Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics

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<g transform="matrix(.012,0,0,-.012,12.925,15.45)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg> and <svg height="15.5125" id="M312" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 19.325001 15.5125" width="19.325001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.363)"><path d="M719 146l-18 -146h-243l-5 18q2 4 17 25t24 36l25 42t26.5 52.5t21 56.5t16 65t5.5 68q0 121 -55.5 194.5t-149.5 73.5q-95 0 -150 -72t-55 -199q0 -54 18.5 -114t36.5 -94t48.5 -82.5t32.5 -51.5l-7 -18h-244l-17 146l27 1q17 -49 31.5 -62t45.5 -13h89l-21 21t-26 26.5
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<g transform="matrix(.012,0,0,-.012,12.925,15.45)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg>: <svg height="11.175" id="M313" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 10.6125 11.175" width="10.6125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg> belongs to <svg height="15.5125" id="M314" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 56.0625 15.5125" width="56.0625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.363)"><path d="M719 146l-18 -146h-243l-5 18q2 4 17 25t24 36l25 42t26.5 52.5t21 56.5t16 65t5.5 68q0 121 -55.5 194.5t-149.5 73.5q-95 0 -150 -72t-55 -199q0 -54 18.5 -114t36.5 -94t48.5 -82.5t32.5 -51.5l-7 -18h-244l-17 146l27 1q17 -49 31.5 -62t45.5 -13h89l-21 21t-26 26.5
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<g transform="matrix(.017,0,0,-.017,23.038,11.363)"><path d="M535 194q0 -90 -71.5 -149.5t-169.5 -59.5t-169.5 59.5t-71.5 149.5v354h58v-354q0 -68 54.5 -113.5t128.5 -45.5t128.5 45.5t54.5 113.5v354h58v-354z" id="x222A"></path></g><g transform="matrix(.017,0,0,-.017,36.789,11.363)"><path d="M719 146l-18 -146h-243l-5 18q2 4 17 25t24 36l25 42t26.5 52.5t21 56.5t16 65t5.5 68q0 121 -55.5 194.5t-149.5 73.5q-95 0 -150 -72t-55 -199q0 -54 18.5 -114t36.5 -94t48.5 -82.5t32.5 -51.5l-7 -18h-244l-17 146l27 1q17 -49 31.5 -62t45.5 -13h89l-21 21t-26 26.5
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<g transform="matrix(.012,0,0,-.012,49.663,15.45)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg> if and only if <svg height="15.0875" id="M315" style="vertical-align:-2.3205pt" version="1.1" viewbox="0 0 25.237499 15.0875" width="25.237499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,12.138)"><path d="M28 461h190v-82q35 51 66 72.5t69 21.5q35 0 58 -22.5t23 -56.5q0 -30 -17.5 -49t-45.5 -19q-37 0 -60 38q-14 23 -28 23q-26 0 -43.5 -30t-17.5 -74v-177q0 -48 13.5 -63.5t58.5 -18.5v-24h-265v24q33 6 43.5 17.5t10.5 42.5v289q0 35 -10.5 47t-44.5 17v24z" id="x1D42B"></path></g><g transform="matrix(.017,0,0,-.017,7.61,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,0,0,-.017,13.492,12.138)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g><g transform="matrix(.017,0,0,-.017,19.288,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg> has a loop.</table>

Abstract and Applied Analysis

fig1

Figure 1

Figure 1: Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics 