Construction of a spiral with given boundary conditions by inversion of the involute of a circle

Abstract: To construct a curve with a monotonic curvature (spiral), and given tangents and curvatures at the ends, the author proposed the following method. From given boundary conditions, the values of two inverse invariants are determined. Then, on some base spiral (initially, a logarithmic spiral was chosen), an arc with the same invariant values is sought for. A linear-fractional map of the found arc solves the problem. It seems that choosing the involute of a circle as the base spiral yields the simplest solution, which we present here.

• The paper demonstrates that inverting a circle’s involute yields an analytic, monotonic spiral that satisfies specific G2 Hermite boundary conditions.
• It employs Möbius transformations and invariant quantities to map a carefully selected involute arc into a canonical frame matching prescribed endpoints.
• The method achieves minimal algebraic complexity, offering robust interpolation and flexibility for CAD applications with various base spiral options.

Construction of Planar Spirals with Prescribed Boundary Data via Inversion of the Involute of a Circle

Problem Setting and Existing Approaches

The interpolation of planar curves with monotonic curvature, commonly referred to as "spirals," under fixed $G^2$ Hermite data (chord endpoints, tangents, and curvatures) constitutes a foundational problem in Computer-Aided Design (CAD). The $G^2$ Hermite spiral problem seeks a curve joining two points with prescribed tangent directions and curvatures at its endpoints, while maintaining monotonic curvature throughout. Traditional strategies include triarc constructions (joining three circular arcs) [Meek & Walton 1996], conic-based interpolants [Frey & Field 2000], rational spirals [Dietz et al. 2008], and involutes of rational PH-curves [Goodman et al. 2009]. These methods either impose geometric limitations or require significant computational manipulation—often lacking general explicit procedures or failing to cover the entire space of admissible boundary data.

A more flexible and general approach is the two-step method introduced in [Kurnosenko 2011], which employs a "base spiral" and a Möbius (linear-fractional) transformation. This framework utilizes the invariance of certain geometric quantities under Möbius maps, thereby reducing the interpolation to matching a small set of invariants.

Methodology: Spiral Construction via the Involute of a Circle

The present paper (2603.29596) proposes a construction in which the base spiral is the involute of a circle, and the interpolating spiral is obtained by inverting an arc of this base spiral via a Möbius map determined by the boundary conditions.

Boundary Invariants and Möbius Equivalence

Given endpoints $A$ and $B$ (placed symmetrically on the $x$-axis), tangent directions $\alpha$, $\beta$, and curvatures $k_1$ and $k_2$, the method first computes two Möbius-invariant quantities:

• The "inverse distance" invariant $Q$
• The angular width $G^2$0

These are given by:

$G^2$1

Here, $G^2$2 and $G^2$3, $G^2$4 are "cumulative" endpoint tangents, accounting for multiple windings.

The existence of a monotonic spiral with the given boundary data is characterized by the conditions $G^2$5 and $G^2$6. The case $G^2$7 corresponds uniquely to biarc solutions (endpoint curvature circles are tangent).

Parameterization of the Involute and Arc Selection

The involute of the circle (radius $G^2$8 for simplicity) is parametrized as follows:

$G^2$9

To ensure increasing curvature, the involute is reflected appropriately.

The crucial step is selecting an arc $A$0, centered symmetrically at $A$1 ($A$2, $A$3), such that the map of this arc under the subsequent Möbius transformation will achieve the prescribed boundary invariants:

1. For a prescribed $A$4 and $A$5, compute $A$6 via:

$A$7

2. Solve for $A$8 satisfying:

$A$9

The solution can be determined numerically and is unique on well-defined search intervals delimited by the roots of $B$0.

Construction of the Final Spiral

Once the appropriate arc is determined, it is mapped (via scaling, translation, and rotation) into a canonical chord coordinate frame, yielding a "base interpolant" with known endpoint data. A Möbius transformation—explicitly determined by boundary data and the base interpolant's endpoint geometry—maps this arc onto a spiral that meets the target $B$1 Hermite conditions. The transformation preserves monotonicity and results in an analytic, closed-form representation for the spiral.

Numerical Illustrations and Comparative Analysis

Several figures in the paper compare the involute-based spiral (E), log-spiral-based spiral (L) [Kurnosenko 2011], and hyperbola-based spiral (H) [Kurnosenko 2011b], under identical $B$2 endpoints. In all scenarios—including those drawn from the Cornu spiral, concentric curvature circles, and the tractrix—this involute approach delivers accurate matches to the prescribed invariants.

In particular, the involute-based spiral yields solutions of minimal algebraic complexity among these classes, with all boundary constraints precisely satisfied. For tractrix data, the method exactly recovers the original curve, since the tractrix is the inverse of the involute.

Theoretical and Practical Implications

The inversion-of-involute construction exhibits several important features:

• Completeness: For any admissible $B$3 Hermite data where a spiral with monotonic curvature exists, the method provides an explicit, constructive solution.
• Simplicity: The arclength, curvature, and spiral segment all admit analytic expressions, enhancing both theoretical transparency and computational efficiency.
• Flexibility: By varying the base spiral (e.g., logarithmic, hyperbolic, or involute), the designer can opt for specific $B$4 profiles or prefer certain arc lengths.

Practical implications in CAD and scientific computing include robust generation of transition curves between geometry elements with guaranteed monotonicity of curvature, eliminating visually and physically undesirable inflection points or curvature reversals. The approach generalizes well to composite curves (e.g., for piecewise-constant curvature applications), and can be extended to open geometric modeling systems requiring precise $B$5 transitions.

Conclusion

This paper presents a thorough geometric and analytic solution to the planar $B$6 Hermite spiral interpolation problem via inversion of the involute of a circle. The method is characterized by the use of Möbius invariants, analytic arc parameterization, and explicit transformation to match boundary data, yielding an effective and minimally complex interpolation strategy covering all admissible $B$7 conditions. Extensions to other base spirals are possible, and the technique offers a systematic pathway for the design of monotonically curved planar segments in geometric modeling and CAD. Future work may include extending this analytic inversion-framework to three-dimensional curve interpolation and spline-based geometric pipelines.