弦長スプライン曲線と正規化スプライン曲線の接線ベクトルについて
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概要
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To obtain a smooth curve that passes through a given set of discrete data points, a cubic spline curve consisting of piecewise parametric cubic polynomials with C^2 continuity at the internal joints are often used, because it is computationally simple. In specifying parameter range, several methods can be chosen. For most practical purposes, the uniform parametrization and the chord length parametrization are popular because of easy implementation. A normalized spline curve is generated by the former parametrization and a chord length spline curve is generated by the latter one. By the way, the usual method to obtain coefficients of a cubic spline curve uses the inverse matrix related to a given set of data points. In other words, the algebraic method using inverse matrix avoids discussing geometric meaning of tangent vectors of a cubic spline curve. In this paper, based on the relation of consecutive three data points, the tangent vector at the internal joint is considered and split into two terms. One is the shape term consisting of positions of data points and another is the adjusting term to preserve C^2 continuity. Then the geometric structure of a tangent vector is explained using those terms. If the distances between adjacent data points in all intervals are equal, a normalized spline curve and a chord length spline curve applied to the same set of data points are identical. But if there are intervals with different distances, both spline curves give different results. However, the logical explanation of this phenomenon is rarely shown. In this paper, the reason is explained mathematically.
- 香川県立保健医療大学の論文