Real and complexified configuration spaces for spherical 4-bar linkages

TL;DR

This work delivers a complete symbolic library for the real and complexified configuration spaces of a spherical 4-bar linkage, expanding Izmestiev’s framework by including all four folding angles for every linkage-length combination and the diagonal-arc polynomial relation. It provides explicit polynomial relations between adjacent and opposite folding angles, in addition to a comprehensive diagonal-length relation derived from a spherical Cayley-Menger determinant, and a rich set of identities and sign conventions. The collection covers numerous special configurations (square, rhombus, cross, Miura variants, isograms, deltoids, conics, elliptic, orthodiagonal) with finite and infinity (degenerate) solution branches, along with a detailed transformation toolkit (switching strips) and a MATLAB visualization/app script. The work culminates in a spherical Grashof-type criterion for motion feasibility and self-intersection checks, enabling practical analysis of degree-4 vertex origami on the sphere with broad relevance to engineering, architecture, and mathematics. The provided parametrizations, complex-domain extensions, and elliptic-function frameworks equip researchers to analyze, simulate, and design spherical 4-bar mechanisms across real and abstract settings.

Abstract

This note is a complete library of symbolic parametrized expressions for both real and complexified configuration spaces of a spherical 4-bar linkage. Building upon the previous work from Izmestiev, (2016, Section 2), this library expands on the expressions by incorporating all four folding angles across all possible linkage length choices, along with the polynomial relation between diagonals (spherical arcs). Furthermore, a complete MATLAB app script is included, enabling visualization and parametrization. The derivations are presented in a detailed manner, ensuring accessibility for researchers across diverse disciplines.

Figures (2)

• Figure 1: (a) A degree-4 single-vertex rigid origami (not necessarily planar as shown here). We label the sector angles counter-clockwise as $\alpha$, $\beta$, $\gamma$, $\delta$; and the folding angles counter-clockwise as $\rho_x$, $\rho_y$, $\rho_z$, $\rho_w$. The tangent of half of these folding angles are $x, ~y, ~z, ~w$. $u$ and $v$ are two spherical diagonals of this vertex, each of which segments the spherical quadrilateral to spherical triangles. (b) and (c) show two non-trivial folded states with the outside edges of the single-vertex drawn on a sphere as arcs of great circles, assuming the panel corresponding to $\gamma$ is fixed when changing the magnitude of $z$. The mountain and valley creases are shown in solid and dashed lines. Generically, for a $z$ there are two sets of folding angles $\{x, ~y, ~w\}$, where the panels corresponding to $\alpha$ and $\beta$ of the two folded states are symmetric with respect to $u$.
• Figure 2: This figure shows a degree-4 single-vertex rigid origami embedded in a 3-dimensional Euclidean coordinate system. Here $\angle AOB = \alpha$, $\angle BOC = \delta$, $\angle COD = \gamma$, $\angle DOA = \beta$. These sector angles $\alpha, ~\beta, ~\gamma, ~\delta$ satisfy the constraint in Proposition \ref{['prop: sector angle range']}. The folding angles on $OA, ~OB, ~OC, ~OD$ are $\rho_x, ~\rho_w, ~\rho_z, ~\rho_y$. Whether these folding angles are all positive or all negative depends on the orientation specified. Triangle $OAD$ is set on the $xy$-plane with $OA=OD=1$. Next we set $\angle OAB = \angle ODC = \pi/2$ to make $O, ~A, ~B, ~C, ~D$ determined under given $\alpha, ~\beta, ~\gamma, ~\delta$. Hence $AB=\tan \alpha$, $CD=\tan \gamma$.

Theorems & Definitions (11)

• Remark 1
• Proposition 1
• proof
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Proposition 8