Fig. 2. Extraction of spline parameters from three-dimensional reconstructions based on high-spatial-resolution magnetic resonance imaging (MRI). (A) Definition of spline parameters. The Bezier equations define the x and y positions of the points along the curve with respect to a parameter t, which ranges from 0 to 1 along the length of the curve given the endpoints (x₁,y₁) and (x₄,y₄), and the corresponding control points (x₂,y₂) and (x₃,y₃), which define the curve. In our implementation, the endpoints lie on the perpendicular axes (y₁=x₄=0) and the control points are perpendicular to the endpoints (x₂=x₁ and y₃=y₄), forcing the tangents to the curve at its endpoint to be perpendicular to the axes. As a consequence, the Bezier equations for the curves become x(t)=(2x₁-3x₃)t³+3(x₃-x₁)t²+x₁ and y(t)=(3y₂-2y₄)t³+(3y₄-6y₂)t²=3y₂t. The spline parameters given in Table 1 are y₂/y₄ and x₃/x₁, given as fractions to make them independent of scale. When the curve is actually constructed, the two endpoints provide y₄ and x₁; once these endpoints are given, all parameters for the curve are defined. See Table 1 for spline parameter values used in the model that define the lines illustrated in B-E. (B) Extraction of ventral spline parameters. A three-dimensional reconstruction of the odontophore as viewed through the jaws (i.e. with the prow seam vertical) is shown. The spline curve is shown as a dark line at the lower right. Only one side is shown here and in C-E since the structure is bilaterally symmetrical. The white spots at the base of the reconstruction and in the anterior parts of C and D are due to cross sections of the I5 muscle. (C) Extraction of anterior spline parameters. A three-dimensional reconstruction of the odontophore as viewed from its ventral surface (i.e. with the prow seam perpendicular to the plane of the figure) is shown. The spline curve is shown as a dark line at the lower right. (D) Extraction of the posterior spline parameters. Same view as C. The spline curve is shown as a dark line at the upper right. (E) Extraction of dorsal spline parameters excluding the ridge from a video recording of the front view of an isolated odontophore. Since the ridge is not discernible in an anaesthetized buccal mass, these parameters could not be extracted from the high-spatial-resolution MRI data. The spline curve is shown as a dark line at the upper right.