On the approximation of plane curves by parametric cubic splines

A plane curveC can be approximated by a parametric cubic spline as follows. Points (x_i,y_i) are chosen in order alongC and a monotonically increasing variable is assigned values _i at the points (x_i,y_i): _i = the cumulative chordal distance from (x₁,y₁). The points (_i,x_i) and (_i,y_i) are then fitted separately by cubic splinesx() andy(), to obtain : (x(),y()). This paper establishes estimates for the errors involved in approximatingC by . It is found that the error in position betweenC and decreases likeh³, whereh is the maximum length of arc between consecutive knots onC. For first derivatives, the error behaves likeh²; for second derivatives, likeh.