Semi-Discrete Height Function in Rigidity

• Semi-discrete height function is a real-valued scalar assigned on a mixed discrete-smooth planar framework that converts a self-stress into a 3D lifting.
• It links the equilibrium condition with integrability via path independence, ensuring that any semi-discrete path produces the same lifting potential.
• The construction extends the classical Maxwell-Cremona correspondence to generate semi-discrete developable surfaces, bridging theory and geometric design.

A semi-discrete height function is a real-valued function attached to a planar semi-discrete framework generated by a discrete sequence of smooth curves. In the formulation of "Stressability of Semi-Discrete Frameworks" (Karpenkov et al., 18 Aug 2025), it assigns to each point $f_i(t)$ a scalar height $H(f_i(t))$ so that the lifted map

$L(i,t)=(f_i(t),H(f_i(t)))$

forms a semi-discrete conjugate surface in $\mathbb{R}^3$. The construction is the semi-discrete analogue of the classical Maxwell-Cremona correspondence: stressability of the planar framework is equivalent to the existence of a lifting, and the height function is the device that converts a self-stress into that lifting.

1. Geometric setting and equilibrium structure

A semi-discrete framework consists of a sequence of smooth planar curves $f_i(t)$, with a discrete direction indexed by $i$ and a smooth direction parametrized by $t$. The framework therefore mixes difference operators in the discrete direction with derivatives in the smooth direction. The central algebraic object is a stress $(\lambda,\mu)$, where $\lambda_i(t)$ is the stress along the smooth direction on the curve $f_i$, and $H(f_i(t))$0 is the stress along the discrete direction between $H(f_i(t))$1 and $H(f_i(t))$2 (Karpenkov et al., 18 Aug 2025).

The self-stress condition is expressed by the difference-differential equilibrium equation

$H(f_i(t))$3

where $H(f_i(t))$4 and $H(f_i(t))$5. This equation plays the role that static equilibrium plays in the classical bar-and-joint setting. In the semi-discrete context, it is simultaneously an equilibrium law and an integrability condition for constructing a global height function.

The motivation is explicitly Maxwell-Cremona-theoretic. In the classical discrete theory, a stressable planar framework is the orthogonal projection of a polyhedral surface in three dimensions. The semi-discrete theory replaces vertices and edges by smooth curves and rulings, and replaces polyhedral liftings by semi-discrete conjugate, hence developable, surfaces.

2. Definition of the semi-discrete height function

The height function is defined relative to an increasing semi-discrete path in parameter space. Let $H(f_i(t))$6, and let

$H(f_i(t))$7

be an increasing semi-discrete path from $H(f_i(t))$8 to $H(f_i(t))$9, with

$L(i,t)=(f_i(t),H(f_i(t)))$0

For such a path, the semi-discrete height function along $L(i,t)=(f_i(t),H(f_i(t)))$1 is

$L(i,t)=(f_i(t),H(f_i(t)))$2

Here the determinant $L(i,t)=(f_i(t),H(f_i(t)))$3 is the signed area form in the plane, encoding orientation and magnitude (Karpenkov et al., 18 Aug 2025).

The formula separates the two directions of the semi-discrete geometry. The sum over $L(i,t)=(f_i(t),H(f_i(t)))$4 collects contributions from the stresses $L(i,t)=(f_i(t),H(f_i(t)))$5 along the smooth curves, evaluated at the discrete transition points of the path. The integral term accumulates the contributions from the discrete-direction stresses $L(i,t)=(f_i(t),H(f_i(t)))$6 along the rulings between neighboring curves.

This definition is initially path-dependent. The crucial issue is whether different increasing semi-discrete paths from $L(i,t)=(f_i(t),H(f_i(t)))$7 to $L(i,t)=(f_i(t),H(f_i(t)))$8 produce the same scalar. The answer is the content of the integrability theorem for self-stresses.

3. Path-independence and the characterization of self-stress

Theorem 3.7 states that if $L(i,t)=(f_i(t),H(f_i(t)))$9 is a self-stress, then $\mathbb{R}^3$0 is independent of the path $\mathbb{R}^3$1, so one may write unambiguously $\mathbb{R}^3$2. Conversely, path-independence for all $\mathbb{R}^3$3 implies that $\mathbb{R}^3$4 is a self-stress (Karpenkov et al., 18 Aug 2025).

This equivalence is the defining structural property of the semi-discrete height function. It converts a formula indexed by paths into a genuine scalar field on the framework. In this sense, the difference-differential equilibrium equation is exactly the compatibility condition needed for lifting.

The result is also the semi-discrete counterpart of a familiar phenomenon in classical rigidity and discrete differential geometry: a stress field gives rise to a potential if and only if the local equilibrium conditions close globally. Here the closure relation is expressed through path-independence in the mixed discrete-smooth parameter domain.

A plausible implication is that the semi-discrete height function should be viewed less as an auxiliary scalar and more as the potential whose existence certifies that the stress system integrates to geometry in $\mathbb{R}^3$5.

4. Liftings, conjugate surfaces, and the Maxwell-Cremona analogue

Once path-independence is established, the lifting is defined by

$\mathbb{R}^3$6

The lifted family of curves, joined along the rulings, forms a semi-discrete conjugate surface; equivalently, its strips are developable. Theorem 4.3 gives the semi-discrete Maxwell-Cremona lifting property: a planar semi-discrete framework is stressable if and only if it is the orthogonal projection of a semi-discrete conjugate surface in $\mathbb{R}^3$7-space (Karpenkov et al., 18 Aug 2025).

The paper also states the converse construction: if one starts with a semi-discrete conjugate surface $\mathbb{R}^3$8, then the corresponding self-stress functions $\mathbb{R}^3$9 and $f_i(t)$0 can be written explicitly in terms of the geometry of the lifted surface. Thus the correspondence is not merely existential. It is constructive in both directions.

The construction is described as natural with respect to affine transformations, and reversing the curve order inverts the construction up to an affine function added to the height. This places the height function within an affine-geometric, rather than purely Euclidean, framework.

Aspect	Classical Maxwell-Cremona	Semi-discrete Maxwell-Cremona
Underlying object	Finite planar bar-joint graphs	Sequence of planar smooth curves
Stress data	Function on edges	Functions $f_i(t)$1 and $f_i(t)$2
Lifted geometry	Polyhedral surface	Semi-discrete developable surface

The comparison identifies the semi-discrete height function as the direct analogue of the classical lifting potential, but adapted to a mixed difference-differential geometry.

5. Existence, liftability, and boundary-force phenomena

The existence of a path-independent semi-discrete height function is equivalent to solvability of the equilibrium equation for $f_i(t)$3. For regular non-degenerate frameworks, this is described as a local linear system of ODEs and is typically solvable unless degenerate (Karpenkov et al., 18 Aug 2025).

The paper isolates several special cases. For a single-curve framework, any configuration with arbitrary boundary forces is stressable. For the one-strip case, consisting of two neighboring curves and no boundary forces, liftability holds if and only if an explicit but nontrivial ODE condition is satisfied; the relevant criterion is given in Equation (5.9). The paper therefore provides a complete characterization for the minimal nontrivial strip geometry.

The discussion of vanishing boundary forces is geometrically significant because boundary data determine whether an internally consistent stress can be supported without external compensation. The paper states that it discusses geometric implications for frameworks with vanishing boundary forces, and it characterizes the liftability of frameworks consisting only of two neighboring curves forming one strip.

For larger systems, the situation is incomplete. For frameworks with two or more strips and vanishing boundary forces, finding the general liftability condition remains open. This places the semi-discrete height function at the center of an active existence problem: the object is fully defined once a self-stress is known, but the classification of when such stresses exist is only partially resolved.

6. Related uses of “height function” in other research areas

The term “height function” appears in several mathematically distinct literatures. On the two-dimensional square lattice, height functions are integer-valued random functions $f_i(t)$4 equipped with Gibbs measures from convex symmetric gradient potentials; in that setting, the main issues are localisation, delocalisation, covariance decay, logarithmic roughness, and the effective temperature gap (Lammers, 2022). On shift-invariant cubic planar graphs, one again studies integer-valued height functions, with delocalisation results for excited and parity potentials (Lammers, 2020).

In tiling theory, height functions arise as potentials derived from flows and tensions on directed graphs. For generalized tilings, a tension produces a potential $f_i(t)$5, and the height function uniquely encodes the tiling, supports flip dynamics, and induces a distributive lattice structure (Bodini et al., 2021). In simplicial topology, a height function can mean a simplicial map $f_i(t)$6, from which one builds section complexes, Reeb complexes, and a spectral sequence computing the homology of $f_i(t)$7 (Vaupel et al., 2022).

In semi-discrete optimal transport, the vector of Laguerre weights $f_i(t)$8 is viewed as a potential or height function in the dual Kantorovich problem. There the “height” parametrizes power diagrams and enforces cell-mass constraints rather than producing a Maxwell-Cremona lifting (Caplan, 2021).

This suggests that “semi-discrete height function” is not a single universal construction. In the geometry of semi-discrete frameworks, it is specifically the lifting potential generated from a self-stress and characterized by path-independence. In statistical mechanics, combinatorics, topology, and optimal transport, the same phrase refers to structurally different scalar fields that serve different roles.

7. Conceptual significance

Within the theory of semi-discrete frameworks, the semi-discrete height function is the object that binds together three layers of structure: equilibrium, integrability, and geometry. Equilibrium is encoded by the self-stress equation; integrability is expressed by path-independence; geometry appears as the lifted semi-discrete conjugate surface in $f_i(t)$9 (Karpenkov et al., 18 Aug 2025).

Its significance lies in making the Maxwell-Cremona paradigm work in a mixed discrete-smooth regime. The function is not merely a bookkeeping device for stresses. It is the constructive mechanism by which algebraic self-stress data become a three-dimensional developable surface whose planar projection recovers the original framework.

For that reason, the semi-discrete height function occupies the same conceptual position in semi-discrete rigidity theory that affine face potentials occupy in classical discrete lifting theory. It is the scalar potential through which a difference-differential statics problem becomes a surface-geometric one.