4-D Visualization of Minkowski Quaternionic Point Set Operations

Abstract

The contribution emphasizes the geometric modeling point of view on Minkowski point set operations. In this paper, the Minkowski product is specified as the quaternionic product. Selected point sets are visualized using double orthogonal projection and perspective projection from four-dimensional to three-dimensional space. In particular, we demonstrate the generation of sets containing circles (Clifford torus, 3-sphere), lines (quadratic cone), or both.

Figures (8)

• Figure 1: A 3-cube in DOP (left) and 4-D perspective (right).
• Figure 2: Clifford torus in DOP and 4-D perspective. The top left figure shows the Minkowski sum $c(u,v)$ generated according to Equations \ref{['eq:c-csum']}, with the orange generating circle $c_1(u)$ and the purple generating circle $c_2(v)$. The top right figure shows the product $d(u,v)$ obtained from Equations \ref{['eq:c-cprod']}, with generating circles $d_1(u)$ (cyan) and $d_2(v)$ (yellow). The bottom row depicts a sequence of rotations between $c(u,v)$ and $d(u,v)$.
• Figure 3: A portion of the quadratic cone $c$ given by Equations \ref{['eq:l-pprod']}, parametrized over the cube $(u,v_1,v_2)\in[-1,1]\times[-1,1]\times[-1,1]$, visualized in DOP (left) and 4-D perspective (right). The black line represents $c_1(u)$, and the blue plane represents $c_2(v_1,v_2)$. The points on $c_1(u)$ and $c_2(v_1,v_2)$ correspond to the parameter values $u=-0.6$, $v_1=0.4$, and $v_2=0.5$. The black point is their product, and it lies on three rulings of $c(u,v_1,v_2)$.
• Figure 4: (Top) The intersection of the quadratic cone with the unit 3-sphere forms a Clifford torus. The highlighted region shows the portion of the torus corresponding to the restricted part of the cone. (Bottom) Orthogonal projections onto $(x,y)$, $(x,w)$, and $(y,w)$, respectively.
• Figure 5: 3-sphere generated by Equations \ref{['eq:hopf']} as the product of the magenta circle $c_1(u)$ and light blue 2-sphere $c_2(v_1,v_2)$ (degenerated to a disk). The portions of tori refer to the cyan circles on the generating 2-sphere for selected choices of the parameter $v_1$.