Points

With these definitions, we can consider Euclidean vectors a, b ∈ ℝ³ and conformal embeddings A, B given by

A = C(a) = a + a²e_∞/2 + e₀   B = C(b) = b + b²e_∞/2 + e₀

This is straightforward in package idiom:

point <- function(x){ as.1vector(x) + sum(x^2)*einf/2 + e0 }

Thus point() takes an R vector of length 3 and returns its conformal embedding. For example, we may translate points (1, 2, 5) and (2, 2, 2) to their conformal equivalent:

a <- c(1,2,5)
b <- c(2,2,2)
point(a)

## Element of a Clifford algebra, equal to
## + 1e_1 + 2e_2 + 5e_3 + 14.5e_4 + 15.5e_5

point(b)

## Element of a Clifford algebra, equal to
## + 2e_1 + 2e_2 + 2e_3 + 5.5e_4 + 6.5e_5

It can be shown that A ⋅ B = −∥a − b∥²/2 [where the dot product is the Clifford inner product, %.%]. Package idiom to verify this would be

c(conformal=drop(point(a) %.% point(b)), Euclidean = -sum((a-b)^2)/2)

## conformal Euclidean 
##        -5        -5

showing that the two results match.

sphere <- function(x,r){ point(x) - r^2*einf/2}
S <- sphere(1:3,5)  # center (1,2,3) radius 5:
S

## Element of a Clifford algebra, equal to
## + 1e_1 + 2e_2 + 3e_3 - 6e_4 - 5e_5

drop(S^2)   # 5^2 = 25

## [1] 25

S*einf*S

## Element of a Clifford algebra, equal to
## - 2e_1 - 4e_2 - 6e_3 - 13e_4 - 15e_5

-S*einf*S/2

## Element of a Clifford algebra, equal to
## + 1e_1 + 2e_2 + 3e_3 + 6.5e_4 + 7.5e_5

point(1:3)

## Element of a Clifford algebra, equal to
## + 1e_1 + 2e_2 + 3e_3 + 6.5e_4 + 7.5e_5

origin <- point(c(0,0,0))
px <- point(c(1,0,0))
py <- point(c(0,1,0))
pz <- point(c(0,0,1))

(S <- origin ^ px ^ py ^ pz)

## Element of a Clifford algebra, equal to
## + 0.5e_1234 - 0.5e_1235 - 0.5e_1245 + 0.5e_1345 - 0.5e_2345

spherestar <- function(...){Reduce(`^`,list(...))}
spherestar(origin, px, py, pz)

## Element of a Clifford algebra, equal to
## + 0.5e_1234 - 0.5e_1235 - 0.5e_1245 + 0.5e_1345 - 0.5e_2345

p <- point(c(1,1,1+sqrt(3))/2)
Mod(p ^ S)

## [1] 1.110223e-16

Planes

A plane is defined as $\hat{\mathbf{n}} + d\mathbf{e}_\infty$, where $\hat{\mathbf{n}}$ is the unit normal and d the distance to the origin.

plane <- function(n,d){ as.1vector(n/sqrt(sum(n^2))) + d*einf}

For example, we consider the plane Π with normal n = (1, 2, 5) and distance 7. We may use plane() above but first need to calculate $\hat{\mathbf{n}}=\mathbf{n}/\left|\mathbf{n}\right|$:

n <- c(1,2,5)
nhat <- n/sqrt(sum(n^2))
d <- 7
Pi <- plane(nhat,d)
Pi

## Element of a Clifford algebra, equal to
## + 0.1825742e_1 + 0.3651484e_2 + 0.9128709e_3 + 7e_4 + 7e_5

Just to check, we may verify that some points known to lie in Π are in fact in its IPNS. We observe that the point $7\hat{\mathbf{n}}\in\Pi$, and also that vectors u = (2, −1, 0) and v = (5, 0, −1) are orthogonal to the normal of Π. So $7\hat{\mathbf{n}} + \alpha\mathbf{u} + \beta\mathbf{v}\in\Pi$ for any real numbers α, β. In R:

u <- c(2,-1,0)
v <- c(5,0,-1)
P1 <- point(d*nhat)                
P2 <- point(d*nhat + 1.3*u + 3.44*v)
P3 <- point(d*nhat - 6.1*u + 1.02*v)

Above we use some made-up values of α, β to generate P1, P2, P3 which are known to lie in plane Π. Then to verify that these points lie in the IPNS of Π we need to take the inner product with the conformal representation of Π:

c(drop(Pi %.% P1),drop(Pi %.% P2),drop(Pi %.% P3))

## [1]  3.191891e-16 -1.110223e-16 -3.330669e-16

Above we see zero (to numerical precision), showing that we do indeed have $7\hat{\mathbf{n}} + \alpha\mathbf{u} + \beta\mathbf{v}\in\operatorname{IPNS}(\Pi)$ for at least these values of α and β. Alternatively, a plane can be thought of as a sphere that touches the point at infinity; thus the OPNS of plane is given by

π^⋆ = P₁ ∧ P₂ ∧ P₃ ∧ e_∞

where P₁, P₂, P₃ are any points that lie in the plane. To illustrate this we will use the three points used in the IPNS above:

Pi2 <- P1 ^ P2 ^ P3 ^ einf

Then for verification we can create another point known to be in the plane and check that this is in the OPNS. Below we will use p4 which is $d\hat{\mathbf{n}} + 7.6\mathbf{u} - 9.23\mathbf{v}$:

p4 <- point(d*nhat + 7.6*u - 9.23*v)
Mod(Pi2 ^ p4)

## [1] 3.005929e-13

Above we see a very small result showing that p4 is indeed in the OPNS of plane Pi2.

Circle

A circle is defined as the intersection of two spheres:

circle <- function(S1,S2){  # IPNS
    S1 ^ S2
}

For example

circle(sphere(1:3,5),sphere(c(1.1,2.1,3.4),6))

## Element of a Clifford algebra, equal to
## - 0.1e_12 + 0.1e_13 + 0.5e_23 - 3.31e_14 - 7.22e_24 - 9.33e_34 - 3.41e_15 -
## 7.32e_25 - 9.73e_35 + 3.91e_45

A circle may be represented in the OPNS by specifying three points that lie on it:

Z^* = P₁ ∧ P₂ ∧ P₃

circlestar <- function(...){  # OPNS; A^B^C
    jj <- list(...)
    stopifnot(length(jj) == dimension)
    Reduce(`^`,lapply(jj,point))
}

(CIRC <- circlestar(c(1,2,3),c(5,6,3),c(8,8,-2)))

## Element of a Clifford algebra, equal to
## + 8e_123 - 38e_124 - 110e_134 - 54e_234 - 42e_125 - 130e_135 - 74e_235 +
## 40e_145 + 68e_245 + 140e_345

verify:

poc  # point on circle, found numerically [chunk omitted]

## Element of a Clifford algebra, equal to
## - 0.7152127e_1 - 0.2498563e_2 + 0.3267822e_3 - 0.159628e_4 + 0.840372e_5

poc ^ CIRC

## Element of a Clifford algebra, equal to
## - 0.003402925e_1234 - 0.003402866e_1235 - 0.001701745e_1245 - 0.00850813e_1345
## - 0.008507714e_2345

Above we see that poc is at least close to the circle from the small magnitude of the terms in the wedge product.

Lines and point pairs

A line is the intersection of two planes; in R:

line <- function(P1,P2){ P1 ^ P2 }

and a “point pair” is the intersection of three spheres; in R:

pointpair <- function(S1,S2,S3){ S1 ^ S2 ^ S3 }

It is not at all obvious that three spheres intersect in a pair of points; and still less obvious that the process is associative. However, we may verify associativity explicitly:

S1 <- sphere(c(3,2,4),3)
S2 <- sphere(c(3,1,4),4)
S3 <- sphere(c(1,3,3),3)
(S1^S2)^S3

## Element of a Clifford algebra, equal to
## - 5e_123 + 31e_124 + 25e_134 - 40.5e_234 + 29e_125 + 25e_135 - 39.5e_235 -
## 10e_145 + 10e_245 - 5e_345

(S1^S2)^S3 == S1^(S2^S3)

## [1] TRUE