Arc Splines and Their Applications

• Arc splines are piecewise planar curves assembled from circular arcs that maintain G1 continuity by sharing endpoints and tangents.
• They are applied in CAD/CAM, path planning, and curve interpolation, often optimized through numerical methods and biarc constructions.
• Advanced approaches include covariance-weighted statistical fitting and symplectic formulations, highlighting their computational and geometric versatility.

Arc splines usually denote piecewise planar curves assembled from circular arcs. In one recent formulation, a polyarc is a sequence of arcs that merely share endpoints, whereas an arc spline is the $G^1$ case in which neighboring arcs share both the join point and the common tangent. In another line of work, the phrase “arc” refers instead to arc length, as in arc-length-parameterized interpolating splines, and not to circular-arc geometry. There is also an unrelated finite-geometric usage in which an arc is a subset of $\mathbb Z_n^2$ containing no three collinear points. The literature therefore uses the same word for distinct mathematical objects, and precision about context is essential (Jeon et al., 2024, Gössner, 11 Aug 2025, Matsegora et al., 19 Jun 2026, Stępień et al., 2015).

1. Terminological scope and principal meanings

In the circular-geometry literature, an arc spline is a piecewise curve made of circular arcs; one source notes that, in the broader literature, arc splines may also include line segments, but its own method is built around circular arcs as the fitting primitives (Jeon et al., 2024). A complementary formulation distinguishes between a polyarc, which is only $G^0$, and an arc spline, which is $G^1$: neighboring circular elements share not only an endpoint but also a common tangent (Gössner, 11 Aug 2025).

This circular-arc meaning should be separated from the problem studied in “Arc-Length Parameterized Interpolating Splines,” which concerns ordinary polynomial interpolating splines whose knot parameters are iteratively corrected to match the cumulative arc lengths of the interpolant. That work is explicitly not about classical circular-arc splines or clothoid splines; the “arc” in its title refers to arc length (Matsegora et al., 19 Jun 2026). A further, unrelated meaning appears in finite geometry, where an arc in $\mathbb Z_n^2$ is a set of points no three of which are collinear; that usage belongs to discrete geometry rather than spline theory (Stępień et al., 2015).

Within geometric design, CAD/CAM, CNC, path planning, and approximation of sampled planar curves, the dominant interpretation is therefore the circular one: an arc spline is a piecewise circular curve, often designed for manufacturability, geometric transparency, and low-cost intersection and trimming operations. The principal technical questions then concern parameterization of individual circular pieces, $G^1$ continuity at joins, branch selection in local interpolation, statistical fitting from noisy data, and optimization of the remaining degrees of freedom.

2. Circular-arc splines as $G^1$ piecewise circular curves

A particularly compact representation uses the endpoints $A,B$ of a circular arc together with its signed enclosed angle $\theta$. If $c=\|B-A\|$ denotes the chord length, then the radius-angle-chord relation is

$\mathbb Z_n^2$0

This makes explicit that a fixed chord supports a one-parameter family of circular arcs indexed by $\mathbb Z_n^2$1 (Gössner, 11 Aug 2025).

The same endpoint formulation yields exact expressions for arc length, area, and tangent. The total arc length is

$\mathbb Z_n^2$2

and for an arc erected over edge vector $\mathbb Z_n^2$3, with unit chord direction $\mathbb Z_n^2$4, the unit tangent at the start point is

$\mathbb Z_n^2$5

Here the symplectic notation $\mathbb Z_n^2$6, with

$\mathbb Z_n^2$7

encodes the quarter-turn operation in the plane (Gössner, 11 Aug 2025).

For a polygonal scaffold $\mathbb Z_n^2$8 with edge vectors $\mathbb Z_n^2$9, an arc can be erected over each edge. If the arc angles $G^0$0 are arbitrary, the result is only a polyarc. $G^0$1 continuity imposes a recurrence involving the polygon’s exterior angles $G^0$2: $G^0$3 Equivalently,

$G^0$4

Hence, over a fixed polygonal chain, all $G^0$5 arc splines form a one-parameter family: once one angle, typically $G^0$6, is chosen, all remaining arc angles are determined (Gössner, 11 Aug 2025).

That one-dimensional freedom can be optimized against explicit global criteria. The cited work studies minimization of total length, total areal deviation from the polygonal curve, and total bending energy, and recommends numerical minimization by golden-section search. The examples show that these objectives generally select different members of the same one-parameter family, so “best” arc spline has no single invariant meaning; it depends on whether length, area fidelity, or bending energy is prioritized (Gössner, 11 Aug 2025).

3. Biarcs as local Hermite primitives

When endpoint positions and endpoint tangent directions are prescribed, a single circular arc is generally insufficient. The standard local primitive is then the biarc, a $G^0$7 planar curve composed of two circular arcs, allowing degenerate cases in which one piece is a straight segment. Biarcs are widely used as building blocks for arc splines, biarc splines, and arc-based least-squares fitting because they preserve the simplicity of circular geometry while solving the planar $G^0$8 Hermite interpolation problem (Bertolazzi et al., 2017).

A robust algebraic formulation reduces the local biarc problem to a single $G^0$9 linear system. Given endpoints $G^1$0 and tangent angles $G^1$1, the construction normalizes by the chord direction $G^1$2, prescribes the join tangent angle by the Matlab-compatible reflection rule

$G^1$3

and solves for the normalized arc lengths $G^1$4. Curvatures then follow from

$G^1$5

Near singular equal-angle configurations, the method replaces case splitting by a Moore–Penrose pseudoinverse, and its stated purpose is to produce a solution that depends smoothly on the input geometry (Bertolazzi et al., 2017).

A more geometric treatment identifies the full one-parameter family of admissible biarcs. For two points $G^1$6 with unit tangents $G^1$7, the admissible join points lie on a joint circle. The biarc angle $G^1$8, defined by

$G^1$9

controls the joint-circle geometry, including its signed radius

$\mathbb Z_n^2$0

This exposes biarc nonuniqueness as a single geometric degree of freedom: choosing one point on the joint circle determines the pair of arcs (Goessner, 31 Oct 2025).

Several join-selection rules are discussed in that framework. The equal-chord biarc is reported as robust; parallel-tangent and J-shaped choices are described as less robust; and a new cubic midpoint biarc chooses the join point as the midpoint of an associated cubic Bézier interpolant, constrained to lie on the joint circle. The examples indicate that equal-chord and cubic-midpoint choices are the most successful in practice, with cubic midpoint improving approximation near sharp polygon angles (Goessner, 31 Oct 2025).

For arc-spline assembly, the significance of biarcs is structural. They are local, $\mathbb Z_n^2$1, and computationally cheap, but they are not $\mathbb Z_n^2$2: curvature is piecewise constant and typically jumps at the internal biarc join and again at joins between adjacent biarcs. This places biarc splines in a distinct regime from curvature-continuous spline families (Bertolazzi et al., 2017).

4. Covariance-weighted approximation of ordered data by $\mathbb Z_n^2$3 arc splines

A recent statistical formulation treats arc-spline fitting as covariance-aware constrained nonlinear least squares. The input is an ordered sequence of planar points, each equipped with a $\mathbb Z_n^2$4D covariance matrix, and the target is a $\mathbb Z_n^2$5-continuous multi-arc spline. The stated motivation is that earlier arc-spline approximation approaches assumed equal point weights, which can cause serious instability when data contain outliers or heteroscedastic noise (Jeon et al., 2024).

A single arc is parameterized by three planar points: two endpoints, the arc nodes $\mathbb Z_n^2$6, and one middle node $\mathbb Z_n^2$7 lying on the arc between them. The middle-node constraint is

$\mathbb Z_n^2$8

which places $\mathbb Z_n^2$9 on the perpendicular bisector of the chord $G^1$0. The data-fit residual for a point $G^1$1 is defined radially, by projecting $G^1$2 to a virtual point $G^1$3 on the current fitted circle and using

$G^1$4

The cost function uses squared Mahalanobis norms, so each point is weighted by the inverse of its own covariance (Jeon et al., 2024).

The multi-arc extension introduces arc nodes $G^1$5 and middle nodes $G^1$6, together with an index-based data association: for each arc node $G^1$7, the closest data-point index $G^1$8 is recomputed, and the points between $G^1$9 and $G^1$0 are assigned to segment $G^1$1. $G^1$2 continuity is enforced by orthogonality conditions between each segment’s tangent at a shared node and the other segment’s radius vector at that node. A further inequality constraint prevents degeneracy by imposing a lower bound on

$G^1$3

The fitting pipeline uses recursive linear approximation for initialization, greedy interval merging with single-arc validity tests, constrained nonlinear least squares for joint optimization, and a $G^1$4-based validation stage that declares segments invalid when too many projected points fall outside their confidence ellipses (Jeon et al., 2024).

The reported real-data example uses lane points from a vehicle experiment in Sejong City, South Korea. On a full trip, 768 lane points are compressed into 28 arc segments for the left lane and 34 for the right lane, corresponding to 57 and 69 control points respectively, while satisfying the reliability-based validity checks. The paper emphasizes that lower RMSE is not the preferred criterion in this setting; the statistical test based on Mahalanobis residuals and a $G^1$5 threshold is treated as the appropriate validity measure (Jeon et al., 2024).

5. Arc-length-parameterized interpolants and higher-order variable-curvature alternatives

Not all research relevant to arc splines uses circular pieces. “Arc-Length Parameterized Interpolating Splines” addresses a different problem: given ordered points $G^1$6, construct a spline that interpolates them exactly while using parameter values equal to the cumulative arc lengths of the spline itself. The method is coordinatewise, applies in any dimension $G^1$7, and in the experiments uses cubic interpolating splines with periodic endpoint conditions. Its core iteration alternates between spline construction and knot correction: $G^1$8 The work repeatedly emphasizes that it is not about circular-arc splines; it is about ordinary polynomial interpolating splines whose parameter is iteratively adjusted to be arc length (Matsegora et al., 19 Jun 2026).

This distinction matters conceptually. Circular arc splines have piecewise constant curvature by construction, whereas the arc-length-parameterized cubic spline remains a polynomial spline whose knot parameters become arc-length-consistent. The cited paper states that “two iterations are sufficient for most inputs,” but it also explicitly leaves convergence and uniqueness as open problems (Matsegora et al., 19 Jun 2026).

A further generalization is provided by second order spiral splines, which are unit-speed planar $G^1$9 interpolants with piecewise quadratic curvature. In that construction, the tangent angle is a $A,B$0 cubic spline,

$A,B$1

and the interpolating curve is

$A,B$2

The paper explicitly positions these splines relative to arc and clothoid splines: $A,B$3 Its fast initializer is based on two tridiagonal linear systems, and the theory applies to convex, sufficiently finely sampled data (Noakes, 2018).

Taken together, these works delineate a hierarchy. Circular arc splines occupy the constant-curvature end of the geometric-design spectrum; arc-length-parameterized interpolating splines alter parameterization rather than local curve type; and second order spiral splines replace piecewise constant curvature by a continuous quadratic curvature model (Matsegora et al., 19 Jun 2026, Noakes, 2018).

6. Symplectic and Minkowski-space formulations

A recent symplectic treatment rewrites planar circular-arc geometry in vector form. With

$A,B$4

one can express the center vector, parametric arc equation, tangent relation, segment area, and bending energy directly through the chord vector $A,B$5, its quarter-turn $A,B$6, and the angle $A,B$7. In this formulation, the endpoint parameterization of a circular arc is especially advantageous for piecewise circular curves, and the resulting $A,B$8 arc splines over a polygon are again a one-parameter family (Gössner, 11 Aug 2025).

The same symplectic vocabulary extends to biarcs. The follow-up work on “Symplectifying Biarcs” uses the skew product $A,B$9 and vector equations for the joint circle, join tangent, arc radii, and arc centers, rather than heavy angle bookkeeping. This suggests a unified coordinate-free language for both one-arc-per-edge arc splines and two-arc Hermite primitives (Goessner, 31 Oct 2025).

An even broader generalization appears in the envelope problem for evolving planar domains. There, arc splines arise from the medial axis transform (MAT) in Minkowski space $\theta$0. A point

$\theta$1

represents an oriented planar circle, and the cyclographic image of a Minkowski-space curve is a planar domain. The key observation is that circular arcs in Minkowski space correspond to MATs of arc spline domains, so approximating evolving MAT branches by Minkowski arcs yields planar envelopes represented by circular arc splines (Vráblíková et al., 24 Nov 2025).

The paper proposes four approximation schemes for the relevant Minkowski-space or planar boundary curves. Their reported characteristics are as follows (Vráblíková et al., 24 Nov 2025):

Method	Continuity / order	Arc count
DAI	$\theta$2, order $\theta$3	$\theta$4
DBI	$\theta$5, order $\theta$6	$\theta$7
IAI	$\theta$8, order $\theta$9	$c=\|B-A\|$0
IBI	$c=\|B-A\|$1, order $c=\|B-A\|$2	$c=\|B-A\|$3

These methods produce an arc-spline superset of the evolving envelope, after which redundant branches are removed by a sweep-line algorithm for circular arcs. The stated advantage is that the geometric flexibility of circular arcs combines naturally with trimming machinery of optimal computational complexity (Vráblíková et al., 24 Nov 2025).

Across these symplectic and Minkowski-space formulations, arc splines remain piecewise circular objects, but their role broadens. They are not only local interpolation tools or low-order approximants of sampled data; they also become a geometrically structured output representation for problems involving polygonal smoothing, Hermite interpolation, statistical fitting, and even envelopes of evolving domains.