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Let $$s \colon I \times J \times K \to \mathbb{R}^4$$ be a parameterization of a hypersurface $$\mathcal{S}$$, where $$I,J,K \subset \mathbb{R}$$ are some intervals. I’m going to show how to draw the cross-section of $$\mathcal{S}$$ by a hyperplane with R.

For the illustration, we consider the tiger:

R1 = 2; R2 = 2; r = 0.5
s <- function(u, v, w){
rbind(
cos(u) * (R1 + r*cos(w)),
sin(u) * (R1 + r*cos(w)),
cos(v) * (R2 + r*sin(w)),
sin(v) * (R2 + r*sin(w))
)
}

Take a hyperplane: $\mathcal{P}\colon \quad \langle \mathbf{a}, \mathbf{x} \rangle = b,$ let $$\vec{\mathbf{n}} = \frac{\mathbf{a}}{\Vert\mathbf{a}\Vert}$$ be a unit normal vector of $$\mathcal{P}$$, and $$\mathbf{x}_0$$ be an arbitrary point in $$\mathcal{P}$$.

a = c(1, 1, 1, 1); b = 2        # plane x+y+z+w = 2
x0 = c(b, b, b, b)/4            # a point in this plane
nrml <- a/sqrt(c(crossprod(a))) # unit normal

Compute a mesh $$\mathcal{M}_0$$ of the isosurface $\bigl(s(u,v,w) - \mathbf{x}_0\bigr) \cdot \vec{\mathbf{n}} = 0.$

library(misc3d)
f <- function(u, v, w){
c(crossprod(s(u, v, w), nrml))
}
u_ <- v_ <- w_ <- seq(0, 2*pi, length.out = 100L)
g <- expand.grid(u = u_, v = v_, w = w_)
voxel <- array(with(g, f(u,v,w)), dim = c(100L,100L,100L))
surf <- computeContour3d(voxel, level = sum(x0*nrml),
x = u_, y = v_, z = w_)
trgls <- makeTriangles(surf)
mesh0 <- misc3d:::t2ve(trgls)

Denote by $$\mathcal{V}\mathcal{S}_0 \subset I \times J \times K$$ the set of vertices of $$\mathcal{M}_0$$, and set $$\mathcal{V}\mathcal{S} = s(\mathcal{V}\mathcal{S}_0) \subset \mathbb{R}^4$$.

VS0 <- mesh0$vb VS <- s(VS0[1L,], VS0[2L,], VS0[3L,])  Let $$R$$ be a rotation in $$\mathbb{R}^4$$ which sends $$\vec{\mathbf{n}} =: \vec{\mathbf{v}}_1$$ to the vector $$(0,0,0,1) =: \vec{\mathbf{v}}_2$$. One can take $$R$$ corresponding to the matrix $\frac{2}{{(\vec{\mathbf{v}}_1+\vec{\mathbf{v}}_2)}' (\vec{\mathbf{v}}_1+\vec{\mathbf{v}}_2)} (\vec{\mathbf{v}}_1+\vec{\mathbf{v}}_2) {(\vec{\mathbf{v}}_1+\vec{\mathbf{v}}_2)}' - I_4.$ rotationMatrix4D <- function(v1, v2){ v1 <- v1 / sqrt(c(crossprod(v1))) v2 <- v2 / sqrt(c(crossprod(v2))) 2*tcrossprod(v1+v2)/c(crossprod(v1+v2)) - diag(4L) } Rot <- rotationMatrix4D(nrml, c(0,0,0,1)) Now define $$\mathcal{V}\mathcal{S}' = R(\mathcal{V}\mathcal{S}) \subset \mathbb{R}^4$$. Then all points in $$\mathcal{V}\mathcal{S}'$$ are equal on their fourth coordinate (up to numerical errors in R): VSprime <- Rot %*% VS head(t(VSprime)) ## [,1] [,2] [,3] [,4] ## [1,] 2.203740 -1.329658 -1.620365 0.9999785 ## [2,] -1.324840 2.206491 -1.657871 1.0002417 ## [3,] -1.320131 2.212244 -1.636339 0.9999972 ## [4,] -1.417790 2.116178 -1.698381 0.9999926 ## [5,] 2.219859 -1.310784 -1.651340 0.9999841 ## [6,] 2.253515 -1.275147 -1.633245 1.0005005 Finally, define $$\mathcal{V}\mathcal{S}'' \subset \mathbb{R}^3$$ as the set obtained by removing the fourth coordinates of the elements of $$\mathcal{V}\mathcal{S}'$$, and define the mesh $$\mathcal{M}$$ whose set of vertices is $$\mathcal{V}\mathcal{S}''$$ and with the same edges as $$\mathcal{M}_0$$: library(rgl) mesh <- tmesh3d( vertices = VSprime[-4L,], indices = mesh0$ib,
homogeneous = FALSE,
normals = ?
)

What about the normals? If you have an implicit equation defining $$\mathcal{S}$$, that is, $$\mathcal{S} = \iota^{-1}(0)$$ with $$\iota\colon \mathbb{R}^4 \to \mathbb{R}$$, then a normal to $$\mathcal{S}$$ at a point $$\mathbf{x} \in \mathbb{R}^4$$ is given by the gradient of $$\iota$$ at $$\mathbf{x}$$. For the tiger, we know an implicit equation, and it is not difficult to get the gradient:

sNormal <- function(XYZT){
x <- XYZT[1L,]; y <- XYZT[2L,]; z <- XYZT[3L,]; t <- XYZT[4L,]
rbind(
x * (1 - R1/sqrt(x^2+y^2)),
y * (1 - R1/sqrt(x^2+y^2)),
z * (1 - R2/sqrt(z^2+t^2)),
t * (1 - R2/sqrt(z^2+t^2))
)
}
Normals <- sNormal(VS)

Once you get the normals:

• project them to the hyperplane $$\mathcal{P}$$;

• apply the rotation $$R$$ to the projected normals;

• remove the fourth coordinates (all equal);

• if necessary, negate the normals.

The projection of $$\mathbf{x} \in \mathbb{R}^4$$ to the hyperplane $$\mathcal{P}$$ is given by $\mathbf{x} - \frac{\langle \mathbf{a}, \mathbf{x} \rangle - b}{\Vert \mathbf{a} \Vert^2} \mathbf{a}.$

# projection onto hyperplane <a,x> = b
projection <- function(a, b, X){
X - tcrossprod(a/c(crossprod(a)), colSums(a*X)-b)
}
mesh <- tmesh3d(
vertices = VSprime[-4L,],
indices = mesh0$ib, homogeneous = FALSE, normals = -t((Rot %*% projection(a, b, Normals))[-4L,]) ) This works: shade3d(mesh, color = "darkmagenta") Here is another way to get the normals. The normal at the point $$s(u,v,w)$$ is $\frac{\partial s}{\partial u}(u,v,w) \times \frac{\partial s}{\partial v}(u,v,w) \times \frac{\partial s}{\partial w}(u,v,w)$ where $$\cdot \times \cdot \times \cdot$$ is the ternary cross-product in $$\mathbb{R}^4$$, defined by $\vec v_1 \times \vec v_2 \times \vec v_3 = \left\vert\begin{matrix} \vec i & \vec j & \vec k & \vec l \\ v_{1x} & v_{1y} & v_{1z} & v_{1t} \\ v_{2x} & v_{2y} & v_{2z} & v_{2t} \\ v_{3x} & v_{3y} & v_{3z} & v_{3t} \end{matrix}\right\vert.$ crossProd4D <- function(v1, v2, v3){ M <- rbind(v1, v2, v3) c(det(M[,-1L]), -det(M[,-2L]), det(M[,-3L]), -det(M[,-4L])) } sNormal <- function(uvw){ u <- uvw[1L]; v <- uvw[2L]; w <- uvw[3L] Du <- c((R1 + r*cos(w))*c(-sin(u),cos(u)), 0, 0) Dv <- c(0, 0, (R2 + r*sin(w))*c(-sin(v),cos(v))) Dw <- r * c(-sin(w)*c(cos(u),sin(u)), cos(w)*c(cos(v),sin(v))) crossProd4D(Du, Dv, Dw) } Normals <- apply(VS0, 2L, sNormal) Then you can calculate the normals in this way and proceed as before: mesh <- tmesh3d( vertices = VSprime[-4L,], indices = mesh0$ib,
homogeneous = FALSE,
normals = t((Rot %*% projection(a, b, Normals))[-4L,])
)

Here is how to do an animation:

b_ <- seq(-11.5, 11.5, length.out = 60L)
open3d(windowRect = c(100, 100, 612, 612), zoom = 0.8)
bg3d(rgb(54, 57, 64, maxColorValue = 255))
view3d(45, 40)
for(i in 1L:length(b_)){
x0 <- rep(b_[i]/4, 4L)
surf <- computeContour3d(voxel, level = sum(x0*nrml),
x = u_, y = v_, z = w_)
trgls <- makeTriangles(surf)
mesh0 <- misc3d:::t2ve(trgls)
VS0 <- mesh0$vb VS <- s(VS0[1L,], VS0[2L,], VS0[3L,]) Normals <- sNormal(VS) mesh <- tmesh3d( vertices = (Rot %*% VS)[-4L,], indices = mesh0$ib,
homogeneous = FALSE,
normals = -t((Rot %*% projection(a, b_[i], Normals))[-4L,])
)
snapshot3d(sprintf("pic%03d.png", i))
clear3d()
}
for(i in 1L:59L){
file.copy(sprintf("pic%03d.png", 60-i), sprintf("pic%03d.png", 60+i))
}
# run gifski
command <- "gifski --fps 12 pic*.png -o slicedTiger.gif"
system(command)
# cleaning
pngfiles <- list.files(pattern = "^pic.*png\$")
file.remove(pngfiles)

## Toroidal hyperboloid

Let’s give another example, a toroidal hyperboloid. This is a quadric with implicit equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} - \frac{t^2}{d^2} = 1,$ and a parameterization of this quadric is $\begin{array}{ccc} s \colon & (0,2\pi) \times (0,2\pi) \times (0, +\infty[ & \longrightarrow & \mathbb{R}^4 \\ & (u,v,w) & \longmapsto & \begin{pmatrix} a \cos u \cosh w \\ b \sin u \cosh w \\ c \cos v \sinh w \\ d \sin v \sinh w \end{pmatrix} \end{array}.$