Involutes in Normed Planes

• Involutes in normed planes are norm-dependent analogues of classical involutes, extending evolute-involute relationships via Minkowski geometry.
• The theory integrates smooth curves, Legendre immersions, and discrete constant-width polygons using curvature concepts and Birkhoff orthogonality.
• Iterative constructions of involutes serve as a symmetrization mechanism, linking spectral, gauge, and nonsmooth extensions for center extraction.

Searching arXiv for recent and foundational papers on involutes in normed planes, constant width, and related curvature/evolute theory. Involutes in normed planes are the norm-dependent analogues of classical involutes in Euclidean differential geometry: a curve, front, or polygon is an involute of another when the latter is its evolute, with all constructions taken relative to a prescribed norm or gauge. In the planar Minkowski setting, the unit ball $U$ and its dual $V$ replace the Euclidean circle, Birkhoff orthogonality replaces metric perpendicularity, and several curvature notions coexist. The resulting theory includes smooth regular curves, Legendre immersions with singularities, polygons of constant Minkowskian width, spectral descriptions via Sturm–Liouville operators, and extensions to gauge planes and even nonsmooth convex disks (Balestro et al., 2017, Balestro et al., 2017, Craizer et al., 2014, Craizer et al., 2016, Balestro et al., 2019, Lángi et al., 2 Sep 2025).

1. Minkowski framework and constant-width geometry

A Minkowski plane is $(\mathbb{R}^2,U)$, where $U$ is a compact, convex, centered set called the unit ball, and $\partial U$ is the Minkowski unit circle. For a convex body or polygon $P$, the support function in a dual direction $f$ is

$h(P)(f)=\sup\{f(p):p\in P\},$

and the width in direction $f$ is

$w(P)(f)=h(P)(f)+h(P)(-f).$

Constant width means that $V$0 is independent of $V$1 (Craizer et al., 2014).

For convex polygons, the constant-width condition admits a concrete characterization. If $V$2 has opposite sides parallel, then $V$3 has constant $V$4-width if and only if $V$5 is homothetic to $V$6. Equivalently, corresponding diagonals of $V$7 and $V$8 are parallel, and

$V$9

for a constant $(\mathbb{R}^2,U)$0, where one may construct $(\mathbb{R}^2,U)$1 by

$(\mathbb{R}^2,U)$2

with center $(\mathbb{R}^2,U)$3 (Craizer et al., 2014).

For smooth strictly convex curves, an analogous normalization is used to place a given convex curve into a Minkowski geometry in which it becomes a constant-width curve. If $(\mathbb{R}^2,U)$4 bounds a strictly convex region $(\mathbb{R}^2,U)$5, the unit ball may be chosen as

$(\mathbb{R}^2,U)$6

and the constant-width relation takes the form

$(\mathbb{R}^2,U)$7

where $(\mathbb{R}^2,U)$8 parametrizes $(\mathbb{R}^2,U)$9 (Craizer, 2013).

These constructions are foundational because involutes in normed planes are especially tractable for constant-width objects: the dual unit ball, the normal directions, and the curvature radius are then linked by explicit formulas.

2. Smooth involutes, evolutes, and curvature-dependent normality

In the smooth theory, involutes are defined in close formal analogy with the Euclidean case, but the geometry depends on the chosen curvature concept and on the unit circle of the norm. If $U$0 is a smooth regular curve parametrized by arc length and $U$1 is an arc-length parametrization of the Minkowski unit circle, the evolute is the locus of curvature centers

$U$2

where $U$3 is the Minkowski curvature radius associated with the circular curvature $U$4, and $U$5 is determined by

$U$6

An involute $U$7 of $U$8 is then a curve whose evolute is $U$9 (Balestro et al., 2017).

A central result is the explicit description of all involutes of a given smooth curve: $\partial U$0 where $\partial U$1 is constant and $\partial U$2 satisfies

$\partial U$3

The derivative of the involute is

$\partial U$4

Accordingly, except at points where $\partial U$5, the involute is regular provided the original curve is regular (Balestro et al., 2017).

This formula shows that, in normed planes, involutes are precisely left parallels of the same evolute. The associated structural statements also persist: the evolute is the envelope of the field of left-normal lines, the evolute is the locus of singularities of the family of left parallels, and ordinary cusps occur at vertices where $\partial U$6 vanishes but $\partial U$7 does not (Balestro et al., 2017).

A related formulation appears in the gauge-plane extension. For a curve $\partial U$8 parametrized by arc length with respect to a gauge $\partial U$9, an involute is written

$P$0

and the reciprocity between evolute and involute holds under sign conditions involving the arc-length curvature $P$1: if $P$2, the evolute of the involute is the original curve, while the reverse-direction statement requires $P$3 (Balestro et al., 2019).

A recurrent source of confusion is the role of “the normal.” In Euclidean geometry, normality and curvature are uniquely tied to the inner product. In normed planes, several curvature types coexist—Minkowski, circular, normal, and arc-length curvature—and the relevant normal field is usually defined via Birkhoff orthogonality, which is generally not symmetric (Balestro et al., 2017, Balestro et al., 2019).

3. Legendre curves and involutes with singularities

The theory extends beyond regular curves to Legendre curves, which are smooth plane curves that may have singular points but still possess a smooth normal field. In a normed plane $P$4, a Legendre curve is a smooth map

$P$5

such that

$P$6

for all $P$7, where $P$8 is the unit circle and $P$9 denotes Birkhoff orthogonality. A Legendre immersion is a Legendre curve for which $f$0 and $f$1 are never simultaneously zero (Balestro et al., 2017).

Using the map $f$2 that assigns to each unit vector its Birkhoff orthogonal direction, one defines

$f$3

Then there exist smooth functions $f$4 such that

$f$5

The pair $f$6 is the curvature pair, and for regular curves the circular curvature is

$f$7

The evolute is

$f$8

whenever $f$9 does not vanish (Balestro et al., 2017).

For a Legendre immersion $h(P)(f)=\sup\{f(p):p\in P\},$0 with curvature pair $h(P)(f)=\sup\{f(p):p\in P\},$1 and $h(P)(f)=\sup\{f(p):p\in P\},$2 nowhere vanishing, the involute family is given explicitly by

$h(P)(f)=\sup\{f(p):p\in P\},$3

where $h(P)(f)=\sup\{f(p):p\in P\},$4. Its curvature pair is

$h(P)(f)=\sup\{f(p):p\in P\},$5

with

$h(P)(f)=\sup\{f(p):p\in P\},$6

The evolute of $h(P)(f)=\sup\{f(p):p\in P\},$7 is $h(P)(f)=\sup\{f(p):p\in P\},$8 (Balestro et al., 2017).

The appearance of $h(P)(f)=\sup\{f(p):p\in P\},$9 is a genuinely non-Euclidean feature. In the Euclidean plane, $f$0; in a general normed plane, $f$1 encodes the geometry of the norm through the differential of the Birkhoff-orthogonality map. This establishes that the absence of an inner product does not prevent a differential-geometric theory of involutes, but it changes the analytic form of the formulas (Balestro et al., 2017).

4. Polygonal involutes and discrete constant-width theory

The polygonal theory provides a direct discrete analogue of the smooth one. For a polygon $f$2 of constant $f$3-width, the Minkowskian normal at the vertex $f$4 is the line

$f$5

The evolute is the polygon whose vertices are the intersections of consecutive normals: $f$6 where the curvature radius $f$7 is determined by

$f$8

Using the dual ball $f$9, one may also write

$w(P)(f)=h(P)(f)+h(P)(-f).$0

so that

$w(P)(f)=h(P)(f)+h(P)(-f).$1

Here $w(P)(f)=h(P)(f)+h(P)(-f).$2 is the curvature center and $w(P)(f)=h(P)(f)+h(P)(-f).$3 is the curvature radius (Craizer et al., 2014).

The central equidistant of $w(P)(f)=h(P)(f)+h(P)(-f).$4 is

$w(P)(f)=h(P)(f)+h(P)(-f).$5

and the evolute of $w(P)(f)=h(P)(f)+h(P)(-f).$6 satisfies

$w(P)(f)=h(P)(f)+h(P)(-f).$7

If

$w(P)(f)=h(P)(f)+h(P)(-f).$8

then an involute $w(P)(f)=h(P)(f)+h(P)(-f).$9 of $V$00, now taken with respect to the dual unit ball $V$01, is

$V$02

where

$V$03

For polygons, the involutes of a given evolute are precisely the equidistants of the original polygon, parametrized by the central equidistant (Craizer et al., 2014).

The discrete theory preserves many properties of the smooth case. It includes polygonal versions of Minkowskian curvature, evolutes, involutes, Barbier’s theorem, and Minkowski-type inequalities. For instance, if $V$04 is a Minkowskian polygonal equidistant of diameter $V$05, then its $V$06-length satisfies

$V$07

and for opposite sides one has

$V$08

The mixed area of polygons $V$09 with parallel sides is

$V$10

and the Minkowski inequality takes the form

$V$11

These statements situate involutes inside a broader discrete affine–Minkowskian geometry (Craizer et al., 2014).

5. Iteration of involutes, area monotonicity, and canonical centers

One of the central developments in the subject is the iterative application of the involute operation. In the smooth constant-width setting, let $V$12 be a convex curve, let the center symmetry set (CSS) be the envelope of its diameters, and let the area evolute (AE) be the locus of the midpoints of diameters: $V$13 For the Minkowski norm constructed from $V$14, the CSS is the evolute of $V$15, and the AE is an involute of the CSS (Craizer, 2013).

If $V$16, define

$V$17

Then the $V$18-involute of $V$19 is

$V$20

and more generally

$V$21

is a family of $V$22-equidistants sharing the same evolute. Moreover,

$V$23

the curves $V$24 are constant $V$25-width curves with curvature radius $V$26, and the evolute of each $V$27 is $V$28 (Craizer, 2013).

Area monotonicity is crucial. The involute $V$29 is contained in the region bounded by $V$30, and the signed areas satisfy

$V$31

In the polygonal theory the corresponding formula is

$V$32

which again forces area decrease under iteration (Craizer, 2013, Craizer et al., 2014).

The iterative process is defined recursively. In the polygonal formulation,

$V$33

The resulting regions are nested: $V$34 and their intersection is a unique point $V$35, the central point of the original polygon. Both sequences converge to this point, and for fixed scalars $V$36,

$V$37

The smooth analogue yields convergence in the $V$38 topology to a unique point $V$39, with the corresponding equidistants converging uniformly to symmetric constant-width curves centered at $V$40 (Craizer et al., 2014, Craizer, 2013).

This identifies iterated involutes as a norm-dependent symmetrization mechanism. A plausible implication is that the involute operator functions not merely as a local differential-geometric construction, but also as a global center-extraction procedure for constant-width objects.

6. Spectral, gauge-theoretic, and nonsmooth extensions

A different analytic perspective arises from closed cycloids in a normed plane. Writing the support function of a curve as

$V$41

its curvature radius is

$V$42

the evolute support function is

$V$43

and the double evolute support function is

$V$44

The associated Sturm–Liouville equation is

$V$45

with $V$46 equal to the support function or the radius of curvature. In this framework, the iteration of involutes of a closed curve of zero dual length converges to a constant curve, because

$V$47

as $V$48 (Craizer et al., 2016).

The gauge-plane generalization removes the symmetry axiom from the norm. With gauge $V$49, unit disk $V$50, associated gauge

$V$51

and Birkhoff orthogonality

$V$52

one retains four curvature types and extends evolutes and involutes to asymmetric geometries. The involute formula

$V$53

and the evolute formula

$V$54

still govern the reciprocity between the two constructions, but right and left normality must be distinguished carefully (Balestro et al., 2019).

The most general geometric definition in the material considered here appears for arbitrary convex disks, without assuming smoothness or strict convexity. Given a convex disk $V$55, a point $V$56, and the oriented tangent line $V$57 at angle $V$58, the involute point $V$59 is defined as the point on $V$60 whose signed normed distance from the contact point $V$61 is $V$62, where $V$63 is the normed arc length along $V$64 from $V$65 to $V$66. The resulting involute is injective; on any interval $V$67 it is convex, and in strictly convex norms the convexity is strict. It also satisfies an increasing chord property on such intervals (Lángi et al., 2 Sep 2025).

These extensions address two common misconceptions. First, involutes in normed planes are not restricted to the smooth strictly convex category; there is a geometric construction for arbitrary convex disks (Lángi et al., 2 Sep 2025). Second, the Euclidean formula does not exhaust the subject: in Legendre and gauge settings, the geometry depends explicitly on Birkhoff orthogonality, curvature pairs, and the norm-dependent factor $V$68 (Balestro et al., 2017, Balestro et al., 2019).