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.

Figures (6)

• Figure 1: Three spirals with the same values of $\left\{c,\alpha,\beta,k_1,k_2\right\}$, but different $\widetilde{\alpha},\widetilde{\beta}$
• Figure 2: The family of chords of the involute \ref{['IncrCrv']}, on which $Q=-0.04$; on the right three selected chords are shown in their own coordinate system
• Figure 3: Graphs $\omega(\theta)$\ref{['eqF']}; the dotted line is the graph for $Q={-\infty}$$\left[\,t_0(\th) = \th\,\right]$
• Figure 4: The boundary conditions are borrowed from an arc of the Cornu spiral (dotted line). The curve, labelled as E, is derived from the involute of a circle. The curves, labelled L (Log. spiral) and H (Hyperbola), are derived from other base spirals, shown at the top.
• Figure 5: The boundary circles of curvature are concentric.