A Coons Patch Spanning a Finite Number of Curves Tested for Variationally Minimizing Its Area

<table>Cosine of the angle between <svg height="11.925" id="M367" style="vertical-align:-0.2457pt" version="1.1" viewbox="0 0 12.4 11.925" width="12.4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path d="M487 676h214v-25q-50 -8 -64 -23q-14 -14 -14 -60v-586h-28l-447 545v-396q0 -58 18 -79q18 -20 76 -27v-25h-226v25q55 7 71 22q17 17 17 62v470q-52 72 -85 72v25h211l349 -430v299q0 61 -18 80q-18 21 -74 26v25z" id="x1D40D"></path></g>
</svg> and <svg height="11.6125" id="M368" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 9.5749998 11.6125" width="9.5749998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path d="M543 24v-24h-234v24q42 2 42 19q0 12 -24 45l-95 134l-23 -24v-114q0 -35 8.5 -45t43.5 -15v-24h-239v24q48 4 48 60v510q0 54 -48 58v24h187v-431l128 128q21 21 21 38q0 18 -35 23l-24 4v23h214v-23q-46 -6 -81 -29q-33 -23 -108 -102l194 -268q10 -15 25 -15z" id="x1D424"></path></g>
</svg> is positive shown for Coons patch in particular for <svg height="15.0875" id="M369" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 41.075001 15.0875" width="41.075001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.8)"><path d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" id="x1D465"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.563,14.887)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
<g transform="matrix(.017,-0,0,-.017,18.162,10.8)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,32.849,10.8)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g>
</svg>. Thus angle between <svg height="11.925" id="M370" style="vertical-align:-0.2457pt" version="1.1" viewbox="0 0 12.4 11.925" width="12.4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path d="M487 676h214v-25q-50 -8 -64 -23q-14 -14 -14 -60v-586h-28l-447 545v-396q0 -58 18 -79q18 -20 76 -27v-25h-226v25q55 7 71 22q17 17 17 62v470q-52 72 -85 72v25h211l349 -430v299q0 61 -18 80q-18 21 -74 26v25z" id="x1D40D"></path></g>
</svg> and <svg height="11.6125" id="M371" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 9.5749998 11.6125" width="9.5749998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path d="M543 24v-24h-234v24q42 2 42 19q0 12 -24 45l-95 134l-23 -24v-114q0 -35 8.5 -45t43.5 -15v-24h-239v24q48 4 48 60v510q0 54 -48 58v24h187v-431l128 128q21 21 21 38q0 18 -35 23l-24 4v23h214v-23q-46 -6 -81 -29q-33 -23 -108 -102l194 -268q10 -15 25 -15z" id="x1D424"></path></g>
</svg> remains within the range <svg height="15" id="M372" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 81.137497 15" width="81.137497" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g><g transform="matrix(.017,-0,0,-.017,12.931,12.162)"><path d="M531 71l-474 214v50l474 215v-56l-416 -184l416 -183v-56zM531 -40h-474v50h474v-50z" id="x2264"></path></g><g transform="matrix(.017,-0,0,-.017,27.634,12.162)"><path d="M501 444q0 -256 -148 -397q-62 -59 -134 -59q-54 0 -88 36.5t-29 124.5q3 51 -6.5 70.5t-29.5 19.5q-14 0 -37 -7l-6 25q42 42 91 42q27 0 42 -21.5t18.5 -48t3.5 -68.5q0 -120 80 -120q127 0 162 337q-23 -17 -60 -32t-67 -15q-87 0 -136 49.5t-49 121.5q0 86 63.5 148
t149.5 62q75 0 127.5 -65t52.5 -203zM423 443q0 106 -42 167t-95 61t-79.5 -38t-26.5 -96q0 -69 43.5 -110.5t111.5 -41.5q52 0 86 27q2 20 2 31z" id="x1D717"></path></g><g transform="matrix(.017,-0,0,-.017,41.165,12.162)"><path d="M531 71l-474 214v50l474 215v-56l-416 -184l416 -183v-56zM531 -40h-474v50h474v-50z" id="x2264"></path></g><g transform="matrix(.017,-0,0,-.017,55.869,12.162)"><path d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43
t61.5 33.5t100 17.5q43 0 131.5 -2.5t129.5 -2.5q23 0 33.5 5t24.5 25z" id="x1D70B"></path></g><g transform="matrix(.017,-0,0,-.017,65.916,12.162)"><path d="M368 703l-264 -866h-60l265 866h59z" id="x2F"></path></g><g transform="matrix(.017,-0,0,-.017,72.919,12.162)"><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>.</table>

Abstract and Applied Analysis

fig6

Figure 6

Figure 6: A Coons Patch Spanning a Finite Number of Curves Tested for Variationally Minimizing Its Area 