Semi-Discrete Conjugate Surfaces

• Semi-discrete conjugate surfaces are defined as surfaces parameterized by one discrete and one smooth variable that satisfy specific conjugacy conditions in their mixed directions.
• The methodology involves both strip-wise and net-based formulations, using differential-difference equations to capture curvature, isothermicity, and developability.
• Key results connect integrable system techniques with applications in minimal surface theory and rigid-ruling deformations in geometric modeling.

Semi-discrete conjugate surfaces are surfaces with one discrete and one smooth parameter for which the mixed smooth/discrete directions satisfy a conjugacy condition. The literature does not present a single universal definition. Instead, closely related formulations appear in semi-discrete conjugate nets $x(k,t)$ on $\mathbb Z\times\mathbb R$, in sequences of smooth curves whose neighboring strips are developable ruled surfaces, and in the analytic theory of hyperbolic differential-difference equations underlying classical conjugate-net geometry (Yasumoto et al., 2017, Rossman et al., 2012, Karpenkov et al., 18 Aug 2025, Smirnov, 23 Jun 2025).

1. Foundational definitions

A standard semi-discrete domain is a subdomain $\mathbb D\subset \mathbb Z\times\mathbb R$, with discrete variable $k\in\mathbb Z$, smooth variable $t\in\mathbb R$, and notation

$$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$

In one formulation, a semi-discrete surface is assumed to be a conjugate net when $\partial x$, $\partial x_1$, and $\Delta x$ lie in a $2$-plane in the ambient $\mathbb Z\times\mathbb R$0; that $\mathbb Z\times\mathbb R$1-plane is the tangent plane of the surface at the edge $\mathbb Z\times\mathbb R$2. In another formulation, a semi-discrete surface is called conjugate when $\mathbb Z\times\mathbb R$3, $\mathbb Z\times\mathbb R$4, and $\mathbb Z\times\mathbb R$5 are linearly dependent (Yasumoto et al., 2017, Rossman et al., 2012).

A strip-wise formulation replaces a pointwise net by a sequence of smooth curves. For

$\mathbb Z\times\mathbb R$6

the strip between neighboring curves is represented by

$\mathbb Z\times\mathbb R$7

A semi-discrete surface is then called conjugate when each such ruled strip is developable. Remark 2.3 gives the equivalent condition that $\mathbb Z\times\mathbb R$8, $\mathbb Z\times\mathbb R$9, and $\mathbb D\subset \mathbb Z\times\mathbb R$0 are linearly dependent for all $\mathbb D\subset \mathbb Z\times\mathbb R$1; in $\mathbb D\subset \mathbb Z\times\mathbb R$2, this is

$\mathbb D\subset \mathbb Z\times\mathbb R$3

This formulation makes explicit that the smooth direction is tangent to each curve, while the discrete direction is carried by rulings between neighboring curves (Karpenkov et al., 18 Aug 2025).

Regularity is expressed by independence of the smooth and discrete directions. In the strip model, the pairs $\mathbb D\subset \mathbb Z\times\mathbb R$4 and $\mathbb D\subset \mathbb Z\times\mathbb R$5 are required to be linearly independent for all $\mathbb D\subset \mathbb Z\times\mathbb R$6. In the net-based literature, the same nondegeneracy appears through tangent-plane and curvature-line assumptions (Karpenkov et al., 18 Aug 2025, Yasumoto et al., 2017).

2. Isothermic, Legendre, and minimal structures

A major branch of the subject places semi-discrete conjugate surfaces inside curvature-line and isothermic geometry. A semi-discrete Legendre immersion $\mathbb D\subset \mathbb Z\times\mathbb R$7 requires three compatibility conditions: $\mathbb D\subset \mathbb Z\times\mathbb R$8, $\mathbb D\subset \mathbb Z\times\mathbb R$9, and $k\in\mathbb Z$0 lie in the tangent plane at $k\in\mathbb Z$1; $k\in\mathbb Z$2, $k\in\mathbb Z$3, and $k\in\mathbb Z$4 lie in one $k\in\mathbb Z$5-dimensional plane; and $k\in\mathbb Z$6 is perpendicular to $k\in\mathbb Z$7. Curvature-line parametrization is then imposed by

$k\in\mathbb Z$8

together with the tangent cross ratio

$k\in\mathbb Z$9

The reality of this cross ratio implies the circularity condition: there is a circle through $t\in\mathbb R$0 and $t\in\mathbb R$1 tangent to $t\in\mathbb R$2 at $t\in\mathbb R$3 and $t\in\mathbb R$4 at $t\in\mathbb R$5 (Yasumoto et al., 2017).

Semi-discrete isothermicity is characterized by factorization of the tangent cross ratio,

$t\in\mathbb R$6

with $t\in\mathbb R$7 depending only on the smooth variable and $t\in\mathbb R$8 only on the discrete variable. In the older minimal-surface formulation, a circular semi-discrete surface is isothermic when there exist positive functions $t\in\mathbb R$9 such that

$$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$0

These two descriptions belong to the same integrable framework of semi-discrete isothermic geometry (Yasumoto et al., 2017, Rossman et al., 2012).

The mixed-area formalism extends curvature theory to pairs of compatible semi-discrete conjugate surfaces. For two semi-discrete conjugate surfaces $$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$1 satisfying

$$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$2

the mixed area is

$$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$3

For a curvature-line parametrized semi-discrete surface $$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$4, Gaussian and mean curvature on edges are defined by

$$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$5

In the same setting, the paper gives explicit principal-curvature formulas and a duality relation between surface and normal,

$$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$6

together with parallel families $$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$7 in $$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$8 and $$x=x(k,t), \qquad x_1=x(k+1,t), \qquad \partial x=\frac{dx}{dt}, \qquad \Delta x=x_1-x.$$9, and

$\partial x$0

in $\partial x$1 and $\partial x$2 (Yasumoto et al., 2017).

Minimality is expressed through duality. If $\partial x$3 and $\partial x$4 are conjugate semi-discrete surfaces, they are dual when

$\partial x$5

A semi-discrete isothermic surface is minimal when $\partial x$6 is inscribed in a sphere. The Weierstrass theorem states that every semi-discrete minimal surface is reconstructed from a semi-discrete holomorphic function $\partial x$7, and conversely every semi-discrete minimal surface arises in that way. In that representation, the dual spherical surface is the inverse stereographic image of $\partial x$8 (Rossman et al., 2012).

3. Hyperbolic differential-difference equations and Darboux-Laplace theory

The analytic counterpart of semi-discrete conjugate geometry is a hyperbolic differential-difference equation. The relevant operator is

$\partial x$9

where $\partial x_1$0, $\partial x_1$1, and $\partial x_1$2. This is the semi-discrete analog of the smooth hyperbolic operator $\partial x_1$3 and the fully discrete hyperbolic operator $\partial x_1$4 (Smirnov, 23 Jun 2025).

The operator admits two first-order factorizations,

$\partial x_1$5

and

$\partial x_1$6

where $\partial x_1$7 and $\partial x_1$8 are the semi-discrete Laplace invariants. Vanishing of $\partial x_1$9 or $\Delta x$0 is equivalent to factorization. Under Laplace transformation, neighboring operators satisfy $\Delta x$1, and the invariants obey one form of the semi-discrete two-dimensional Toda lattice (Smirnov, 23 Jun 2025).

This framework is directly tied to conjugate-net theory. The paper starts from the classical fact that a parametrization of a surface in $\Delta x$2 whose second fundamental form is diagonal is a conjugate net, and such nets satisfy a linear hyperbolic second-order PDE. In the semi-discrete case, the same role is played by a scalar hyperbolic differential-difference equation; the paper states that a semi-discrete surface coordinate function $\Delta x$3 in an affine or projective ambient space is typically governed componentwise by exactly such an equation (Smirnov, 23 Jun 2025).

For finite Laplace series, the theory becomes explicit. If forward transforms terminate at $\Delta x$4 and backward transforms at $\Delta x$5, then the general solution of $\Delta x$6 has the form

$\Delta x$7

where $\Delta x$8 is an arbitrary function of the discrete variable and $\Delta x$9 is an arbitrary function of the continuous variable. The paper also proves a semi-discrete Darboux determinant formula, giving the general solution as a single determinant built from derivatives of $2$0, shifts of $2$1, basis functions $2$2, $2$3, and a gauge factor $2$4 (Smirnov, 23 Jun 2025).

A plausible implication is that finite-Laplace-series semi-discrete conjugate surfaces form the directly solvable subclass of the theory: the coordinate functions are reconstructed from arbitrary smooth data in the continuous direction and arbitrary shifted data in the discrete direction.

4. Developable strips, frameworks, and liftings

A geometric application of the strip model is the semi-discrete Maxwell-Cremona correspondence. A planar semi-discrete framework is a map

$2$5

that is, a discrete sequence of smooth planar curves $2$6. A stress consists of functions $2$7 along the curves and $2$8 along the strip direction between $2$9 and $\mathbb Z\times\mathbb R$00. Writing the tangential force as

$\mathbb Z\times\mathbb R$01

the local equilibrium equation is

$\mathbb Z\times\mathbb R$02

or equivalently

$\mathbb Z\times\mathbb R$03

A pair $\mathbb Z\times\mathbb R$04 satisfying this equation is a self-stress (Karpenkov et al., 18 Aug 2025).

The paper defines a semi-discrete height function $\mathbb Z\times\mathbb R$05 by summing discrete jump terms and continuous strip integrals along an increasing semi-discrete path. Theorem 3.4 states that $\mathbb Z\times\mathbb R$06 is a self-stress if and only if $\mathbb Z\times\mathbb R$07 is independent of the chosen path. This produces a lifting

$\mathbb Z\times\mathbb R$08

For neighboring curves, the lifted vectors

$\mathbb Z\times\mathbb R$09

are linearly dependent, so each strip is developable and $\mathbb Z\times\mathbb R$10 is a semi-discrete conjugate surface (Karpenkov et al., 18 Aug 2025).

The principal theorem states both directions of the correspondence: if $\mathbb Z\times\mathbb R$11 is a self-stress for a framework $\mathbb Z\times\mathbb R$12, then its semi-discrete lifting is a semi-discrete conjugate surface in $\mathbb Z\times\mathbb R$13; conversely, if $\mathbb Z\times\mathbb R$14 is a semi-discrete conjugate surface in $\mathbb Z\times\mathbb R$15 and its orthogonal projection $\mathbb Z\times\mathbb R$16 to $\mathbb Z\times\mathbb R$17 is regular, then $\mathbb Z\times\mathbb R$18 is a stressable framework. In the abstract, this is summarized as the statement that stressable semi-discrete frameworks in the plane are precisely the orthogonal projections of semi-discrete conjugate surfaces in $\mathbb Z\times\mathbb R$19-space (Karpenkov et al., 18 Aug 2025).

The same paper records further geometric consequences. Frameworks with vanishing boundary forces $\mathbb Z\times\mathbb R$20 have liftings with planar boundary curves. For a one-strip framework with no boundary forces, liftability is characterized by an explicit equation, labeled (4.1), involving derivatives of the two boundary curves (Karpenkov et al., 18 Aug 2025).

5. Globally developable nets and rigid-ruling deformations

A more specialized theory studies globally developable semi-discrete conjugate nets arising from crease-rule patterns. The basic object is a sequence of smooth curves

$\mathbb Z\times\mathbb R$21

with ruled strips between adjacent curves. A single ruled patch is written as

$\mathbb Z\times\mathbb R$22

where $\mathbb Z\times\mathbb R$23 is the directrix and $\mathbb Z\times\mathbb R$24 is the unit ruling direction. Developability is imposed by

$\mathbb Z\times\mathbb R$25

The paper interprets these objects simultaneously as curved-crease origami and as developable semi-discrete conjugate nets (Mundilova, 6 Mar 2026).

The central deformation problem is rigid-ruling folding: a continuous family of non-trivial folded states preserving the rulings, equivalently a conjugate-net-preserving isometry. For a single crease, the folded state is governed by

$\mathbb Z\times\mathbb R$26

$\mathbb Z\times\mathbb R$27

$\mathbb Z\times\mathbb R$28

For multiple creases, compatibility across a common strip is encoded by equality of the ruling curvature

$\mathbb Z\times\mathbb R$29

The paper states that the ruling curvature determines the bend configuration of a developed patch up to Euclidean motion (Mundilova, 6 Mar 2026).

For a pair of creases, the main theorem gives necessary and sufficient conditions for a rigid-ruling folding motion. With $\mathbb Z\times\mathbb R$30 and $\mathbb Z\times\mathbb R$31 defined from the crease data, the conditions are

$\mathbb Z\times\mathbb R$32

and

$\mathbb Z\times\mathbb R$33

The paper also gives integrated versions $\mathbb Z\times\mathbb R$34, $\mathbb Z\times\mathbb R$35, and $\mathbb Z\times\mathbb R$36. It proves a local-to-global assembly lemma: if each pair of adjacent creases with their incident surfaces can undergo a rigid-ruling folding motion, then the entire regular crease-rule pattern can too (Mundilova, 6 Mar 2026).

Several structural restrictions follow. If a candidate crease-rule pattern with rigid-ruling motion contains one constant fold-angle crease, then all creases are constant fold-angle creases. A Combescure transformation preserves existence of folded states, existence of rigid-ruling folding motions, and whether a crease is planar or constant fold-angle. The paper also derives explicit nonlinear third-order ODEs for appending new compatible creases in the cylindrical and conical cases, and notes that tangent-developable strips reduce by a semi-discrete Combescure transformation to the conical case. In each setting, the appended crease generally depends on three free initial values (Mundilova, 6 Mar 2026).

6. Related traditions, neighboring theories, and scope

The phrase “conjugate surface” is not used uniformly across the literature. In semi-discrete differential geometry, it usually refers to a parametrization or strip geometry, not necessarily to a transformed partner surface. One paper explicitly notes that it does not develop a separate theory explicitly called “semi-discrete conjugate surfaces” in the classical sense of pairs of conjugate surfaces, even though its basic objects are semi-discrete conjugate nets and its mixed-area formalism is defined for two semi-discrete conjugate surfaces satisfying

$\mathbb Z\times\mathbb R$37

(Yasumoto et al., 2017).

In the smooth CMC tradition, by contrast, “conjugate” often refers to sister-surface correspondences rather than conjugate-net parametrization. The Daniel sister correspondence establishes an isometric duality between minimal immersions

$\mathbb Z\times\mathbb R$38

and $\mathbb Z\times\mathbb R$39-immersions

$\mathbb Z\times\mathbb R$40

with the phase-rotation relations

$\mathbb Z\times\mathbb R$41

This is structurally relevant to semi-discrete conjugate-surface theory, but it belongs to a different, fully smooth usage of “conjugate” (Manzano et al., 2018).

A separate integrable bridge comes from the discretization principle via permutability. Starting from a smooth surface and two commuting transforms, one obtains a Bianchi quadrilateral and then a discrete net by evaluating transformed surfaces at a fixed point. The paper emphasizes that if one lets one variable remain the smooth parameter and only iterates one transform direction discretely, one gets a family $\mathbb Z\times\mathbb R$42 depending smoothly on $\mathbb Z\times\mathbb R$43 and discretely on $\mathbb Z\times\mathbb R$44. It does not develop this semi-discrete geometry, but presents the mechanism as an obvious precursor (Cho et al., 23 Mar 2026).

Two neighboring theories supply additional context. “Principal binets” are fully discrete, not semi-discrete, but they separate conjugacy from orthogonality by using a pair of primal and dual conjugate nets, Möbius/Laguerre/Lie lifts, and a consistency principle on higher-dimensional lattices. Their direct contribution to semi-discrete theory is therefore structural rather than formal (Affolter et al., 2024). Likewise, the paper on explicit semi-discrete surfaces built from Jacobi elliptic functions does not define semi-discrete conjugate surfaces in the standard net-theoretic sense, but studies an adjacent integrable class of semi-discrete surfaces and discrete $\mathbb Z\times\mathbb R$45-surfaces, with the edge relation

$\mathbb Z\times\mathbb R$46

as a central compatibility formula (Kajiwara et al., 2024).

Taken together, these strands show that semi-discrete conjugate surfaces occupy an interface between projective conjugacy, curvature-line and isothermic geometry, developable-strip kinematics, and hyperbolic differential-difference equations. The field is technically unified by mixed smooth/discrete compatibility, but not by a single canonical definition.