This paper presents a new study of the geometric structure of 3D spinal curves. The spine is considered as an heterogeneous beam, compound of vertebrae and intervertebral discs. The spine is modeled as a deformable wire along which vertebrae are beads rotating about the wire. 3D spinal curves are compound of plane regions connected together by zones of transition. The 3D spinal curve is uniquely flexed along the plane regions. The angular offsets between adjacent regions are concentrated at level of the middle zones of transition, so illustrating the heterogeneity of the spinal geometric structure. The plane regions along the 3D spinal curve must satisfy two criteria: (i) a criterion of minimum distance between the curve and the regional plane and (ii) a criterion controlling that the curve is continuously plane at the level of the region. The geometric structure of each 3D spinal curve is characterized by the sizes and orientations of regional planes, by the parameters representing flexed regions and by the sizes and functions of zones of transition. Spinal curves of asymptomatic subjects show three plane regions corresponding to spinal curvatures: lumbar, thoracic and cervical curvatures. In some scoliotic spines, four plane regions may be detected. 1. Introduction Sagittal balance of asymptomatic subjects and patients has been studied from sagittal radiographic images [1–3]. Spinal curves show generally three regions with opposite curvatures: lumbar, thoracic, and cervical curvatures. Adjacent regional curves are bounded either by specific vertebrae in case of clinical applications or by points of inflexion when the sagittal curve is represented by a mathematical function [4, 5]. Angles of regional curvatures (Cobb angles) may be directly measured upon radiographic images or calculated from the sagittal curve equation. The mathematical equation of the sagittal spinal projection has been obtained from using a new technique [6]. The projected spinal structure is delimited by two border curves. Each of them is represented by a B-spline curve passing a restricted number of points recorded along the border. The sagittal spinal curve representing the beam projection is defined as the median curve drawn between the two border lines. It is modeled by a B-spline curve. In sagittal projections, each spinal region showing homogeneous curvature is described by parameters defining its shape and orientation [7]. The technique defined for sagittal radiographic projections has been extended to frontal images of scoliotic spines [8], frontal and sagittal radiographic
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