Geodesic
Local interpolation curve with a prescribed degree of smoothness which preserves the constant sign of curvature
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 23 (1983) no. 2, pp. 290-300

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A local interpolation method is described, whereby the curve is kept monotonic and its curvature sign fixed, provided that the initial points enable such a curve to be constructed. The algorithm allows the straight parts on the curve to be separated and provides continuity of the derivatives of a given degree. It is shown that, if the function $f^{(q)}(x)$ is continuous in the interval $[a,b]$, $q=0,1,2$, then the interpolation function of the appropriate degree of smoothness converges to the function $f(x)$ on a sequence of meshesat least at the rat $\|\Delta\|^q$, where $\|\Delta\|=\max_i|\Delta x_i|$. 
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     author = {I. A. Rumyantsev},
     title = {Local interpolation curve with a prescribed degree of smoothness which preserves the constant sign of curvature},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {290--300},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_1983_23_2_a4/}
}
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I. A. Rumyantsev. Local interpolation curve with a prescribed degree of smoothness which preserves the constant sign of curvature. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 23 (1983) no. 2, pp. 290-300. http://geodesic.mathdoc.fr/item/ZVMMF_1983_23_2_a4/