The use of Cornu spirals in drawing planar curves of controlled curvature
Journal of Computational and Applied Mathematics
IBM Journal of Research and Development
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Inflection points and singularities on planar rational cubic curve segments
Computer Aided Geometric Design
Planar G2 transition curves composed of cubic Bézier spiral segments
Journal of Computational and Applied Mathematics
Geometric modeling: techniques, applications, systems and tools
Geometric modeling: techniques, applications, systems and tools
Technical section: Rational cubic spline interpolation with shape control
Computers and Graphics
G2 Pythagorean hodograph quintic transition between two circles with shape control
Computer Aided Geometric Design
Transition between concentric or tangent circles with a single segment of G2 PH quintic curve
Computer Aided Geometric Design
G2 cubic transition between two circles with shape control
Journal of Computational and Applied Mathematics
An involute spiral that matches G2 Hermite data in the plane
Computer Aided Geometric Design
Incenter subdivision scheme for curve interpolation
Computer Aided Geometric Design
Admissible regions for rational cubic spirals matching G2 Hermite data
Computer-Aided Design
Fitting G2 multispiral transition curve joining two straight lines
Computer-Aided Design
Matching admissible G2 Hermite data by a biarc-based subdivision scheme
Computer Aided Geometric Design
Computer Aided Geometric Design
Employing pythagorean hodograph curves for artistic patterns
Acta Cybernetica
C1 Hermite interpolation with PH curves by boundary data modification
Journal of Computational and Applied Mathematics
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The paper derives a spiral condition, for a single Pythagorean hodograph quintic transition curve of G^2 contact, between two circles with one circle inside the other. A spiral is free of local curvature extrema, making spiral designs an interesting mathematical problem with importance for both physical and aesthetic applications. In the construction of highways or railway routes in particular, it is often desirable to have a transition curve from circle to circle. Here, we treat an open problem on planar quintic spiral segments, called transition curve elements, examine techniques for curve design using the new results, and derive lower and upper bounds for the distance between the two circles. The proposed method is applicable for non-tangent and non-concentric circles.