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See the ", StyleBox["Mathematica ", FontSlant->"Italic"], "Help browser or wolfram.com for more information on using ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell["\<\ This notebook and those for the other seven chapters use a common stylesheet \ that has been imported into the notebook. This style sheet supports \ chapters, section, subsections, numbered equations, numbered figures and \ citations. A separate copy of the stylesheet can be download from the same \ page where this notebook was downloaded. \ \>", "Text"], Cell[TextData[{ StyleBox["COPYRIGHT ISSUES:", FontWeight->"Bold"], " The authors reserve all copyrights associated with this work. Any of \ the material appearing in these notebooks (such as the polyhedral meshes in \ chapter 7) can be used and modified without restriction as long as the use is \ non\[Hyphen]commercial. We simply ask that you acknowledge the authors when \ using material from these notebooks. For those readers interested in \ commercial use of the material in these notebooks, please contact \ jwarren@cs.rice.edu." }], "Text"], Cell[CellGroupData[{ Cell["Helper functions associated with previous chapters", "Subsubsection"], Cell[BoxData[ \(makeGenFun[coeff_, a_, x_] := Sum[coeff\[LeftDoubleBracket]i\[RightDoubleBracket] x\^\(i - 1 + a\), {i, 1, Length[coeff]}]\)], "Input", InitializationCell->True], Cell[BoxData[ \(makeCoeff[genFun_, {min_, max_}, k_, x_] := With[{expFun = Together[genFun]}, \n\t Table[Coefficient[expFun, x, i], {i, \(2\^k\) min, \(2\^k\) max}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(subdivide[s_, p_, k_] := If[k \[Equal] 0, p, subdivide[s, s*\((p /. {x \[Rule] x\^2})\), k - 1]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(makeGenFun[coeff_, {a_, b_}, x_, y_] := Sum[coeff\[LeftDoubleBracket]i, j\[RightDoubleBracket] \(x\^\(i - 1 + a\)\) y\^\(j - 1 + b\), {i, \(Dimensions[ coeff]\)\[LeftDoubleBracket]1\[RightDoubleBracket]}, {j, \ \(Dimensions[coeff]\)\[LeftDoubleBracket]2\[RightDoubleBracket]}]\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(makeCoeff[genFun_, {{minX_, maxX_}, {minY_, maxY_}}, k_, x_, y_] := With[{expFun = Expand[genFun]}, \n\t Table[With[{rowY = Table[Coefficient[expFun, y, j], {j, \(2\^k\) minY, \(2\^k\) maxY}]}, \n\t\t\t\tTable[ Coefficient[rowY, x, i], {i, \(2\^k\) minX, \(2\^k\) maxX}]]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(subdivide2D[s_, p_, k_] := If[k \[Equal] 0, p, subdivide2D[s, s*\((p /. {x \[Rule] x\^2, y \[Rule] y\^2})\), k - 1]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(norm[t_, k_] := With[{coeff = Transpose[ Partition[ CoefficientList[t], {2\^k, 2\^k}, {2\^k, 2\^k}, {1, 1}, 0], {3, 4, 1, 2}]}, \[IndentingNewLine]Max[ Apply[Plus, Abs[coeff], {2, 4}]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(plotGen2D[genFun_, {{minX_, maxX_}, {minY_, maxY_}}, k_, x_, y_, opts___] := Module[{expFun = Expand[genFun], rowY}, \[IndentingNewLine]plotCoeff3D[ Table[rowY = Coefficient[expFun, y, j]; \[IndentingNewLine]Table[{i\/2\^k, j\/2\^k, Coefficient[rowY, x, i]}, {i, \(2\^k\) minX, \(2\^k\) maxX}], {j, \(2\^k\) minY, \(2\^k\) maxY}], opts]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(plotCoeff3D[p_, plotArgs___] := Show[Graphics3D[\[IndentingNewLine]Table[ Polygon[{p\[LeftDoubleBracket]i, j\[RightDoubleBracket], p\[LeftDoubleBracket]i + 1, j\[RightDoubleBracket], p\[LeftDoubleBracket]i + 1, j + 1\[RightDoubleBracket], p\[LeftDoubleBracket]i, j + 1\[RightDoubleBracket]}], {i, \(Dimensions[ p]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - 1}, {j, \(Dimensions[ p]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - 1}]], Join[{plotArgs}, {Axes \[Rule] True}]]\)], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Subdivision for polyharmonic splines", "Section"], Cell[CellGroupData[{ Cell["The radial basis for polyharmonic splines", "Subsection"], Cell["\<\ General definition of radial basis functions for polyharmonic splines\ \>", "Text"], Cell[BoxData[ \(\[ScriptC][ m_, \[ScriptX]_, \[ScriptY]_] := \(\(\((\[ScriptX]\^2 + \ \[ScriptY]\^2)\)\^\(m - 1\)\) Log[\[ScriptX]\^2 + \ \[ScriptY]\^2]\)\/\(\(2\^\(2 m\)\) \(\((\(\((m - 1)\)!\))\)\^2\) \[Pi]\)\)], \ "Input", InitializationCell->True, CellTags->"def polyharm radial"], Cell[TextData[{ "Observe that for ", Cell[BoxData[ \(\[ScriptC][1, 0, 0]\)]], " is unbounded. 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.30091 L s .02767 .31501 m .03135 .31362 L s .0236 .34073 m .02728 .33935 L s .02153 .35374 m .02522 .35237 L s .01945 .36687 m .02315 .3655 L s .01736 .3801 m .02105 .37874 L s .01311 .4069 m .01681 .40555 L s .01096 .42047 m .01466 .41913 L s .00879 .43415 m .0125 .43282 L s .00661 .44794 m .01032 .44663 L s .00218 .47589 m .00589 .47459 L s .25 Mabswid .03716 .25514 m 0 .48963 L s 0 .48963 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s .40529 .59895 m .03716 .25514 L s .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .40529 .59895 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s 0 0 m 1 0 L 1 .81114 L 0 .81114 L closepath clip newpath .5 Mabswid .745 .61 .708 r .37632 .74671 .39806 .80578 .43793 .75562 .4163 .69866 Metetra .732 .607 .718 r .4163 .69866 .43793 .75562 .47739 .71337 .45587 .65832 Metetra .717 .604 .729 r .45587 .65832 .47739 .71337 .5167 .6781 .49529 .62475 Metetra .702 .6 .74 r .49529 .62475 .5167 .6781 .55608 .649 .53478 .59704 Metetra .686 .597 .75 r .53478 .59704 .55608 .649 .59573 .62538 .57454 .57442 Metetra .669 .592 .76 r .57454 .57442 .59573 .62538 .63584 .6067 .61475 .55623 Metetra .651 .585 .767 r .61475 .55623 .63584 .6067 .67657 .59263 .65557 .54205 Metetra .632 .576 .773 r .65557 .54205 .67657 .59263 .71811 .58308 .69714 .53174 Metetra .61 .563 .777 r .69714 .53174 .71811 .58308 .76067 .57823 .73965 .52553 Metetra .584 .547 .778 r .73965 .52553 .76067 .57823 .80447 .57855 .78333 .52393 Metetra .554 .526 .778 r .78333 .52393 .80447 .57855 .84982 .58471 .82846 .52772 Metetra .52 .501 .775 r .82846 .52772 .84982 .58471 .89708 .59757 .87539 .53786 Metetra .481 .472 .77 r .87539 .53786 .89708 .59757 .9467 .61819 .92456 .55545 Metetra .439 .441 .764 r .92456 .55545 .9467 .61819 .99922 .6478 .97651 .58172 Metetra .755 .629 .721 r .35412 .69495 .37632 .74671 .4163 .69866 .39427 .6488 Metetra .74 .628 .734 r .39427 .6488 .4163 .69866 .45587 .65832 .43401 .61028 Metetra .725 .626 .747 r .43401 .61028 .45587 .65832 .49529 .62475 .4736 .57837 Metetra .708 .624 .761 r .4736 .57837 .49529 .62475 .53478 .59704 .51327 .55211 Metetra .691 .622 .774 r .51327 .55211 .53478 .59704 .57454 .57442 .55322 .5306 Metetra .675 .62 .786 r .55322 .5306 .57454 .57442 .61475 .55623 .59362 .51302 Metetra .658 .615 .795 r .59362 .51302 .61475 .55623 .65557 .54205 .6346 .49878 Metetra .639 .608 .802 r .6346 .49878 .65557 .54205 .69714 .53174 .67631 .4877 Metetra .618 .596 .806 r .67631 .4877 .69714 .53174 .73965 .52553 .71889 .48004 Metetra .591 .578 .807 r .71889 .48004 .73965 .52553 .78333 .52393 .76255 .47646 Metetra .559 .555 .806 r .76255 .47646 .78333 .52393 .82846 .52772 .80758 .47788 Metetra .521 .526 .802 r .80758 .47788 .82846 .52772 .87539 .53786 .85431 .48537 Metetra .476 .493 .796 r .85431 .48537 .87539 .53786 .92456 .55545 .90318 .5001 Metetra .427 .457 .788 r .90318 .5001 .92456 .55545 .97651 .58172 .95472 .52333 Metetra .764 .651 .736 r .33134 .64964 .35412 .69495 .39427 .6488 .37173 .60516 Metetra .749 .65 .751 r .37173 .60516 .39427 .6488 .43401 .61028 .41169 .56829 Metetra .731 .65 .768 r .41169 .56829 .43401 .61028 .4736 .57837 .45151 .53801 Metetra .713 .65 .785 r .45151 .53801 .4736 .57837 .51327 .55211 .49141 .51327 Metetra .695 .65 .801 r .49141 .51327 .51327 .55211 .55322 .5306 .53162 .49301 Metetra .678 .651 .816 r .53162 .49301 .55322 .5306 .59362 .51302 .57228 .47619 Metetra .663 .65 .827 r .57228 .47619 .59362 .51302 .6346 .49878 .61352 .46198 Metetra .647 .645 .834 r .61352 .46198 .6346 .49878 .67631 .4877 .65544 .45008 Metetra .627 .634 .838 r .65544 .45008 .67631 .4877 .71889 .48004 .69819 .44084 Metetra .601 .615 .838 r .69819 .44084 .71889 .48004 .76255 .47646 .74194 .43515 Metetra .566 .589 .836 r .74194 .43515 .76255 .47646 .80758 .47788 .78697 .43416 Metetra .523 .556 .831 r .78697 .43416 .80758 .47788 .85431 .48537 .83362 .43909 Metetra .472 .518 .824 r .83362 .43909 .85431 .48537 .90318 .5001 .88233 .45116 Metetra .415 .476 .814 r .88233 .45116 .90318 .5001 .95472 .52333 .93362 .47166 Metetra .775 .673 .752 r .30787 .61002 .33134 .64964 .37173 .60516 .34856 .56689 Metetra .757 .674 .77 r .34856 .56689 .37173 .60516 .41169 .56829 .38881 .53145 Metetra .737 .675 .79 r .38881 .53145 .41169 .56829 .45151 .53801 .42889 .50267 Metetra .716 .676 .81 r .42889 .50267 .45151 .53801 .49141 .51327 .46908 .47947 Metetra .695 .679 .83 r .46908 .47947 .49141 .51327 .53162 .49301 .50958 .46062 Metetra .678 .683 .848 r .50958 .46062 .53162 .49301 .57228 .47619 .55057 .44479 Metetra .665 .687 .861 r .55057 .44479 .57228 .47619 .61352 .46198 .59214 .43073 Metetra .655 .687 .868 r .59214 .43073 .61352 .46198 .65544 .45008 .63438 .4179 Metetra .639 .677 .869 r .63438 .4179 .65544 .45008 .69819 .44084 .67738 .40687 Metetra .613 .656 .868 r .67738 .40687 .69819 .44084 .74194 .43515 .72131 .39893 Metetra .575 .626 .865 r .72131 .39893 .74194 .43515 .78697 .43416 .76645 .39554 Metetra .526 .589 .861 r .76645 .39554 .78697 .43416 .83362 .43909 .81314 .39805 Metetra .468 .546 .853 r .81314 .39805 .83362 .43909 .88233 .45116 .86182 .40773 Metetra .4 .497 .842 r .86182 .40773 .88233 .45116 .93362 .47166 .91301 .42586 Metetra .785 .698 .767 r .28361 .57543 .30787 .61002 .34856 .56689 .32467 .53324 Metetra .765 .699 .789 r .32467 .53324 .34856 .56689 .38881 .53145 .36525 .49886 Metetra .742 .7 .812 r .36525 .49886 .38881 .53145 .42889 .50267 .40566 .47132 Metetra .716 .702 .836 r .40566 .47132 .42889 .50267 .46908 .47947 .44616 .44952 Metetra .69 .705 .859 r .44616 .44952 .46908 .47947 .50958 .46062 .487 .43218 Metetra .67 .712 .879 r .487 .43218 .50958 .46062 .55057 .44479 .52836 .41762 Metetra .663 .722 .893 r .52836 .41762 .55057 .44479 .59214 .43073 .57033 .4039 Metetra .663 .728 .897 r .57033 .4039 .59214 .43073 .63438 .4179 .61296 .38995 Metetra .654 .719 .896 r .61296 .38995 .63438 .4179 .67738 .40687 .65628 .37684 Metetra .628 .695 .893 r .65628 .37684 .67738 .40687 .72131 .39893 .70048 .36657 Metetra .586 .663 .891 r .70048 .36657 .72131 .39893 .76645 .39554 .74583 .36091 Metetra .53 .622 .888 r .74583 .36091 .76645 .39554 .81314 .39805 .79267 .36129 Metetra .462 .574 .881 r .79267 .36129 .81314 .39805 .86182 .40773 .84146 .36898 Metetra .382 .518 .869 r .84146 .36898 .86182 .40773 .91301 .42586 .8927 .3852 Metetra .797 .723 .783 r .25844 .54535 .28361 .57543 .32467 .53324 .29996 .50356 Metetra .774 .725 .807 r .29996 .50356 .32467 .53324 .36525 .49886 .34095 .46974 Metetra .746 .725 .832 r .34095 .46974 .36525 .49886 .40566 .47132 .38173 .44292 Metetra .715 .725 .858 r .38173 .44292 .40566 .47132 .44616 .44952 .42258 .42209 Metetra .682 .726 .883 r .42258 .42209 .44616 .44952 .487 .43218 .46378 .40602 Metetra .656 .73 .904 r .46378 .40602 .487 .43218 .52836 .41762 .50553 .39294 Metetra .653 .745 .917 r .50553 .39294 .52836 .41762 .57033 .4039 .54795 .37991 Metetra .673 .761 .916 r .54795 .37991 .57033 .4039 .61296 .38995 .59101 .36448 Metetra .671 .751 .91 r .59101 .36448 .61296 .38995 .65628 .37684 .63473 .34907 Metetra .642 .725 .908 r .63473 .34907 .65628 .37684 .70048 .36657 .67928 .33669 Metetra .595 .691 .91 r .67928 .33669 .70048 .36657 .74583 .36091 .72494 .3292 Metetra .532 .65 .91 r .72494 .3292 .74583 .36091 .79267 .36129 .77206 .32798 Metetra .452 .599 .907 r .77206 .32798 .79267 .36129 .84146 .36898 .82108 .33422 Metetra .358 .536 .894 r .82108 .33422 .84146 .36898 .8927 .3852 .87254 .34911 Metetra .809 .75 .797 r .23225 .51946 .25844 .54535 .29996 .50356 .27433 .47745 Metetra .784 .751 .823 r .27433 .47745 .29996 .50356 .34095 .46974 .31582 .44349 Metetra .753 .75 .85 r .31582 .44349 .34095 .46974 .38173 .44292 .35704 .41664 Metetra .716 .746 .875 r .35704 .41664 .38173 .44292 .42258 .42209 .3983 .39592 Metetra .678 .74 .897 r .3983 .39592 .42258 .42209 .46378 .40602 .43989 .38018 Metetra .645 .736 .915 r .43989 .38018 .46378 .40602 .50553 .39294 .48203 .3678 Metetra .64 .743 .923 r .48203 .3678 .50553 .39294 .54795 .37991 .52489 .35573 Metetra .686 .769 .915 r .52489 .35573 .54795 .37991 .59101 .36448 .56842 .33851 Metetra .681 .756 .908 r .56842 .33851 .59101 .36448 .63473 .34907 .6126 .32157 Metetra .648 .736 .913 r .6126 .32157 .63473 .34907 .67928 .33669 .65758 .30801 Metetra .597 .709 .922 r .65758 .30801 .67928 .33669 .72494 .3292 .70365 .29956 Metetra .526 .671 .929 r .70365 .29956 .72494 .3292 .77206 .32798 .75116 .2975 Metetra .435 .619 .929 r .75116 .2975 .77206 .32798 .82108 .33422 .80056 .303 Metetra .323 .55 .915 r .80056 .303 .82108 .33422 .87254 .34911 .85238 .31721 Metetra .824 .779 .81 r .20488 .49771 .23225 .51946 .27433 .47745 .24765 .45479 Metetra .797 .779 .837 r .24765 .45479 .27433 .47745 .31582 .44349 .28974 .41994 Metetra .763 .775 .863 r .28974 .41994 .31582 .44349 .35704 .41664 .33151 .39218 Metetra .724 .767 .887 r .33151 .39218 .35704 .41664 .3983 .39592 .37326 .37048 Metetra .685 .753 .903 r .37326 .37048 .3983 .39592 .43989 .38018 .4153 .35358 Metetra .656 .739 .911 r .4153 .35358 .43989 .38018 .48203 .3678 .45788 .33969 Metetra .653 .729 .905 r .45788 .33969 .48203 .3678 .52489 .35573 .50115 .32584 Metetra .671 .725 .89 r .50115 .32584 .52489 .35573 .56842 .33851 .54513 .30953 Metetra .67 .735 .899 r .54513 .30953 .56842 .33851 .6126 .32157 .5898 .29323 Metetra .642 .733 .915 r .5898 .29323 .6126 .32157 .65758 .30801 .63529 .27998 Metetra .589 .717 .933 r .63529 .27998 .65758 .30801 .70365 .29956 .68186 .27165 Metetra .51 .685 .946 r .68186 .27165 .70365 .29956 .75116 .2975 .72988 .26965 Metetra .403 .632 .948 r .72988 .26965 .75116 .2975 .80056 .303 .77978 .27517 Metetra .27 .555 .929 r .77978 .27517 .80056 .303 .85238 .31721 .83211 .28943 Metetra .84 .81 .821 r .17616 .4803 .20488 .49771 .24765 .45479 .21976 .43581 Metetra .812 .81 .849 r .21976 .43581 .24765 .45479 .28974 .41994 .26259 .39935 Metetra .777 .804 .875 r .26259 .39935 .28974 .41994 .33151 .39218 .30502 .36989 Metetra .738 .792 .896 r .30502 .36989 .33151 .39218 .37326 .37048 .34737 .34631 Metetra .7 .773 .908 r .34737 .34631 .37326 .37048 .4153 .35358 .38995 .32726 Metetra .673 .753 .91 r .38995 .32726 .4153 .35358 .45788 .33969 .43302 .31104 Metetra .663 .735 .903 r .43302 .31104 .45788 .33969 .50115 .32584 .47673 .29563 Metetra .663 .726 .896 r .47673 .29563 .50115 .32584 .54513 .30953 .52115 .27996 Metetra .654 .731 .906 r .52115 .27996 .54513 .30953 .5898 .29323 .56632 .26512 Metetra .624 .735 .927 r .56632 .26512 .5898 .29323 .63529 .27998 .61236 .25313 Metetra .567 .726 .949 r .61236 .25313 .63529 .27998 .68186 .27165 .65951 .24583 Metetra .477 .696 .964 r .65951 .24583 .68186 .27165 .72988 .26965 .70813 .24467 Metetra .35 .637 .961 r .70813 .24467 .72988 .26965 .77978 .27517 .75865 .25096 Metetra .191 .547 .93 r .75865 .25096 .77978 .27517 .83211 .28943 .81163 .26596 Metetra .858 .844 .83 r .14583 .46772 .17616 .4803 .21976 .43581 .19044 .42106 Metetra .829 .844 .858 r .19044 .42106 .21976 .43581 .26259 .39935 .23416 .38241 Metetra .795 .838 .885 r .23416 .38241 .26259 .39935 .30502 .36989 .27738 .35068 Metetra .756 .825 .906 r .27738 .35068 .30502 .36989 .34737 .34631 .32045 .32472 Metetra .719 .805 .918 r .32045 .32472 .34737 .34631 .38995 .32726 .36367 .30324 Metetra .69 .782 .922 r .36367 .30324 .38995 .32726 .43302 .31104 .4073 .28483 Metetra .671 .763 .92 r .4073 .28483 .43302 .31104 .47673 .29563 .45151 .26822 Metetra .657 .753 .92 r .45151 .26822 .47673 .29563 .52115 .27996 .49643 .25283 Metetra .637 .752 .931 r .49643 .25283 .52115 .27996 .56632 .26512 .54212 .23924 Metetra .597 .751 .951 r .54212 .23924 .56632 .26512 .61236 .25313 .58875 .2288 Metetra .528 .739 .972 r .58875 .2288 .61236 .25313 .65951 .24583 .63654 .22301 Metetra .418 .702 .981 r .63654 .22301 .65951 .24583 .70813 .24467 .68584 .22327 Metetra .265 .629 .962 r .68584 .22327 .70813 .24467 .75865 .25096 .73709 .23091 Metetra .078 .516 .901 r .73709 .23091 .75865 .25096 .81163 .26596 .79087 .24728 Metetra .876 .88 .835 r .11359 .4607 .14583 .46772 .19044 .42106 .15939 .41136 Metetra .848 .883 .865 r .15939 .41136 .19044 .42106 .23416 .38241 .20417 .37007 Metetra .813 .878 .893 r .20417 .37007 .23416 .38241 .27738 .35068 .24834 .33572 Metetra .775 .866 .916 r .24834 .33572 .27738 .35068 .32045 .32472 .29225 .30718 Metetra .737 .848 .932 r .29225 .30718 .32045 .32472 .36367 .30324 .33622 .2833 Metetra .703 .826 .941 r .33622 .2833 .36367 .30324 .4073 .28483 .38051 .26296 Metetra .675 .807 .946 r .38051 .26296 .4073 .28483 .45151 .26822 .42532 .24531 Metetra .647 .795 .953 r .42532 .24531 .45151 .26822 .49643 .25283 .4708 .23001 Metetra .612 .788 .966 r .4708 .23001 .49643 .25283 .54212 .23924 .51708 .21742 Metetra .554 .779 .982 r .51708 .21742 .54212 .23924 .58875 .2288 .56435 .20848 Metetra .461 .754 .993 r .56435 .20848 .58875 .2288 .63654 .22301 .61284 .20438 Metetra .321 .697 .98 r .61284 .20438 .63654 .22301 .68584 .22327 .66291 .20641 Metetra .136 .594 .925 r .66291 .20641 .68584 .22327 .73709 .23091 .71501 .21587 Metetra 0 .451 .821 r .71501 .21587 .73709 .23091 .79087 .24728 .76972 .23415 Metetra .891 .918 .835 r .07904 .46021 .11359 .4607 .15939 .41136 .12625 .40772 Metetra .863 .923 .866 r .12625 .40772 .15939 .41136 .20417 .37007 .17228 .36345 Metetra .829 .922 .897 r .17228 .36345 .20417 .37007 .24834 .33572 .21756 .32626 Metetra .79 .914 .924 r .21756 .32626 .24834 .33572 .29225 .30718 .26246 .29509 Metetra .748 .899 .946 r .26246 .29509 .29225 .30718 .33622 .2833 .30731 .2689 Metetra .708 .881 .961 r .30731 .2689 .33622 .2833 .38051 .26296 .35238 .24683 Metetra .668 .863 .973 r .35238 .24683 .38051 .26296 .42532 .24531 .3979 .22824 Metetra .624 .847 .985 r .3979 .22824 .42532 .24531 .4708 .23001 .44405 .21292 Metetra .565 .83 .995 r .44405 .21292 .4708 .23001 .51708 .21742 .49102 .20113 Metetra .478 .803 1 r .49102 .20113 .51708 .21742 .56435 .20848 .539 .19357 Metetra .349 .751 .983 r .539 .19357 .56435 .20848 .61284 .20438 .58829 .19122 Metetra .172 .654 .926 r .58829 .19122 .61284 .20438 .66291 .20641 .63923 .19524 Metetra 0 .513 .818 r .63923 .19524 .66291 .20641 .71501 .21587 .69229 .20688 Metetra 0 .347 .675 r .69229 .20688 .71501 .21587 .76972 .23415 .74807 .22755 Metetra .9 .953 .826 r .0417 .46738 .07904 .46021 .12625 .40772 .09056 .41135 Metetra .87 .962 .858 r .09056 .41135 .12625 .40772 .17228 .36345 .13806 .3638 Metetra .834 .965 .89 r .13806 .3638 .17228 .36345 .21756 .32626 .18465 .32362 Metetra .792 .961 .919 r .18465 .32362 .21756 .32626 .26246 .29509 .23071 .28977 Metetra .744 .95 .945 r .23071 .28977 .26246 .29509 .30731 .2689 .2766 .26137 Metetra .693 .934 .966 r .2766 .26137 .30731 .2689 .35238 .24683 .32259 .23767 Metetra .635 .914 .981 r .32259 .23767 .35238 .24683 .3979 .22824 .36895 .21819 Metetra .565 .888 .989 r .36895 .21819 .3979 .22824 .44405 .21292 .4159 .20279 Metetra .473 .851 .985 r .4159 .20279 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.6545 .5147 Metetra .644 .613 .804 r .6545 .5147 .67496 .55635 .71602 .54619 .69568 .50342 Metetra .628 .602 .805 r .69568 .50342 .71602 .54619 .75802 .54001 .73767 .4946 Metetra .606 .582 .8 r .73767 .4946 .75802 .54001 .80122 .53899 .78066 .48929 Metetra .574 .551 .791 r .78066 .48929 .80122 .53899 .84605 .54523 .82498 .48932 Metetra .53 .508 .776 r .82498 .48932 .84605 .54523 .8932 .56193 .87119 .49753 Metetra .473 .453 .755 r .87119 .49753 .8932 .56193 .94371 .59384 .92015 .51812 Metetra .404 .389 .729 r .92015 .51812 .94371 .59384 .99922 .6478 .97324 .55708 Metetra .745 .603 .7 r .35579 .66048 .37729 .72347 .4174 .66307 .39574 .60986 Metetra .728 .613 .729 r .39574 .60986 .4174 .66307 .45664 .61951 .43498 .57371 Metetra .711 .625 .759 r .43498 .57371 .45664 .61951 .49561 .58747 .47407 .54714 Metetra .695 .638 .789 r .47407 .54714 .49561 .58747 .53468 .56316 .51336 .52668 Metetra .681 .651 .814 r .51336 .52668 .53468 .56316 .5741 .54394 .55305 .50994 Metetra .671 .662 .832 r .55305 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.835 r .38984 .50586 .41274 .53697 .45202 .51439 .42937 .48601 Metetra .686 .707 .864 r .42937 .48601 .45202 .51439 .49158 .49666 .46924 .47006 Metetra .672 .718 .883 r .46924 .47006 .49158 .49666 .53159 .48158 .5096 .45596 Metetra .666 .727 .895 r .5096 .45596 .53159 .48158 .57216 .46778 .55055 .4425 Metetra .664 .732 .9 r .55055 .4425 .57216 .46778 .61332 .45453 .59211 .4291 Metetra .663 .733 .902 r .59211 .4291 .61332 .45453 .65511 .44161 .6343 .4156 Metetra .661 .731 .901 r .6343 .4156 .65511 .44161 .69757 .42925 .67715 .40214 Metetra .653 .722 .899 r .67715 .40214 .69757 .42925 .74081 .4182 .72073 .38933 Metetra .635 .704 .896 r .72073 .38933 .74081 .4182 .78502 .40987 .76516 .37842 Metetra .599 .672 .891 r .76516 .37842 .78502 .40987 .83055 .40663 .81077 .37155 Metetra .537 .618 .881 r .81077 .37155 .83055 .40663 .87805 .41196 .85809 .37203 Metetra .437 .535 .859 r .85809 .37203 .87805 .41196 .92856 .43088 .90808 .38461 Metetra .766 .696 .785 r .28632 .53717 .31035 .57086 .35034 .53269 .32652 .50238 Metetra .732 .707 .827 r .32652 .50238 .35034 .53269 .38984 .50586 .36626 .47801 Metetra .701 .717 .862 r .36626 .47801 .38984 .50586 .42937 .48601 .40607 .45982 Metetra .678 .726 .885 r .40607 .45982 .42937 .48601 .46924 .47006 .44627 .44484 Metetra .667 .733 .899 r .44627 .44484 .46924 .47006 .5096 .45596 .48701 .43115 Metetra .663 .738 .906 r .48701 .43115 .5096 .45596 .55055 .4425 .52836 .41767 Metetra .663 .741 .908 r .52836 .41767 .55055 .4425 .59211 .4291 .57033 .40399 Metetra .664 .742 .908 r .57033 .40399 .59211 .4291 .6343 .4156 .61296 .39 Metetra .664 .741 .908 r .61296 .39 .6343 .4156 .67715 .40214 .65624 .37579 Metetra .66 .737 .907 r .65624 .37579 .67715 .40214 .72073 .38933 .70022 .3618 Metetra .647 .725 .906 r .70022 .3618 .72073 .38933 .76516 .37842 .74501 .3491 Metetra .616 .701 .905 r .74501 .3491 .76516 .37842 .81077 .37155 .79086 .33971 Metetra .555 .656 .903 r .79086 .33971 .81077 .37155 .85809 .37203 .83826 .3368 Metetra .451 .579 .89 r .83826 .3368 .85809 .37203 .90808 .38461 .88809 .34496 Metetra .772 .724 .808 r .26139 .50804 .28632 .53717 .32652 .50238 .3019 .47501 Metetra .732 .73 .849 r .3019 .47501 .32652 .50238 .36626 .47801 .34195 .45187 Metetra .696 .733 .879 r .34195 .45187 .36626 .47801 .40607 .45982 .3821 .43443 Metetra .673 .736 .897 r .3821 .43443 .40607 .45982 .44627 .44484 .42266 .41979 Metetra .664 .739 .906 r .42266 .41979 .44627 .44484 .48701 .43115 .46378 .40607 Metetra .662 .741 .909 r .46378 .40607 .48701 .43115 .52836 .41767 .50553 .39231 Metetra .663 .742 .909 r .50553 .39231 .52836 .41767 .57033 .40399 .54794 .37824 Metetra .664 .743 .909 r .54794 .37824 .57033 .40399 .61296 .39 .591 .36385 Metetra .665 .743 .908 r .591 .36385 .61296 .39 .65624 .37579 .63474 .34912 Metetra .663 .742 .908 r .63474 .34912 .65624 .37579 .70022 .3618 .67917 .33434 Metetra .653 .735 .909 r .67917 .33434 .70022 .3618 .74501 .3491 .72439 .32049 Metetra .625 .718 .913 r .72439 .32049 .74501 .3491 .79086 .33971 .77061 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.33434 .72439 .32049 .70321 .29183 Metetra .628 .725 .917 r .70321 .29183 .72439 .32049 .77061 .3095 .74989 .28001 Metetra .569 .695 .926 r .74989 .28001 .77061 .3095 .81829 .30447 .798 .274 Metetra .451 .629 .93 r .798 .274 .81829 .30447 .86824 .3099 .84832 .27828 Metetra .793 .775 .837 r .20853 .45989 .23549 .48239 .27645 .4496 .2501 .42586 Metetra .742 .767 .873 r .2501 .42586 .27645 .4496 .3169 .42657 .29106 .40184 Metetra .7 .756 .895 r .29106 .40184 .3169 .42657 .35743 .40911 .33202 .38359 Metetra .674 .746 .905 r .33202 .38359 .35743 .40911 .39837 .39429 .37337 .36815 Metetra .664 .742 .908 r .37337 .36815 .39837 .39429 .43989 .38027 .4153 .35363 Metetra .662 .74 .908 r .4153 .35363 .43989 .38027 .48205 .36612 .45789 .33906 Metetra .663 .741 .908 r .45789 .33906 .48205 .36612 .52489 .35161 .50116 .32414 Metetra .664 .741 .908 r .50116 .32414 .52489 .35161 .5684 .33682 .54512 .30889 Metetra .665 .741 .907 r .54512 .30889 .5684 .33682 .6126 .32166 .5898 .29328 Metetra .663 .741 .908 r .5898 .29328 .6126 .32166 .65751 .30635 .6352 .27761 Metetra .653 .738 .912 r .6352 .27761 .65751 .30635 .70321 .29183 .68141 .26284 Metetra .624 .728 .921 r .68141 .26284 .70321 .29183 .74989 .28001 .72864 .25095 Metetra .559 .701 .936 r .72864 .25095 .74989 .28001 .798 .274 .7773 .24507 Metetra .424 .635 .943 r .7773 .24507 .798 .274 .84832 .27828 .82819 .24975 Metetra .813 .804 .843 r .18031 .44098 .20853 .45989 .2501 .42586 .22273 .40413 Metetra .757 .789 .879 r .22273 .40413 .2501 .42586 .29106 .40184 .26433 .37796 Metetra .71 .77 .899 r .26433 .37796 .29106 .40184 .33202 .38359 .30583 .35811 Metetra .681 .754 .907 r .30583 .35811 .33202 .38359 .37337 .36815 .34764 .34152 Metetra .667 .745 .908 r .34764 .34152 .37337 .36815 .4153 .35363 .39 .32621 Metetra .663 .741 .908 r .39 .32621 .4153 .35363 .45789 .33906 .43301 .31109 Metetra .663 .74 .907 r .43301 .31109 .45789 .33906 .50116 .32414 .47673 .29572 Metetra .664 .74 .907 r .47673 .29572 .50116 .32414 .54512 .30889 .52115 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.62276 Metetra .741 .695 .808 r .42542 .62276 .44891 .65706 .49021 .63297 .46714 .59423 Metetra .731 .673 .794 r .46714 .59423 .49021 .63297 .53185 .60984 .50909 .56569 Metetra .708 .644 .783 r .50909 .56569 .53185 .60984 .5739 .5895 .55132 .54006 Metetra .666 .608 .78 r .55132 .54006 .5739 .5895 .61654 .57397 .59403 .5219 Metetra .611 .572 .786 r .59403 .5219 .61654 .57397 .66 .56424 .63762 .51386 Metetra .564 .555 .802 r .63762 .51386 .66 .56424 .70449 .55944 .68241 .51348 Metetra .536 .56 .826 r .68241 .51348 .70449 .55944 .7501 .5576 .72847 .51627 Metetra .526 .58 .852 r .72847 .51627 .7501 .5576 .79685 .55695 .77574 .51929 Metetra .526 .604 .874 r .77574 .51929 .79685 .55695 .84472 .55632 .82417 .5213 Metetra .531 .626 .891 r .82417 .5213 .84472 .55632 .89369 .55515 .87373 .52194 Metetra .538 .645 .903 r .87373 .52194 .89369 .55515 .94374 .55318 .9244 .52117 Metetra .748 .742 .847 r .2766 .67415 .30218 .70011 .3429 .67585 .31782 .64808 Metetra .754 .734 .834 r .31782 .64808 .3429 .67585 .38399 .65007 .35944 .61982 Metetra .76 .723 .818 r .35944 .61982 .38399 .65007 .42542 .62276 .4014 .5889 Metetra .763 .707 .799 r .4014 .5889 .42542 .62276 .46714 .59423 .44362 .55491 Metetra .76 .684 .778 r .44362 .55491 .46714 .59423 .50909 .56569 .486 .51817 Metetra .74 .648 .758 r .486 .51817 .50909 .56569 .55132 .54006 .52843 .48212 Metetra .68 .591 .749 r .52843 .48212 .55132 .54006 .59403 .5219 .57109 .45785 Metetra .578 .521 .754 r .57109 .45785 .59403 .5219 .63762 .51386 .61474 .4549 Metetra .502 .492 .779 r .61474 .4549 .63762 .51386 .68241 .51348 .65986 .46406 Metetra .476 .515 .818 r .65986 .46406 .68241 .51348 .72847 .51627 .70641 .47427 Metetra .478 .554 .856 r .70641 .47427 .72847 .51627 .77574 .51929 .75423 .482 Metetra .49 .592 .883 r .75423 .482 .77574 .51929 .82417 .5213 .80323 .48688 Metetra .505 .622 .901 r .80323 .48688 .82417 .5213 .87373 .52194 .85339 .48925 Metetra .519 .646 .913 r .85339 .48925 .87373 .52194 .9244 .52117 .90467 .48954 Metetra .756 .754 .851 r .25023 .64914 .2766 .67415 .31782 .64808 .29198 .62163 Metetra .766 .749 .837 r .29198 .62163 .31782 .64808 .35944 .61982 .33416 .59134 Metetra .777 .741 .82 r .33416 .59134 .35944 .61982 .4014 .5889 .37671 .5573 Metetra .788 .728 .796 r .37671 .5573 .4014 .5889 .44362 .55491 .41956 .51795 Metetra .798 .707 .764 r .41956 .51795 .44362 .55491 .486 .51817 .46256 .47071 Metetra .798 .672 .726 r .46256 .47071 .486 .51817 .52843 .48212 .50541 .41311 Metetra .715 .586 .711 r .50541 .41311 .52843 .48212 .57109 .45785 .5479 .37292 Metetra .505 .435 .711 r .5479 .37292 .57109 .45785 .61474 .4549 .59143 .38479 Metetra .367 .379 .741 r .59143 .38479 .61474 .4549 .65986 .46406 .63701 .41466 Metetra .374 .458 .819 r .63701 .41466 .65986 .46406 .70641 .47427 .68407 .4346 Metetra .409 .535 .872 r .68407 .4346 .70641 .47427 .75423 .482 .73241 .44694 Metetra .446 .589 .902 r .73241 .44694 .75423 .482 .80323 .48688 .78196 .45416 Metetra .476 .626 .918 r .78196 .45416 .80323 .48688 .85339 .48925 .83267 .45778 Metetra .499 .651 .927 r .83267 .45778 .85339 .48925 .90467 .48954 .88455 .45871 Metetra .763 .768 .858 r .22297 .62539 .25023 .64914 .29198 .62163 .26525 .59704 Metetra .775 .767 .845 r .26525 .59704 .29198 .62163 .33416 .59134 .308 .56566 Metetra .79 .764 .829 r .308 .56566 .33416 .59134 .37671 .5573 .35116 .53003 Metetra .809 .758 .805 r .35116 .53003 .37671 .5573 .41956 .51795 .39468 .48793 Metetra .833 .747 .769 r .39468 .48793 .41956 .51795 .46256 .47071 .43845 .43444 Metetra .854 .724 .718 r .43845 .43444 .46256 .47071 .50541 .41311 .48212 .36078 Metetra .641 .59 .782 r .48212 .36078 .50541 .41311 .5479 .37292 .52489 .34623 Metetra .664 .741 .907 r .52489 .34623 .5479 .37292 .59143 .38479 .56832 .33145 Metetra .072 .247 .737 r .56832 .33145 .59143 .38479 .63701 .41466 .61413 .37657 Metetra .197 .419 .851 r .61413 .37657 .63701 .41466 .68407 .4346 .66156 .40196 Metetra .319 .539 .908 r .66156 .40196 .68407 .4346 .73241 .44694 .71033 .41622 Metetra .397 .603 .93 r .71033 .41622 .73241 .44694 .78196 .45416 .76035 .4242 Metetra .447 .64 .939 r .76035 .4242 .78196 .45416 .83267 .45778 .81159 .42807 Metetra .481 .664 .943 r .81159 .42807 .83267 .45778 .88455 .45871 .86404 .429 Metetra .765 .784 .868 r .19473 .60304 .22297 .62539 .26525 .59704 .23752 .57456 Metetra .779 .787 .859 r .23752 .57456 .26525 .59704 .308 .56566 .2808 .54325 Metetra .796 .791 .847 r .2808 .54325 .308 .56566 .35116 .53003 .32453 .50813 Metetra .819 .798 .832 r .32453 .50813 .35116 .53003 .39468 .48793 .36865 .46761 Metetra .85 .809 .81 r .36865 .46761 .39468 .48793 .43845 .43444 .41304 .4191 Metetra .891 .837 .786 r .41304 .4191 .43845 .43444 .48212 .36078 .45744 .36011 Metetra .853 .975 .868 r .45744 .36011 .48212 .36078 .52489 .34623 .5012 .31876 Metetra .287 .563 .931 r .5012 .31876 .52489 .34623 .56832 .33145 .54526 .33008 Metetra .579 .003 0 r .54526 .33008 .56832 .33145 .61413 .37657 .59118 .35963 Metetra 0 .416 .819 r .59118 .35963 .61413 .37657 .66156 .40196 .63882 .37912 Metetra .224 .578 .943 r .63882 .37912 .66156 .40196 .71033 .41622 .68791 .3909 Metetra .356 .636 .96 r .68791 .3909 .71033 .41622 .76035 .4242 .73835 .39748 Metetra .426 .665 .961 r .73835 .39748 .76035 .4242 .81159 .42807 .79008 .40037 Metetra .469 .683 .959 r .79008 .40037 .81159 .42807 .86404 .429 .84308 .40053 Metetra .764 .799 .881 r .16542 .58199 .19473 .60304 .23752 .57456 .20869 .554 Metetra .777 .806 .876 r .20869 .554 .23752 .57456 .2808 .54325 .25247 .52374 Metetra .793 .818 .872 r .25247 .52374 .2808 .54325 .32453 .50813 .2967 .49072 Metetra .813 .838 .869 r .2967 .49072 .32453 .50813 .36865 .46761 .34131 .45454 Metetra .835 .875 .873 r .34131 .45454 .36865 .46761 .41304 .4191 .38617 .4155 Metetra .835 .956 .892 r .38617 .4155 .41304 .4191 .45744 .36011 .43108 .37719 Metetra .43108 .37719 .45744 .36011 .5012 .31876 .47587 .35094 Metetra .356 0 .014 r .47587 .35094 .5012 .31876 .54526 .33008 .52107 .34655 Metetra .417 0 0 r .52107 .34655 .54526 .33008 .59118 .35963 .56752 .35454 Metetra 0 .471 .713 r .56752 .35454 .59118 .35963 .63882 .37912 .6155 .3636 Metetra .195 .639 .945 r .6155 .3636 .63882 .37912 .68791 .3909 .66496 .37007 Metetra .344 .681 .979 r .66496 .37007 .68791 .3909 .73835 .39748 .71583 .37358 Metetra .42 .697 .978 r .71583 .37358 .73835 .39748 .79008 .40037 .76804 .37448 Metetra .465 .706 .973 r .76804 .37448 .79008 .40037 .84308 .40053 .82159 .37318 Metetra .759 .812 .895 r .13497 .56193 .16542 .58199 .20869 .554 .1787 .53486 Metetra .769 .823 .895 r .1787 .53486 .20869 .554 .25247 .52374 .22293 .50616 Metetra .781 .84 .897 r .22293 .50616 .25247 .52374 .2967 .49072 .26762 .47583 Metetra .791 .866 .904 r .26762 .47583 .2967 .49072 .34131 .45454 .31269 .44421 Metetra .793 .909 .921 r .31269 .44421 .34131 .45454 .38617 .4155 .35803 .41254 Metetra .739 .965 .937 r .35803 .41254 .38617 .4155 .43108 .37719 .40355 .38388 Metetra .44 .874 .8 r .40355 .38388 .43108 .37719 .47587 .35094 .44926 .36296 Metetra .44926 .36296 .47587 .35094 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.935 r .37492 .38243 .40355 .38388 .44926 .36296 .42149 .36332 Metetra .357 .83 .867 r .42149 .36332 .44926 .36296 .4955 .35253 .46867 .35027 Metetra .267 .77 .867 r .46867 .35027 .4955 .35253 .54275 .35002 .51676 .34236 Metetra .292 .766 .933 r .51676 .34236 .54275 .35002 .5913 .35076 .56603 .33752 Metetra .356 .768 .98 r .56603 .33752 .5913 .35076 .64125 .35169 .6166 .33383 Metetra .413 .763 .995 r .6166 .33383 .64125 .35169 .69261 .35151 .66855 .3301 Metetra .455 .755 .994 r .66855 .3301 .69261 .35151 .74536 .34982 .72189 .3257 Metetra .485 .75 .988 r .72189 .3257 .74536 .34982 .79947 .34661 .77662 .32037 Metetra .74 .828 .919 r .07049 .52317 .10334 .54247 .14751 .51647 .11508 .49822 Metetra .743 .84 .924 r .11508 .49822 .14751 .51647 .19218 .48944 .16019 .47268 Metetra .744 .857 .932 r .16019 .47268 .19218 .48944 .23731 .46162 .20579 .44689 Metetra .738 .877 .945 r .20579 .44689 .23731 .46162 .28285 .43361 .25183 .42143 Metetra .719 .9 .96 r .25183 .42143 .28285 .43361 .32873 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Flatten[{eq1, eq2}]]\)\[LeftDoubleBracket]1\[RightDoubleBracket]; \ \[IndentingNewLine]\[ScriptP][x_, y_] = \[ScriptP][x, y] /. ans // Simplify; \[IndentingNewLine]\[ScriptP]]\)\(\[IndentingNewLine]\) \)\)], "Input", InitializationCell->True], Cell["Example of scattered data interpolant on uniform grid", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"p", "=", RowBox[{"(", GridBox[{ {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "1", "1", "1", "1", "0", "0"}, {"0", "0", "1", "1", "1", "1", "0", "0"}, {"0", "0", "1", "1", "1", "1", "0", "0"}, {"0", "0", "1", "1", "1", "1", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0"} }], ")"}]}], ",", "\[ScriptP]"}], "}"}], ",", "\[IndentingNewLine]", \(\[ScriptP] = makeInterp[\[IndentingNewLine]{{0, 0}, {1, 0}, {2, 0}, {3, 0}, {4, 0}, {5, 0}, {6, 0}, {7, 0}, \[IndentingNewLine]{0, 1}, {1, 1}, {2, 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