe_¥x) (e₀xe_¥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(a²b-b²a)e_¥ + (a-b)e₀ - ½(a²-b²)e_¥0 .
[ Proof : (a + ½a²e_¥ + e₀)(b + ½b²e_¥ + e₀) = ab - ½a²be_¥ - be₀ + ½ab²e_¥ + ½b²e₀e_¥ + ae₀ + ½a²e_¥e₀
    = a¿b + aÙb - be₀ + ½(ab²-a²b)e_¥ + ½b²(-1-e_¥0) + ae₀ + ½a²(-1+e_¥0)
    = -½(a-b)² + aÙb +½(a²b-b²a)e_¥ + (a-b)e₀ - ½(a²-b²)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 l₁a+l₂b represents a point   (ie. is null) only if (a-b)²=0 .
    aÙb = aÙb +½(a²b-b²a)e_¥ + (a-b)e₀ - ½(a²-b²)e_¥0.
    For N=3, our multivectors now require 2⁵=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 + ae₀ + be_¥ where x Î Â^N but this appears less useful than the decomposition a_e¥x_e¥ + b_e¥e_¥ where x_e¥²=0 and e⁰¿x_e¥=1 ; achieved by a_e¥ = e⁰¿y = -e_¥¿y ; b_e¥ = ½(e_¥¿y)^-1y² ; and x_e¥ = - ½(e⁰¿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 enough e from y to turn it null. If this has nonzero e₀ coordinate then we can  rescale to give the normalised e-horodrop D_e^~(y). with unit e₀ coordinate. For e=e_¥ we can think of moving from scaled hyper(anti)sphere dual y=l(c+e₀+½(c²±r²)e_¥) to the embedded centre point c as a natural definition of f₀^-1 f₀. This fails for y=x+be_¥ where x Î U^N and so e⁰¿y=0, which we can intrepret as representing a particular "directed infinity" ¥x .

    For general null e then provided e¿y ¹ 0 we can remove ½(e¿y)^-1y² e and rescale for unit e₀ coordinate equivalent to yey rescaled.
    For general unit e with (e¿y)² ³ e²y² we can remove e²(-e¿y ± ((e¿y)²-e²y²)^½) e and reach H^N_e¥ at two alternative nullvectors having opposite e coordinate and typically corresponding to x⁴ greater and less than 1.    
    If we obtain unrescalable l(x+be_¥) we interpret it as ¥x .
    Clearly D_e¥(y) = (e⁰¿y) f₀((e⁰¿y)^-1 ^_e¥0(y)) = (e⁰¿y) f₀f₀^-1(y) if y has nonzero e₀ coordinate.   f₀^-1 D_e¥ for y with nonzero e₀ coordinate thus tricvially consists of dividing the U^N compobnent by the e₀ coo0rdinate, independant of the e_¥ coordinate.s

    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(e⁰¿x)^-1 corresponding (for "normalised" null x with e⁰¿x=1) to obtain
    L_e¥^±(x) = x + e₀ + ½(x² -/+ 1)e_¥   =   x + ½(e_--e₊) -/+ ½(e_-+e₊) + ½x²e_¥   =   x -/+ e_± + ½x²e_¥ . which we can think of as moving from scaled null point embedding lx = lf₀(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 = r² } where r>0 corresponds to the Â_N+1,1 equation x¿c = -½ r².
    Writing s = c-½r²e_¥ = c + e₀ +½(c²-r²)e_¥ = c + ½(1+c²-r²)e_- + ½(-1+c²-r²)e₊ we have s² = r² ; e⁰¿s=1 , and  x¿s=0 Û x lies on the hypersphere.
[ Proof :  x¿s = x¿(c-½r²e_¥) = -½ r² - ½r²x¿e_¥ = 0.  .]
    So for any plussquare s Î Â^N+1,1 satisfying  e⁰¿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=(s²)^½ ; and we can associate any hyperblade s^*=se_¥0i^-1 with an N-D hypersphere provided e⁰¿s¹1 and s²>0. In particular, (e₀-½e_¥)^*=-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+e₀ +½(c²-r²)e_¥ satisfies s²=r² ; 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^~ + e₀ + ½(c²+2rc¿b^~ + r²)e_¥) ¿(ce_¥-e_¥0) = c²e_¥ + r(b^~¿c)e_¥+c + e₀ - ½(c²+2rc¿b^~ + r²)e_¥ = c+e₀+½(c²-r²)e_¥  .]

    To what does the horosphere solution set for x¿s = 0 correspond if s²>0 but e⁰¿s=0 rather than 1?
    s = ^_e¥0(s) + ¯_e¥0(s) = s+(s¿e^¥)e_¥ when e⁰¿s=0 so s = |s|(n-(s¿e₀/|s|)e_¥) where n is a unit vector in Â^N and |s|=(s²)^½=(s²)^½.
    Thus x¿s=0 Û x¿(n-(s¿e₀/|s|)e_¥) = 0 Û x¿n - e₀¿(s¿e₀/|s|)e_¥ = 0 Û x¿n + (s¿e₀/|s|) = 0   Û x¿n + d = 0   where d = (s¿e₀/|s|) = s^~¿e₀.
    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 s² = r² , the squared radius of the U^N hypersphere to which s is dual, as the squadius of plussquare U^N% point s. For s²<0 we have a negative squadius   corresponding to the squared radius of an antihypersphere ( { x : x²=-r² } ) 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^*=a₀Ùa₁Ù....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_iÙs^*=0 for i=0,1,..N so a₀,a₁,...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 a₀,a₁,...a_N. We can thus represent the hyperplane containing a₁,a₂,...a_N by the blade e_¥Ùa₁Ù...Ùa_N.

    If s₁^* and s₂^* represent spheres then  s₁¿s₂   =   ½(r₁²+r₂² - (c₁-c₂)² )     .
    This may initially seem a disappointing inner product since we might instead have wished for the closest distance between the spheres , but it has a definite geometric interpretation nonetheless. If a is a point common to both spheres then the cosine of the angle subtended by the "radial normals" (a-c₁) and (a-c₂) is given (via the traditional triangular cosine rule) as ½(r₁² +r₂²-(c₁-c₂)²)(r₁r₂)^-1 . Thus s₁¿s₂ is r₁r₂ times the cosine of the angle subtended by the radial normals at any common point. This result also holds when one of s₁ or s₂ represents a hyperplane.

    s₁Ùs₂ = c₁Ùc₂ + e₀d + ½e_¥c + ½e_¥0( c₁²-c₂²+r₂²-r₁²)     where c º (c₁²-r₁²)c₂ - (c₂²-r₂²)c₁ and d º c₂-c₁.
    (s₁Ùs₂)²   =   (s₁¿s₂)² - r₁²r₂²   =   ¼( (r₁²-r₂²)² - 2(c₁-c₂)²(r₁²+r₂²) + (c₁-c₂)⁴ ) .
[ Proof : (s₁Ùs₂)²   =   (c₁Ùc₂)² + c¿d + ¼(c₁²-c₂²+r₂²-r₁²)² Set c₁=0 so that c=-r₁²c₂ and d=c₂ . We then have
    (s₁Ùs₂)² = -r₁²c₂² + ¼(c₂²-r₂²+r₁²)²   =   ¼c₂⁴ + ¼(r₁²-r₂²)² + -½c₂²(r₁²+r₂²)   . Replacing c₂ with c₂-c₁ gives
    ¼( (r₁²-r₂²)² - 2(c₁-c₂)²(r₁²+r₂²) + (c₁-c₂)⁴ ) which rearranges as
    ¼((r₁²+r₂²)² - 2(c₁-c₂)²(r₁²+r₂²) + (c₁-c₂)⁴ - 4r₁²r₂²)   =   (½(r₁²+r₂² - (c₁-c₂)² ))² - r₁²r₂²  .]

    With s₁=x so r₁=0 we have x¿s₂ = ½(r₂²-(c₂-x)²) which  is positive only when x lies outside s₂^* and zero only (for Euclidean U^N) when it lies on it. Also (xÙs₂)²= ¼(r₂²-(c₂-x)²)² is zero if x lies in hypersphere s₂^* . More generally, the sign of (s₁Ùs₂)² indicates where s₁^* intersects s₂^*. Neagtive implies an interesction and zero implies tnagential contact.

    s+le_¥ is dual to a hypersphere with centre c and squadius r²-2l.
    s+le₀ = (1+l)( (1+l)^-1c + e₀ + ½e_¥(1+l)^-1(c²-r²))   is dual to a hypersphere with centre (1+l)^-1c and squadius (1+l)^-2(r²-lc²).

    It is easily verified that (ls₁+(1-l)s₂)^* represents a hypersphare of centre lc₁ + (1-l)c₂ and squadius l(l-1)(c₁+c₂)² + lr₁²+(1-l)r₂².
[ Proof : ls₁+(1-l)s₂   =   lc₁+(1-l)c₂ + e₀ + ½e_¥(lc₁²+(1-l)c₂²-lr₁²-(1-l)r₂²)   =   c²+e₀+½e_¥(c² - (-lc₁²-(1-l)c₂²+lr₁²+(1-l)r₂²+c²)) dual to a hypersphere centre c=lc₁+(1-l)c₂ of squadius
    (-lc₁²-(1-l)c₂²+lr₁²+(1-l)r₂²+c²)
      =   (l(l-1)c₁²+l(1-l)c₂²+2l(1-l)(c₁¿c₂) + lr₁²+(1-l)r₂² )
      =   l(l-1)(c₁²+c₂²+2(c₁¿c₂)) + lr₁²+(1-l)r₂² )
      =   l(l-1)(c₁+c₂)² + lr₁²+(1-l)r₂²  .]

    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^Ne_N is represented as x=x+e₀+½(X²-(x^N)²) in Â_N,1^% which represents the dual of a hypersphere centre x and radius 2^½x^N .
    If x²<0 this sphere encloses 0, if x²=0 it includes 0. The Â^N,1 squaredlength x² represents the Â^N+1,0 squeperation of 0 from the spherical surface.

k-planes

    Let a₀,a₁,..a_k be k+1 1-vector horopoints in Â_N+1,1 corresponding to points a₀,a₁,..,a_k in Â^N and set Â^N k-blade
    a_k º (a₁-a₀)Ù(a₂-a₀)Ù...Ù(a_k-a₀) . We have
    e_¥Ùa₀Ùa₁Ù...Ùa_k = e_¥Ù(a₀ + e₀)Ùa_k .
[ Proof : e_¥Ù(a₀+½a₀²e_¥+e₀)Ùa₁Ù...Ùa_k = (e_¥Ùa₀+e_¥0)Ùa₁Ù...Ùa_k = (e_¥a₀+e_¥0)Ù(a₁+½a₁²e_¥+e₀)Ùa₃Ù...Ùa_k = (e_¥a₀+e_¥0)Ù(a₁+e₀)Ùa₃Ù...Ùa_k = ((e_¥a₀)Ùa₁+e_¥0Ùa₁+e_¥Ùa₁Ùe₀)Ùa₃Ù...Ùa_k = (e_¥a₀Ùa₁+e_¥0Ù(a₁-a₀))Ùa₃Ù...Ùa_k = ...
    = (e_¥Ùa₀Ù...Ùa_k + e_¥0Ù(a₁-a₀)Ù...Ù(a_k-a₀)) = e_¥Ù(a₀Ù(a₁-a₀)Ù...Ù(a_k-a₀) + e₀Ù(a₁-a₀)Ù...Ù(a_k-a₀))  .]
    Also,  (e_¥Ùa₀Ùa₁Ù...Ùa_k)²   =   a_k²      =   (-1)^½k(k-1)(k!V_k)² where V_k is the volume (content) of the k-simplex.
[ Proof : For a₀=0 we trivially have (e_¥Ùe₀Ùa_k)² = (-e_¥0a_k)² = a_k². More generally
    (e_¥Ù(e₀+a₀)Ùa_k)² = (½e_¥(a₀Ùa_k)-e_¥0a_k)² = (-e_¥0(½e_¥(a₀Ùa_k)+a_k)² = (½e_¥(a₀Ùa_k)+a_k)² = (½(e_¥(a₀Ùa_k)a_k + a_ke_¥(a₀Ùa_k))+a_k²)
    = (½e_¥((a₀Ùa_k)a_k + a_k^#(a₀Ùa_k))+a_k²) = a_k² since we know result is scalar  .]

    Now, x + e₀ + ½ x²e_¥ Î e_¥Ù(a₀ + e₀)Ùa_k Û (x-a₀)Ùa_k = 0 . Thus
    e_¥Ùa₀Ùa₁Ù...Ùa_k = e_¥Ùa₀Ùa_k = -i² (a₀¿(e_¥a_k^*))^* represents the Â^N k-plane containing a₀,a₁,...a_k. .
[ Proof :   (x + e₀ + ½ x²e_¥)Ùe_¥Ù(a₀ + e₀)Ùa_k = 0 Û e_¥Ù(x + e₀)Ù(a₀ + e₀)Ùa_k = 0 Û e_¥Ù(xÙa₀Ùa_k - e₀(x-a₀)Ùa_k) = 0 .
    Also, e_¥Ùa₀Ùa_k = -(a₀¿((e_¥Ùa_k)^*))^-* = -(a₀¿((e_¥Ak)^*))^-* = -(a₀¿(e_¥Ake_¥0i^-1))^-* = -(a₀¿(e_¥a_ki^-1))^-*  .]
    In particular e_¥0Ùb_k = e_¥0b_k represents the k-plane through 0 with tangent k-blade b_k.
    If a₀¿a_k=0 we have e_¥Ù(a₀ + e₀)Ùa_k =   (e_¥(a₀ + e₀)+1)a_k and it is sometimes convenient to express the (k+2)-blade in product form as (e_¥(e₀+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+e₀)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+e₀)Ùd = (e_¥(a+e₀)+1)Ùd = (e_¥(a+e₀)+1)d represents the 1-plane (line) through a and a+ld

    The unit plusquare 2-blade e_¥Ùa = e_¥Ù(a+e₀) = e_¥(a+e₀) + 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+e₀+½a²e_¥ . It is this invertible 0-plane representation that is "output" by the meet and join operations, eg. the meet of lines e_¥Ù(a+e₀)Ùu and e_¥Ù(a+e₀)Ùv is (e_¥Ù(a+e₀)Ùu) Ç (e_¥Ù(a+e₀)Ùv) = e_¥Ù(a+e₀) .

    We can recover the tangent a_k from A_k+2 = e_¥Ù(a₀ + e₀)Ùa_k via   a_k = e_¥0¿