Mercurial > hg > xsplines
changeset 0:11ac175098d1
talk: init
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new file mode 100644 --- /dev/null +++ b/talk/img/FirstBlends.eps @@ -0,0 +1,1921 @@ +%!PS-Adobe-3.0 EPSF-2.0 +%%Title: Maple plot +%%Creator: Maple +%%Pages: 1 +%%BoundingBox: 93 85 500 660 +%%DocumentNeededResources: font Helvetica +%%EndComments +20 dict begin +gsave +/drawborder false def +/m {moveto} def +/l {lineto} def +/C {setrgbcolor} def +/Y /setcmykcolor where { %%ifelse Use built-in operator + /setcmykcolor get + }{ %%ifelse Emulate setcmykcolor with setrgbcolor + { %%def + 1 sub 3 { %%repeat + 3 index add neg dup 0 lt { pop 0 } if 3 1 roll + } repeat setrgbcolor + } bind + } ifelse def +/G {setgray} def +/S {stroke} def +/NP {newpath} def +%%%%This draws a filled polygon and avoids bugs/features +%%%% of some postscript interpreters +%%%%GHOSTSCRIPT: has a bug in reversepath - removing +%%%%the call to reversepath is a sufficient work around +/P {gsave fill grestore reversepath stroke} def +%%%%This function is needed for drawing text. It takes two arguments: a +%%%%string and a rotation. It outputs width and height. +/stringbbox { + gsave NP 0 0 m % Set origin for easy calculations. + dup rotate exch % Set orientation, and put string on top. + false charpath flattenpath + 0 exch sub rotate % Undo rotation. + pathbbox 4 2 roll pop pop % Get dimens. of our string. + 1.1 mul cvi exch 1.1 mul cvi exch % Pad some. + grestore} def +/thin 3 def +/medium 7 def +/thick 16 def +/boundarythick 20 def % thichess of bounding box +%%IncludeResource: font Helvetica +72.000000 72.000000 translate +/defaultTitleFont {/Helvetica findfont 219 scalefont setfont} def +/defaultFont {/Helvetica findfont 154 scalefont setfont} def +0.0936 0.0936 scale +1 setlinejoin +1 setlinecap +0.0 setgray +/inch {72 mul} def +/fheight 0.35 inch neg def + 0 G +medium setlinewidth +[] 0 setdash defaultFont + 0.59 G +NP 4055 6110 m 4055 6230 l +S +NP 4055 5998 m 4055 6110 l +S +NP 4055 5881 m 4055 5998 l +S +NP 4055 5771 m 4055 5881 l +S +NP 4055 5648 m 4055 5771 l +S +NP 4055 5541 m 4055 5648 l +S +NP 4055 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+NP 474 3323 m 500 3323 l +S +NP 449 3461 m 500 3461 l +S +NP 474 3599 m 500 3599 l +S +NP 474 3738 m 500 3738 l +S +NP 474 3876 m 500 3876 l +S +NP 474 4015 m 500 4015 l +S +NP 449 4153 m 500 4153 l +S +NP 474 4292 m 500 4292 l +S +NP 474 4430 m 500 4430 l +S +NP 474 4569 m 500 4569 l +S +NP 474 4707 m 500 4707 l +S +NP 449 4846 m 500 4846 l +S +NP 474 4984 m 500 4984 l +S +NP 474 5123 m 500 5123 l +S +NP 474 5261 m 500 5261 l +S +NP 474 5399 m 500 5399 l +S +NP 449 5538 m 500 5538 l +S +NP 474 5676 m 500 5676 l +S +NP 474 5815 m 500 5815 l +S +NP 474 5953 m 500 5953 l +S +NP 474 6092 m 500 6092 l +S +NP 449 6230 m 500 6230 l +S +medium setlinewidth +NP 500 692 m 4500 692 l +S +thin setlinewidth +NP 588 673 m 588 692 l +S +NP 766 673 m 766 692 l +S +NP 944 654 m 944 692 l +S +NP 1122 673 m 1122 692 l +S +NP 1299 673 m 1299 692 l +S +NP 1477 673 m 1477 692 l +S +NP 1655 673 m 1655 692 l +S +NP 1833 654 m 1833 692 l +S +NP 2011 673 m 2011 692 l +S +NP 2188 673 m 2188 692 l +S +NP 2366 673 m 2366 692 l +S +NP 2544 673 m 2544 692 l +S +NP 2722 654 m 2722 692 l +S +NP 2900 673 m 2900 692 l +S +NP 3077 673 m 3077 692 l +S +NP 3255 673 m 3255 692 l +S +NP 3433 673 m 3433 692 l +S +NP 3611 654 m 3611 692 l +S +NP 3788 673 m 3788 692 l +S +NP 3966 673 m 3966 692 l +S +NP 4144 673 m 4144 692 l +S +NP 4322 673 m 4322 692 l +S +NP 4500 654 m 4500 692 l +S +medium setlinewidth +(0) dup 0 stringbbox 692 exch sub exch 427 exch sub exch m show +(0.1) dup 0 stringbbox 2 idiv 1384 exch sub exch 427 exch sub exch m show +(0.2) dup 0 stringbbox 2 idiv 2076 exch sub exch 427 exch sub exch m show +(0.3) dup 0 stringbbox 2 idiv 2769 exch sub exch 427 exch sub exch m show +(0.4) dup 0 stringbbox 2 idiv 3461 exch sub exch 427 exch sub exch m show +(0.5) dup 0 stringbbox 2 idiv 4153 exch sub exch 427 exch sub exch m show +(0.6) dup 0 stringbbox 2 idiv 4846 exch sub exch 427 exch sub exch m show +(0.7) dup 0 stringbbox 2 idiv 5538 exch sub exch 427 exch sub exch m show +(0.8) dup 0 stringbbox 2 idiv 6230 exch sub exch 427 exch sub exch m show +(2) dup 0 stringbbox 637 exch sub exch 2 idiv 944 exch sub exch m show +(4) dup 0 stringbbox 637 exch sub exch 2 idiv 1833 exch sub exch m show +(6) dup 0 stringbbox 637 exch sub exch 2 idiv 2722 exch sub exch m show +(8) dup 0 stringbbox 637 exch sub exch 2 idiv 3611 exch sub exch m show +(10) dup 0 stringbbox 637 exch sub exch 2 idiv 4500 exch sub exch m show +(t) dup 0 stringbbox 249 exch sub exch 2 idiv 2500 exch sub exch m show +boundarythick setlinewidth +% The following draws a box around the plot, +% if the variable drawborder is true +drawborder { +/bd boundarythick 2 idiv def +[] 0 setdash +NP bd bd m bd 6923 bd sub l +5000 bd sub 6923 bd sub l +5000 bd sub bd l +bd bd l S +} if % end of if to draw the border + +showpage +grestore +end +%%EOF
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new file mode 100644 --- /dev/null +++ b/talk/img/after.eps @@ -0,0 +1,711 @@ +%!PS-Adobe-3.0 EPSF-2.0 +%%Title: Maple plot +%%Creator: Maple +%%Pages: 1 +%%BoundingBox: 93 85 496 660 +%%DocumentNeededResources: font Helvetica +%%EndComments +20 dict begin +gsave +/drawborder false def +/m {moveto} def +/l {lineto} def +/C {setrgbcolor} def +/Y /setcmykcolor where { %%ifelse Use built-in operator + /setcmykcolor get + }{ %%ifelse Emulate setcmykcolor with setrgbcolor + { %%def + 1 sub 3 { %%repeat + 3 index add neg dup 0 lt { pop 0 } if 3 1 roll + } repeat setrgbcolor + } bind + } ifelse def +/G {setgray} def +/S {stroke} def +/NP {newpath} def +%%%%This draws a filled polygon and avoids bugs/features +%%%% of some postscript interpreters +%%%%GHOSTSCRIPT: has a bug in reversepath - removing +%%%%the call to reversepath is a sufficient work around +/P {gsave fill grestore reversepath stroke} def +%%%%This function is needed for drawing text. It takes two arguments: a +%%%%string and a rotation. It outputs width and height. +/stringbbox { + gsave NP 0 0 m % Set origin for easy calculations. + dup rotate exch % Set orientation, and put string on top. + false charpath flattenpath + 0 exch sub rotate % Undo rotation. + pathbbox 4 2 roll pop pop % Get dimens. of our string. + 1.1 mul cvi exch 1.1 mul cvi exch % Pad some. + grestore} def +/thin 3 def +/medium 7 def +/thick 16 def +/boundarythick 20 def % thichess of bounding box +%%IncludeResource: font Helvetica +72.000000 72.000000 translate +/defaultTitleFont {/Helvetica findfont 219 scalefont setfont} def +/defaultFont {/Helvetica findfont 154 scalefont setfont} def +0.0936 0.0936 scale +1 setlinejoin +1 setlinecap +0.0 setgray +/inch {72 mul} def +/fheight 0.35 inch neg def + 0 G +medium setlinewidth +[] 0 setdash defaultFont + 0.3 G +NP 2500 6110 m 2500 6230 l +S +NP 2500 5998 m 2500 6110 l +S +NP 2500 5881 m 2500 5998 l +S +NP 2500 5771 m 2500 5881 l +S +NP 2500 5648 m 2500 5771 l +S +NP 2500 5541 m 2500 5648 l +S +NP 2500 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1588 m 1197 1739 l +S +NP 1080 1323 m 1159 1588 l +S +NP 1000 1112 m 1080 1323 l +S +NP 921 950 m 1000 1112 l +S +NP 842 834 m 921 950 l +S +NP 772 764 m 842 834 l +S +NP 702 721 m 772 764 l +S +NP 662 707 m 702 721 l +S +NP 622 698 m 662 707 l +S +NP 582 694 m 622 698 l +S +NP 542 692 m 582 694 l +S +NP 500 692 m 542 692 l +S + 0 G +medium setlinewidth +NP 500 692 m 500 6230 l +S +thin setlinewidth +NP 449 692 m 500 692 l +S +NP 474 913 m 500 913 l +S +NP 474 1135 m 500 1135 l +S +NP 474 1356 m 500 1356 l +S +NP 474 1578 m 500 1578 l +S +NP 449 1799 m 500 1799 l +S +NP 474 2021 m 500 2021 l +S +NP 474 2243 m 500 2243 l +S +NP 474 2464 m 500 2464 l +S +NP 474 2686 m 500 2686 l +S +NP 449 2907 m 500 2907 l +S +NP 474 3129 m 500 3129 l +S +NP 474 3350 m 500 3350 l +S +NP 474 3572 m 500 3572 l +S +NP 474 3793 m 500 3793 l +S +NP 449 4015 m 500 4015 l +S +NP 474 4236 m 500 4236 l +S +NP 474 4458 m 500 4458 l +S +NP 474 4679 m 500 4679 l +S +NP 474 4901 m 500 4901 l +S +NP 449 5123 m 500 5123 l +S +NP 474 5344 m 500 5344 l +S +NP 474 5566 m 500 5566 l +S +NP 474 5787 m 500 5787 l +S +NP 474 6009 m 500 6009 l +S +NP 449 6230 m 500 6230 l +S +medium setlinewidth +NP 500 692 m 4500 692 l +S +thin setlinewidth +NP 500 654 m 500 692 l +S +NP 633 673 m 633 692 l +S +NP 766 673 m 766 692 l +S +NP 900 673 m 900 692 l +S +NP 1033 673 m 1033 692 l +S +NP 1166 654 m 1166 692 l +S +NP 1300 673 m 1300 692 l +S +NP 1433 673 m 1433 692 l +S +NP 1566 673 m 1566 692 l +S +NP 1700 673 m 1700 692 l +S +NP 1833 654 m 1833 692 l +S +NP 1966 673 m 1966 692 l +S +NP 2100 673 m 2100 692 l +S +NP 2233 673 m 2233 692 l +S +NP 2366 673 m 2366 692 l +S +NP 2500 654 m 2500 692 l +S +NP 2633 673 m 2633 692 l +S +NP 2766 673 m 2766 692 l +S +NP 2900 673 m 2900 692 l +S +NP 3033 673 m 3033 692 l +S +NP 3166 654 m 3166 692 l +S +NP 3300 673 m 3300 692 l +S +NP 3433 673 m 3433 692 l +S +NP 3566 673 m 3566 692 l +S +NP 3700 673 m 3700 692 l +S +NP 3833 654 m 3833 692 l +S +NP 3966 673 m 3966 692 l +S +NP 4100 673 m 4100 692 l +S +NP 4233 673 m 4233 692 l +S +NP 4366 673 m 4366 692 l +S +NP 4500 654 m 4500 692 l +S +medium setlinewidth +(0) dup 0 stringbbox 692 exch sub exch 427 exch sub exch m show +(0.2) dup 0 stringbbox 2 idiv 1799 exch sub exch 427 exch sub exch m show +(0.4) dup 0 stringbbox 2 idiv 2907 exch sub exch 427 exch sub exch m show +(0.6) dup 0 stringbbox 2 idiv 4015 exch sub exch 427 exch sub exch m show +(0.8) dup 0 stringbbox 2 idiv 5123 exch sub exch 427 exch sub exch m show +(1) dup 0 stringbbox 2 idiv 6230 exch sub exch 427 exch sub exch m show +(2) dup 0 stringbbox 637 exch sub exch pop 500 exch m show +(3) dup 0 stringbbox 637 exch sub exch 2 idiv 1166 exch sub exch m show +(4) dup 0 stringbbox 637 exch sub exch 2 idiv 1833 exch sub exch m show +(5) dup 0 stringbbox 637 exch sub exch 2 idiv 2500 exch sub exch m show +(6) dup 0 stringbbox 637 exch sub exch 2 idiv 3166 exch sub exch m show +(7) dup 0 stringbbox 637 exch sub exch 2 idiv 3833 exch sub exch m show +(8) dup 0 stringbbox 637 exch sub exch 2 idiv 4500 exch sub exch m show +(t) dup 0 stringbbox 249 exch sub exch 2 idiv 2500 exch sub exch m show +boundarythick setlinewidth +% The following draws a box around the plot, +% if the variable drawborder is true +drawborder { +/bd boundarythick 2 idiv def +[] 0 setdash +NP bd bd m bd 6923 bd sub l +5000 bd sub 6923 bd sub l +5000 bd sub bd l +bd bd l S +} if % end of if to draw the border + +showpage +grestore +end +%%EOF
new file mode 100644 --- /dev/null +++ b/talk/img/before.eps @@ -0,0 +1,699 @@ +%!PS-Adobe-3.0 EPSF-2.0 +%%Title: Maple plot +%%Creator: Maple +%%Pages: 1 +%%BoundingBox: 93 85 496 660 +%%DocumentNeededResources: font Helvetica +%%EndComments +20 dict begin +gsave +/drawborder false def +/m {moveto} def +/l {lineto} def +/C {setrgbcolor} def +/Y /setcmykcolor where { %%ifelse Use built-in operator + /setcmykcolor get + }{ %%ifelse Emulate setcmykcolor with setrgbcolor + { %%def + 1 sub 3 { %%repeat + 3 index add neg dup 0 lt { pop 0 } if 3 1 roll + } repeat setrgbcolor + } bind + } ifelse def +/G {setgray} def +/S {stroke} def +/NP {newpath} def +%%%%This draws a filled polygon and avoids bugs/features +%%%% of some postscript interpreters +%%%%GHOSTSCRIPT: has a bug in reversepath - removing +%%%%the call to reversepath is a sufficient work around +/P {gsave fill grestore reversepath stroke} def +%%%%This function is needed for drawing text. It takes two arguments: a +%%%%string and a rotation. It outputs width and height. +/stringbbox { + gsave NP 0 0 m % Set origin for easy calculations. + dup rotate exch % Set orientation, and put string on top. + false charpath flattenpath + 0 exch sub rotate % Undo rotation. + pathbbox 4 2 roll pop pop % Get dimens. of our string. + 1.1 mul cvi exch 1.1 mul cvi exch % Pad some. + grestore} def +/thin 3 def +/medium 7 def +/thick 16 def +/boundarythick 20 def % thichess of bounding box +%%IncludeResource: font Helvetica +72.000000 72.000000 translate +/defaultTitleFont {/Helvetica findfont 219 scalefont setfont} def +/defaultFont {/Helvetica findfont 154 scalefont setfont} def +0.0936 0.0936 scale +1 setlinejoin +1 setlinecap +0.0 setgray +/inch {72 mul} def +/fheight 0.35 inch neg def + 0 G +medium setlinewidth +[] 0 setdash defaultFont + 0.3 G +NP 2500 6110 m 2500 6230 l +S +NP 2500 5998 m 2500 6110 l +S +NP 2500 5881 m 2500 5998 l +S +NP 2500 5771 m 2500 5881 l +S +NP 2500 5648 m 2500 5771 l +S +NP 2500 5541 m 2500 5648 l +S +NP 2500 5425 m 2500 5541 l +S +NP 2500 5307 m 2500 5425 l +S +NP 2500 5193 m 2500 5307 l +S +NP 2500 5073 m 2500 5193 l +S +NP 2500 4959 m 2500 5073 l +S +NP 2500 4843 m 2500 4959 l +S +NP 2500 4731 m 2500 4843 l +S +NP 2500 4613 m 2500 4731 l +S +NP 2500 4505 m 2500 4613 l +S +NP 2500 4385 m 2500 4505 l +S +NP 2500 4273 m 2500 4385 l +S +NP 2500 4148 m 2500 4273 l +S +NP 2500 4036 m 2500 4148 l +S +NP 2500 3920 m 2500 4036 l +S +NP 2500 3801 m 2500 3920 l +S +NP 2500 3687 m 2500 3801 l +S +NP 2500 3580 m 2500 3687 l +S +NP 2500 3458 m 2500 3580 l +S +NP 2500 3341 m 2500 3458 l +S +NP 2500 3232 m 2500 3341 l +S +NP 2500 3113 m 2500 3232 l +S +NP 2500 2999 m 2500 3113 l +S +NP 2500 2880 m 2500 2999 l +S +NP 2500 2771 m 2500 2880 l +S +NP 2500 2648 m 2500 2771 l +S +NP 2500 2542 m 2500 2648 l +S +NP 2500 2417 m 2500 2542 l +S +NP 2500 2312 m 2500 2417 l +S +NP 2500 2197 m 2500 2312 l +S +NP 2500 2077 m 2500 2197 l +S +NP 2500 1957 m 2500 2077 l +S +NP 2500 1851 m 2500 1957 l +S +NP 2500 1731 m 2500 1851 l +S +NP 2500 1614 m 2500 1731 l +S +NP 2500 1496 m 2500 1614 l +S +NP 2500 1383 m 2500 1496 l +S +NP 2500 1273 m 2500 1383 l +S +NP 2500 1155 m 2500 1273 l +S +NP 2500 1036 m 2500 1155 l +S +NP 2500 918 m 2500 1036 l +S +NP 2500 813 m 2500 918 l +S +NP 2500 692 m 2500 813 l +S + 0.11 G +NP 4437 693 m 4500 692 l +S +NP 4358 701 m 4437 693 l +S +NP 4322 710 m 4358 701 l +S +NP 4286 725 m 4322 710 l +S +NP 4251 745 m 4286 725 l +S +NP 4215 771 m 4251 745 l +S +NP 4135 858 m 4215 771 l +S +NP 4056 988 m 4135 858 l +S +NP 3982 1151 m 4056 988 l +S +NP 3908 1357 m 3982 1151 l +S +NP 3825 1643 m 3908 1357 l +S +NP 3743 2001 m 3825 1643 l +S +NP 3706 2185 m 3743 2001 l +S +NP 3669 2380 m 3706 2185 l +S +NP 3631 2585 m 3669 2380 l +S +NP 3594 2796 m 3631 2585 l +S +NP 3556 3016 m 3594 2796 l +S +NP 3518 3233 m 3556 3016 l +S +NP 3479 3444 m 3518 3233 l +S +NP 3441 3641 m 3479 3444 l +S +NP 3362 3990 m 3441 3641 l +S +NP 3284 4238 m 3362 3990 l +S +NP 3246 4316 m 3284 4238 l +S +NP 3208 4365 m 3246 4316 l +S +NP 3189 4379 m 3208 4365 l +S +NP 3170 4384 m 3189 4379 l +S +NP 3151 4381 m 3170 4384 l +S +NP 3132 4371 m 3151 4381 l +S +NP 3061 4267 m 3132 4371 l +S +NP 2991 4066 m 3061 4267 l +S +NP 2950 3911 m 2991 4066 l +S +NP 2910 3731 m 2950 3911 l +S +NP 2870 3530 m 2910 3731 l +S +NP 2829 3313 m 2870 3530 l +S +NP 2790 3095 m 2829 3313 l +S +NP 2751 2872 m 2790 3095 l +S +NP 2712 2650 m 2751 2872 l +S +NP 2674 2434 m 2712 2650 l +S +NP 2637 2241 m 2674 2434 l +S +NP 2601 2059 m 2637 2241 l +S +NP 2565 1889 m 2601 2059 l +S +NP 2529 1733 m 2565 1889 l +S +NP 2451 1441 m 2529 1733 l +S +NP 2372 1207 m 2451 1441 l +S +NP 2297 1029 m 2372 1207 l +S +NP 2222 894 m 2297 1029 l +S +NP 2142 794 m 2222 894 l +S +NP 2063 734 m 2142 794 l +S +NP 2027 717 m 2063 734 l +S +NP 1991 705 m 2027 717 l +S +NP 1955 698 m 1991 705 l +S +NP 1919 694 m 1955 698 l +S +NP 1838 692 m 1919 694 l +S +NP 1756 692 m 1838 692 l +S +NP 1616 692 m 1756 692 l +S +NP 1451 692 m 1616 692 l +S +NP 1312 692 m 1451 692 l +S +NP 1159 692 m 1312 692 l +S +NP 1000 692 m 1159 692 l +S +NP 842 692 m 1000 692 l +S +NP 702 692 m 842 692 l +S +NP 542 692 m 702 692 l +S +NP 500 692 m 542 692 l +S + 0.41 G +NP 4358 692 m 4500 692 l +S +NP 4215 692 m 4358 692 l +S +NP 4056 692 m 4215 692 l +S +NP 3908 692 m 4056 692 l +S +NP 3825 692 m 3908 692 l +S +NP 3743 694 m 3825 692 l +S +NP 3706 699 m 3743 694 l +S +NP 3669 707 m 3706 699 l +S +NP 3631 719 m 3669 707 l +S +NP 3594 738 m 3631 719 l +S +NP 3518 800 m 3594 738 l +S +NP 3441 899 m 3518 800 l +S +NP 3362 1044 m 3441 899 l +S +NP 3284 1234 m 3362 1044 l +S +NP 3208 1466 m 3284 1234 l +S +NP 3132 1753 m 3208 1466 l +S +NP 3096 1907 m 3132 1753 l +S +NP 3061 2074 m 3096 1907 l +S +NP 3026 2252 m 3061 2074 l +S +NP 2991 2441 m 3026 2252 l +S +NP 2950 2666 m 2991 2441 l +S +NP 2910 2897 m 2950 2666 l +S +NP 2870 3127 m 2910 2897 l +S +NP 2829 3353 m 2870 3127 l +S +NP 2790 3559 m 2829 3353 l +S +NP 2751 3751 m 2790 3559 l +S +NP 2712 3922 m 2751 3751 l +S +NP 2674 4070 m 2712 3922 l +S +NP 2601 4274 m 2674 4070 l +S +NP 2529 4374 m 2601 4274 l +S +NP 2510 4383 m 2529 4374 l +S +NP 2490 4383 m 2510 4383 l +S +NP 2480 4380 m 2490 4383 l +S +NP 2470 4375 m 2480 4380 l +S +NP 2461 4367 m 2470 4375 l +S +NP 2451 4358 m 2461 4367 l +S +NP 2411 4301 m 2451 4358 l +S +NP 2372 4213 m 2411 4301 l +S +NP 2297 3964 m 2372 4213 l +S +NP 2222 3624 m 2297 3964 l +S +NP 2182 3419 m 2222 3624 l +S +NP 2142 3201 m 2182 3419 l +S +NP 2103 2976 m 2142 3201 l +S +NP 2063 2750 m 2103 2976 l +S +NP 2027 2547 m 2063 2750 l +S +NP 1991 2349 m 2027 2547 l +S +NP 1955 2161 m 1991 2349 l +S +NP 1919 1984 m 1955 2161 l +S +NP 1838 1634 m 1919 1984 l +S +NP 1756 1353 m 1838 1634 l +S +NP 1686 1158 m 1756 1353 l +S +NP 1616 1001 m 1686 1158 l +S +NP 1534 863 m 1616 1001 l +S +NP 1451 771 m 1534 863 l +S +NP 1416 745 m 1451 771 l +S +NP 1382 726 m 1416 745 l +S +NP 1347 712 m 1382 726 l +S +NP 1312 702 m 1347 712 l +S +NP 1236 693 m 1312 702 l +S +NP 1159 692 m 1236 693 l +S +NP 1000 692 m 1159 692 l +S +NP 842 692 m 1000 692 l +S +NP 702 692 m 842 692 l +S +NP 542 692 m 702 692 l +S +NP 500 692 m 542 692 l +S + 0.3 G +NP 4358 692 m 4500 692 l +S +NP 4215 692 m 4358 692 l +S +NP 4056 692 m 4215 692 l +S +NP 3908 692 m 4056 692 l +S +NP 3743 692 m 3908 692 l +S +NP 3594 692 m 3743 692 l +S +NP 3441 692 m 3594 692 l +S +NP 3284 692 m 3441 692 l +S +NP 3132 692 m 3284 692 l +S +NP 3061 696 m 3132 692 l +S +NP 2991 710 m 3061 696 l +S +NP 2910 749 m 2991 710 l +S +NP 2829 824 m 2910 749 l +S +NP 2751 936 m 2829 824 l +S +NP 2674 1092 m 2751 936 l +S +NP 2601 1278 m 2674 1092 l +S +NP 2529 1506 m 2601 1278 l +S +NP 2490 1652 m 2529 1506 l +S +NP 2451 1814 m 2490 1652 l +S +NP 2411 1993 m 2451 1814 l +S +NP 2372 2187 m 2411 1993 l +S +NP 2334 2385 m 2372 2187 l +S +NP 2297 2593 m 2334 2385 l +S +NP 2259 2806 m 2297 2593 l +S +NP 2222 3022 m 2259 2806 l +S +NP 2182 3246 m 2222 3022 l +S +NP 2142 3461 m 2182 3246 l +S +NP 2103 3664 m 2142 3461 l +S +NP 2063 3847 m 2103 3664 l +S +NP 1991 4122 m 2063 3847 l +S +NP 1919 4304 m 1991 4122 l +S +NP 1899 4337 m 1919 4304 l +S +NP 1878 4361 m 1899 4337 l +S +NP 1868 4370 m 1878 4361 l +S +NP 1858 4377 m 1868 4370 l +S +NP 1848 4382 m 1858 4377 l +S +NP 1838 4384 m 1848 4382 l +S +NP 1817 4381 m 1838 4384 l +S +NP 1797 4370 m 1817 4381 l +S +NP 1777 4350 m 1797 4370 l +S +NP 1756 4321 m 1777 4350 l +S +NP 1686 4159 m 1756 4321 l +S +NP 1616 3907 m 1686 4159 l +S +NP 1575 3723 m 1616 3907 l +S +NP 1534 3516 m 1575 3723 l +S +NP 1492 3294 m 1534 3516 l +S +NP 1451 3061 m 1492 3294 l +S +NP 1416 2862 m 1451 3061 l +S +NP 1382 2664 m 1416 2862 l +S +NP 1347 2470 m 1382 2664 l +S +NP 1312 2283 m 1347 2470 l +S +NP 1274 2087 m 1312 2283 l +S +NP 1236 1905 m 1274 2087 l +S +NP 1197 1739 m 1236 1905 l +S +NP 1159 1588 m 1197 1739 l +S +NP 1080 1323 m 1159 1588 l +S +NP 1000 1112 m 1080 1323 l +S +NP 921 950 m 1000 1112 l +S +NP 842 834 m 921 950 l +S +NP 772 764 m 842 834 l +S +NP 702 721 m 772 764 l +S +NP 662 707 m 702 721 l +S +NP 622 698 m 662 707 l +S +NP 582 694 m 622 698 l +S +NP 542 692 m 582 694 l +S +NP 500 692 m 542 692 l +S + 0 G +medium setlinewidth +NP 500 692 m 500 6230 l +S +thin setlinewidth +NP 449 692 m 500 692 l +S +NP 474 913 m 500 913 l +S +NP 474 1135 m 500 1135 l +S +NP 474 1356 m 500 1356 l +S +NP 474 1578 m 500 1578 l +S +NP 449 1799 m 500 1799 l +S +NP 474 2021 m 500 2021 l +S +NP 474 2243 m 500 2243 l +S +NP 474 2464 m 500 2464 l +S +NP 474 2686 m 500 2686 l +S +NP 449 2907 m 500 2907 l +S +NP 474 3129 m 500 3129 l +S +NP 474 3350 m 500 3350 l +S +NP 474 3572 m 500 3572 l +S +NP 474 3793 m 500 3793 l +S +NP 449 4015 m 500 4015 l +S +NP 474 4236 m 500 4236 l +S +NP 474 4458 m 500 4458 l +S +NP 474 4679 m 500 4679 l +S +NP 474 4901 m 500 4901 l +S +NP 449 5123 m 500 5123 l +S +NP 474 5344 m 500 5344 l +S +NP 474 5566 m 500 5566 l +S +NP 474 5787 m 500 5787 l +S +NP 474 6009 m 500 6009 l +S +NP 449 6230 m 500 6230 l +S +medium setlinewidth +NP 500 692 m 4500 692 l +S +thin setlinewidth +NP 500 654 m 500 692 l +S +NP 633 673 m 633 692 l +S +NP 766 673 m 766 692 l +S +NP 900 673 m 900 692 l +S +NP 1033 673 m 1033 692 l +S +NP 1166 654 m 1166 692 l +S +NP 1300 673 m 1300 692 l +S +NP 1433 673 m 1433 692 l +S +NP 1566 673 m 1566 692 l +S +NP 1700 673 m 1700 692 l +S +NP 1833 654 m 1833 692 l +S +NP 1966 673 m 1966 692 l +S +NP 2100 673 m 2100 692 l +S +NP 2233 673 m 2233 692 l +S +NP 2366 673 m 2366 692 l +S +NP 2500 654 m 2500 692 l +S +NP 2633 673 m 2633 692 l +S +NP 2766 673 m 2766 692 l +S +NP 2900 673 m 2900 692 l +S +NP 3033 673 m 3033 692 l +S +NP 3166 654 m 3166 692 l +S +NP 3300 673 m 3300 692 l +S +NP 3433 673 m 3433 692 l +S +NP 3566 673 m 3566 692 l +S +NP 3700 673 m 3700 692 l +S +NP 3833 654 m 3833 692 l +S +NP 3966 673 m 3966 692 l +S +NP 4100 673 m 4100 692 l +S +NP 4233 673 m 4233 692 l +S +NP 4366 673 m 4366 692 l +S +NP 4500 654 m 4500 692 l +S +medium setlinewidth +(0) dup 0 stringbbox 692 exch sub exch 427 exch sub exch m show +(0.2) dup 0 stringbbox 2 idiv 1799 exch sub exch 427 exch sub exch m show +(0.4) dup 0 stringbbox 2 idiv 2907 exch sub exch 427 exch sub exch m show +(0.6) dup 0 stringbbox 2 idiv 4015 exch sub exch 427 exch sub exch m show +(0.8) dup 0 stringbbox 2 idiv 5123 exch sub exch 427 exch sub exch m show +(1) dup 0 stringbbox 2 idiv 6230 exch sub exch 427 exch sub exch m show +(2) dup 0 stringbbox 637 exch sub exch pop 500 exch m show +(3) dup 0 stringbbox 637 exch sub exch 2 idiv 1166 exch sub exch m show +(4) dup 0 stringbbox 637 exch sub exch 2 idiv 1833 exch sub exch m show +(5) dup 0 stringbbox 637 exch sub exch 2 idiv 2500 exch sub exch m show +(6) dup 0 stringbbox 637 exch sub exch 2 idiv 3166 exch sub exch m show +(7) dup 0 stringbbox 637 exch sub exch 2 idiv 3833 exch sub exch m show +(8) dup 0 stringbbox 637 exch sub exch 2 idiv 4500 exch sub exch m show +(t) dup 0 stringbbox 249 exch sub exch 2 idiv 2500 exch sub exch m show +boundarythick setlinewidth +% The following draws a box around the plot, +% if the variable drawborder is true +drawborder { +/bd boundarythick 2 idiv def +[] 0 setdash +NP bd bd m bd 6923 bd sub l +5000 bd sub 6923 bd sub l +5000 bd sub bd l +bd bd l S +} if % end of if to draw the border + +showpage +grestore +end +%%EOF
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new file mode 100644 --- /dev/null +++ b/talk/img/fakebsplines.eps @@ -0,0 +1,1261 @@ +%!PS-Adobe-3.0 EPSF-2.0 +%%Title: Maple plot +%%Creator: Maple +%%Pages: 1 +%%BoundingBox: 93 95 493 655 +%%DocumentNeededResources: font Helvetica +%%EndComments +20 dict begin +gsave +/drawborder false def +/m {moveto} def +/l {lineto} def +/C {setrgbcolor} def +/Y /setcmykcolor where { %%ifelse Use built-in operator + /setcmykcolor get + }{ %%ifelse Emulate setcmykcolor with setrgbcolor + { %%def + 1 sub 3 { %%repeat + 3 index add neg dup 0 lt { pop 0 } if 3 1 roll + } repeat setrgbcolor + } bind + } ifelse def +/G {setgray} def +/S {stroke} def +/NP {newpath} def +%%%%This draws a filled polygon and avoids bugs/features +%%%% of some postscript interpreters +%%%%GHOSTSCRIPT: has a bug in reversepath - removing +%%%%the call to reversepath is a sufficient work around +/P {gsave fill grestore reversepath stroke} def +%%%%This function is needed for drawing text. It takes two arguments: a +%%%%string and a rotation. It outputs width and height. +/stringbbox { + gsave NP 0 0 m % Set origin for easy calculations. + dup rotate exch % Set orientation, and put string on top. + false charpath flattenpath + 0 exch sub rotate % Undo rotation. + pathbbox 4 2 roll pop pop % Get dimens. of our string. + 1.1 mul cvi exch 1.1 mul cvi exch % Pad some. + grestore} def +/thin 3 def +/medium 7 def +/thick 16 def +/boundarythick 20 def % thichess of bounding box +%%IncludeResource: font Helvetica +72.000000 72.000000 translate +/defaultTitleFont {/Helvetica findfont 219 scalefont setfont} def +/defaultFont {/Helvetica findfont 154 scalefont setfont} def +0.0936 0.0936 scale +1 setlinejoin +1 setlinecap +0.0 setgray +/inch {72 mul} def +/fheight 0.35 inch neg def + 0 G +medium setlinewidth +[] 0 setdash defaultFont + 0.41 G +NP 4494 798 m 4497 798 l +S +NP 4491 798 m 4494 798 l +S +NP 4489 798 m 4491 798 l +S +NP 4483 798 m 4489 798 l +S +NP 4478 798 m 4483 798 l +S +NP 4467 798 m 4478 798 l +S +NP 4456 798 m 4467 798 l +S +NP 4434 798 m 4456 798 l +S +NP 4413 798 m 4434 798 l +S +NP 4372 798 m 4413 798 l +S +NP 4332 798 m 4372 798 l +S +NP 4247 798 m 4332 798 l +S +NP 4168 798 m 4247 798 l +S +NP 4079 798 m 4168 798 l +S +NP 4002 798 m 4079 798 l +S +NP 3918 798 m 4002 798 l +S +NP 3833 798 m 3918 798 l +S +NP 3792 798 m 3833 798 l +S +NP 3750 799 m 3792 798 l +S +NP 3729 801 m 3750 799 l +S +NP 3707 807 m 3729 801 l +S +NP 3686 819 m 3707 807 l +S +NP 3664 839 m 3686 819 l +S +NP 3623 904 m 3664 839 l +S +NP 3582 1015 m 3623 904 l +S +NP 3540 1183 m 3582 1015 l +S +NP 3498 1414 m 3540 1183 l +S +NP 3457 1696 m 3498 1414 l +S +NP 3417 2046 m 3457 1696 l +S +NP 3396 2264 m 3417 2046 l +S +NP 3374 2506 m 3396 2264 l +S +NP 3353 2771 m 3374 2506 l +S +NP 3331 3058 m 3353 2771 l +S +NP 3312 3335 m 3331 3058 l +S +NP 3292 3623 m 3312 3335 l +S +NP 3273 3918 m 3292 3623 l +S +NP 3253 4214 m 3273 3918 l +S +NP 3232 4536 m 3253 4214 l +S +NP 3210 4844 m 3232 4536 l +S +NP 3188 5132 m 3210 4844 l +S +NP 3167 5392 m 3188 5132 l +S +NP 3126 5785 m 3167 5392 l +S +NP 3086 6034 m 3126 5785 l +S +NP 3075 6075 m 3086 6034 l +S +NP 3063 6104 m 3075 6075 l +S +NP 3052 6119 m 3063 6104 l +S +NP 3041 6121 m 3052 6119 l +S +NP 3035 6117 m 3041 6121 l +S +NP 3030 6109 m 3035 6117 l +S +NP 3024 6099 m 3030 6109 l +S +NP 3018 6085 m 3024 6099 l +S +NP 3007 6047 m 3018 6085 l +S +NP 2996 5997 m 3007 6047 l +S +NP 2955 5717 m 2996 5997 l +S +NP 2915 5297 m 2955 5717 l +S +NP 2894 5034 m 2915 5297 l +S +NP 2873 4747 m 2894 5034 l +S +NP 2852 4443 m 2873 4747 l +S +NP 2831 4129 m 2852 4443 l +S +NP 2810 3805 m 2831 4129 l +S +NP 2788 3485 m 2810 3805 l +S +NP 2767 3175 m 2788 3485 l +S +NP 2745 2881 m 2767 3175 l +S +NP 2704 2371 m 2745 2881 l +S +NP 2662 1949 m 2704 2371 l +S +NP 2624 1631 m 2662 1949 l +S +NP 2586 1371 m 2624 1631 l +S +NP 2542 1140 m 2586 1371 l +S +NP 2498 977 m 2542 1140 l +S +NP 2455 877 m 2498 977 l +S +NP 2413 824 m 2455 877 l +S +NP 2373 803 m 2413 824 l +S +NP 2334 798 m 2373 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+NP 4478 798 m 4483 798 l +S +NP 4467 798 m 4478 798 l +S +NP 4456 798 m 4467 798 l +S +NP 4434 798 m 4456 798 l +S +NP 4413 798 m 4434 798 l +S +NP 4372 798 m 4413 798 l +S +NP 4332 798 m 4372 798 l +S +NP 4247 798 m 4332 798 l +S +NP 4168 798 m 4247 798 l +S +NP 4079 798 m 4168 798 l +S +NP 4002 798 m 4079 798 l +S +NP 3918 798 m 4002 798 l +S +NP 3833 798 m 3918 798 l +S +NP 3750 798 m 3833 798 l +S +NP 3664 798 m 3750 798 l +S +NP 3582 798 m 3664 798 l +S +NP 3498 798 m 3582 798 l +S +NP 3417 798 m 3498 798 l +S +NP 3374 799 m 3417 798 l +S +NP 3331 812 m 3374 799 l +S +NP 3312 825 m 3331 812 l +S +NP 3292 846 m 3312 825 l +S +NP 3273 874 m 3292 846 l +S +NP 3253 913 m 3273 874 l +S +NP 3210 1038 m 3253 913 l +S +NP 3167 1225 m 3210 1038 l +S +NP 3126 1461 m 3167 1225 l +S +NP 3086 1757 m 3126 1461 l +S +NP 3041 2170 m 3086 1757 l +S +NP 2996 2686 m 3041 2170 l +S +NP 2976 2951 m 2996 2686 l +S +NP 2955 3233 m 2976 2951 l +S +NP 2935 3528 m 2955 3233 l +S +NP 2915 3832 m 2935 3528 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2704 1914 l +S +NP 2643 2550 m 2662 2328 l +S +NP 2624 2791 m 2643 2550 l +S +NP 2605 3048 m 2624 2791 l +S +NP 2586 3320 m 2605 3048 l +S +NP 2564 3644 m 2586 3320 l +S +NP 2542 3977 m 2564 3644 l +S +NP 2520 4310 m 2542 3977 l +S +NP 2498 4635 m 2520 4310 l +S +NP 2476 4933 m 2498 4635 l +S +NP 2455 5208 m 2476 4933 l +S +NP 2434 5456 m 2455 5208 l +S +NP 2413 5669 m 2434 5456 l +S +NP 2373 5962 m 2413 5669 l +S +NP 2334 6108 m 2373 5962 l +S +NP 2323 6120 m 2334 6108 l +S +NP 2312 6120 m 2323 6120 l +S +NP 2307 6116 m 2312 6120 l +S +NP 2302 6108 m 2307 6116 l +S +NP 2296 6098 m 2302 6108 l +S +NP 2291 6085 m 2296 6098 l +S +NP 2270 6002 m 2291 6085 l +S +NP 2248 5875 m 2270 6002 l +S +NP 2207 5515 m 2248 5875 l +S +NP 2166 5026 m 2207 5515 l +S +NP 2144 4730 m 2166 5026 l +S +NP 2123 4416 m 2144 4730 l +S +NP 2101 4092 m 2123 4416 l +S +NP 2080 3766 m 2101 4092 l +S +NP 2060 3472 m 2080 3766 l +S +NP 2040 3188 m 2060 3472 l +S +NP 2021 2917 m 2040 3188 l +S +NP 2001 2662 m 2021 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+S +NP 4281 779 m 4281 798 l +S +NP 4427 779 m 4427 798 l +S +medium setlinewidth +(0) dup 0 stringbbox 798 exch sub exch 427 exch sub exch m show +(0.1) dup 0 stringbbox 2 idiv 1597 exch sub exch 427 exch sub exch m show +(0.2) dup 0 stringbbox 2 idiv 2395 exch sub exch 427 exch sub exch m show +(0.3) dup 0 stringbbox 2 idiv 3194 exch sub exch 427 exch sub exch m show +(0.4) dup 0 stringbbox 2 idiv 3992 exch sub exch 427 exch sub exch m show +(0.5) dup 0 stringbbox 2 idiv 4791 exch sub exch 427 exch sub exch m show +(0.6) dup 0 stringbbox 2 idiv 5589 exch sub exch 427 exch sub exch m show +(2) dup 0 stringbbox 744 exch sub exch 2 idiv 1227 exch sub exch m show +(4) dup 0 stringbbox 744 exch sub exch 2 idiv 1954 exch sub exch m show +(6) dup 0 stringbbox 744 exch sub exch 2 idiv 2681 exch sub exch m show +(8) dup 0 stringbbox 744 exch sub exch 2 idiv 3409 exch sub exch m show +(10) dup 0 stringbbox 744 exch sub exch 2 idiv 4136 exch sub exch m show +(t) dup 0 stringbbox 355 exch sub exch 2 idiv 2500 exch sub exch m show +boundarythick setlinewidth +% The following draws a box around the plot, +% if the variable drawborder is true +drawborder { +/bd boundarythick 2 idiv def +[] 0 setdash +NP bd bd m bd 6923 bd sub l +5000 bd sub 6923 bd sub l +5000 bd sub bd l +bd bd l S +} if % end of if to draw the border + +showpage +grestore +end +%%EOF
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It takes two arguments: a +%%%%string and a rotation. It outputs width and height. +/stringbbox { + gsave NP 0 0 m % Set origin for easy calculations. + dup rotate exch % Set orientation, and put string on top. + false charpath flattenpath + 0 exch sub rotate % Undo rotation. + pathbbox 4 2 roll pop pop % Get dimens. of our string. + 1.1 mul cvi exch 1.1 mul cvi exch % Pad some. + grestore} def +/thin 3 def +/medium 7 def +/thick 16 def +/boundarythick 20 def % thichess of bounding box +%%IncludeResource: font Helvetica +72.000000 72.000000 translate +/defaultTitleFont {/Helvetica findfont 219 scalefont setfont} def +/defaultFont {/Helvetica findfont 154 scalefont setfont} def +0.0936 0.0936 scale +1 setlinejoin +1 setlinecap +0.0 setgray +/inch {72 mul} def +/fheight 0.35 inch neg def + 0 G +medium setlinewidth +[] 0 setdash defaultFont + 0.41 G +NP 4494 1266 m 4497 1266 l +S +NP 4491 1266 m 4494 1266 l +S +NP 4489 1266 m 4491 1266 l +S +NP 4483 1266 m 4489 1266 l +S +NP 4478 1266 m 4483 1266 l +S +NP 4467 1266 m 4478 1266 l +S +NP 4456 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1266 m 510 1266 l +S +NP 505 1266 m 508 1266 l +S +NP 502 1266 m 505 1266 l +S + 0 G +medium setlinewidth +NP 500 692 m 500 6230 l +S +thin setlinewidth +NP 474 877 m 500 877 l +S +NP 474 1072 m 500 1072 l +S +NP 449 1266 m 500 1266 l +S +NP 474 1460 m 500 1460 l +S +NP 474 1654 m 500 1654 l +S +NP 474 1849 m 500 1849 l +S +NP 474 2043 m 500 2043 l +S +NP 449 2237 m 500 2237 l +S +NP 474 2431 m 500 2431 l +S +NP 474 2625 m 500 2625 l +S +NP 474 2820 m 500 2820 l +S +NP 474 3014 m 500 3014 l +S +NP 449 3208 m 500 3208 l +S +NP 474 3402 m 500 3402 l +S +NP 474 3597 m 500 3597 l +S +NP 474 3791 m 500 3791 l +S +NP 474 3985 m 500 3985 l +S +NP 449 4179 m 500 4179 l +S +NP 474 4374 m 500 4374 l +S +NP 474 4568 m 500 4568 l +S +NP 474 4762 m 500 4762 l +S +NP 474 4956 m 500 4956 l +S +NP 449 5151 m 500 5151 l +S +NP 474 5345 m 500 5345 l +S +NP 474 5539 m 500 5539 l +S +NP 474 5733 m 500 5733 l +S +NP 474 5928 m 500 5928 l +S +NP 449 6122 m 500 6122 l +S +medium setlinewidth +NP 500 1266 m 4500 1266 l +S +thin setlinewidth +NP 645 1247 m 645 1266 l +S +NP 790 1247 m 790 1266 l +S +NP 936 1247 m 936 1266 l +S +NP 1081 1247 m 1081 1266 l +S +NP 1227 1228 m 1227 1266 l +S +NP 1372 1247 m 1372 1266 l +S +NP 1518 1247 m 1518 1266 l +S +NP 1663 1247 m 1663 1266 l +S +NP 1809 1247 m 1809 1266 l +S +NP 1954 1228 m 1954 1266 l +S +NP 2100 1247 m 2100 1266 l +S +NP 2245 1247 m 2245 1266 l +S +NP 2390 1247 m 2390 1266 l +S +NP 2536 1247 m 2536 1266 l +S +NP 2681 1228 m 2681 1266 l +S +NP 2827 1247 m 2827 1266 l +S +NP 2972 1247 m 2972 1266 l +S +NP 3118 1247 m 3118 1266 l +S +NP 3263 1247 m 3263 1266 l +S +NP 3409 1228 m 3409 1266 l +S +NP 3554 1247 m 3554 1266 l +S +NP 3700 1247 m 3700 1266 l +S +NP 3845 1247 m 3845 1266 l +S +NP 3990 1247 m 3990 1266 l +S +NP 4136 1228 m 4136 1266 l +S +NP 4281 1247 m 4281 1266 l +S +NP 4427 1247 m 4427 1266 l +S +medium setlinewidth +(0) dup 0 stringbbox 1266 exch sub exch 427 exch sub exch m show +(0.2) dup 0 stringbbox 2 idiv 2237 exch sub exch 427 exch sub exch m show +(0.4) dup 0 stringbbox 2 idiv 3208 exch sub exch 427 exch sub exch m show +(0.6) dup 0 stringbbox 2 idiv 4179 exch sub exch 427 exch sub exch m show +(0.8) dup 0 stringbbox 2 idiv 5151 exch sub exch 427 exch sub exch m show +(1) dup 0 stringbbox 2 idiv 6122 exch sub exch 427 exch sub exch m show +(2) dup 0 stringbbox 1211 exch sub exch 2 idiv 1227 exch sub exch m show +(4) dup 0 stringbbox 1211 exch sub exch 2 idiv 1954 exch sub exch m show +(6) dup 0 stringbbox 1211 exch sub exch 2 idiv 2681 exch sub exch m show +(8) dup 0 stringbbox 1211 exch sub exch 2 idiv 3409 exch sub exch m show +(10) dup 0 stringbbox 1211 exch sub exch 2 idiv 4136 exch sub exch m show +(t) dup 0 stringbbox 823 exch sub exch 2 idiv 2500 exch sub exch m show +boundarythick setlinewidth +% The following draws a box around the plot, +% if the variable drawborder is true +drawborder { +/bd boundarythick 2 idiv def +[] 0 setdash +NP bd bd m bd 6923 bd sub l +5000 bd sub 6923 bd sub l +5000 bd sub bd l +bd bd l S +} if % end of if to draw the border + +showpage +grestore +end +%%EOF
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new file mode 100644 --- /dev/null +++ b/talk/img/waydown.eps @@ -0,0 +1,713 @@ +%!PS-Adobe-3.0 EPSF-2.0 +%%Title: Maple plot +%%Creator: Maple +%%Pages: 1 +%%BoundingBox: 84 131 496 660 +%%DocumentNeededResources: font Helvetica +%%EndComments +20 dict begin +gsave +/drawborder false def +/m {moveto} def +/l {lineto} def +/C {setrgbcolor} def +/Y /setcmykcolor where { %%ifelse Use built-in operator + /setcmykcolor get + }{ %%ifelse Emulate setcmykcolor with setrgbcolor + { %%def + 1 sub 3 { %%repeat + 3 index add neg dup 0 lt { pop 0 } if 3 1 roll + } repeat setrgbcolor + } bind + } ifelse def +/G {setgray} def +/S {stroke} def +/NP {newpath} def +%%%%This draws a filled polygon and avoids bugs/features +%%%% of some postscript interpreters +%%%%GHOSTSCRIPT: has a bug in reversepath - removing +%%%%the call to reversepath is a sufficient work around +/P {gsave fill grestore reversepath stroke} def +%%%%This function is needed for drawing text. It takes two arguments: a +%%%%string and a rotation. It outputs width and height. +/stringbbox { + gsave NP 0 0 m % Set origin for easy calculations. + dup rotate exch % Set orientation, and put string on top. + false charpath flattenpath + 0 exch sub rotate % Undo rotation. + pathbbox 4 2 roll pop pop % Get dimens. of our string. + 1.1 mul cvi exch 1.1 mul cvi exch % Pad some. + grestore} def +/thin 3 def +/medium 7 def +/thick 16 def +/boundarythick 20 def % thichess of bounding box +%%IncludeResource: font Helvetica +72.000000 72.000000 translate +/defaultTitleFont {/Helvetica findfont 219 scalefont setfont} def +/defaultFont {/Helvetica findfont 154 scalefont setfont} def +0.0936 0.0936 scale +1 setlinejoin +1 setlinecap +0.0 setgray +/inch {72 mul} def +/fheight 0.35 inch neg def + 0 G +medium setlinewidth +[] 0 setdash defaultFont + 0.3 G +NP 2500 6130 m 2500 6230 l +S +NP 2500 6037 m 2500 6130 l +S +NP 2500 5939 m 2500 6037 l +S +NP 2500 5848 m 2500 5939 l +S +NP 2500 5745 m 2500 5848 l +S +NP 2500 5656 m 2500 5745 l +S +NP 2500 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2500 2581 l +S +NP 2500 2383 m 2500 2481 l +S +NP 2500 2285 m 2500 2383 l +S +NP 2500 2191 m 2500 2285 l +S +NP 2500 2099 m 2500 2191 l +S +NP 2500 2001 m 2500 2099 l +S +NP 2500 1901 m 2500 2001 l +S +NP 2500 1803 m 2500 1901 l +S +NP 2500 1715 m 2500 1803 l +S +NP 2500 1615 m 2500 1715 l +S + 0.11 G +NP 4437 1616 m 4500 1615 l +S +NP 4358 1623 m 4437 1616 l +S +NP 4322 1630 m 4358 1623 l +S +NP 4286 1642 m 4322 1630 l +S +NP 4251 1659 m 4286 1642 l +S +NP 4215 1681 m 4251 1659 l +S +NP 4135 1754 m 4215 1681 l +S +NP 4056 1862 m 4135 1754 l +S +NP 3982 1998 m 4056 1862 l +S +NP 3908 2169 m 3982 1998 l +S +NP 3825 2408 m 3908 2169 l +S +NP 3743 2706 m 3825 2408 l +S +NP 3706 2859 m 3743 2706 l +S +NP 3669 3022 m 3706 2859 l +S +NP 3631 3193 m 3669 3022 l +S +NP 3594 3368 m 3631 3193 l +S +NP 3556 3552 m 3594 3368 l +S +NP 3518 3733 m 3556 3552 l +S +NP 3479 3908 m 3518 3733 l +S +NP 3441 4073 m 3479 3908 l +S +NP 3362 4363 m 3441 4073 l +S +NP 3284 4570 m 3362 4363 l +S +NP 3246 4635 m 3284 4570 l +S +NP 3208 4676 m 3246 4635 l +S +NP 3189 4687 m 3208 4676 l +S +NP 3170 4692 m 3189 4687 l +S +NP 3160 4691 m 3170 4692 l +S +NP 3151 4686 m 3160 4691 l +S +NP 3141 4678 m 3151 4686 l +S +NP 3132 4667 m 3141 4678 l +S +NP 3061 4488 m 3132 4667 l +S +NP 2991 4176 m 3061 4488 l +S +NP 2950 3953 m 2991 4176 l +S +NP 2910 3708 m 2950 3953 l +S +NP 2870 3454 m 2910 3708 l +S +NP 2829 3201 m 2870 3454 l +S +NP 2790 2968 m 2829 3201 l +S +NP 2751 2749 m 2790 2968 l +S +NP 2712 2547 m 2751 2749 l +S +NP 2674 2357 m 2712 2547 l +S +NP 2601 2030 m 2674 2357 l +S +NP 2529 1726 m 2601 2030 l +S +NP 2451 1470 m 2529 1726 l +S +NP 2372 1335 m 2451 1470 l +S +NP 2353 1319 m 2372 1335 l +S +NP 2334 1308 m 2353 1319 l +S +NP 2316 1303 m 2334 1308 l +S +NP 2297 1303 m 2316 1303 l +S +NP 2278 1308 m 2297 1303 l +S +NP 2259 1316 m 2278 1308 l +S +NP 2240 1329 m 2259 1316 l +S +NP 2222 1345 m 2240 1329 l +S +NP 2142 1434 m 2222 1345 l +S +NP 2063 1525 m 2142 1434 l +S +NP 2027 1558 m 2063 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+S +NP 2910 3998 m 2950 3647 l +S +NP 2870 4359 m 2910 3998 l +S +NP 2829 4714 m 2870 4359 l +S +NP 2790 5034 m 2829 4714 l +S +NP 2751 5320 m 2790 5034 l +S +NP 2712 5568 m 2751 5320 l +S +NP 2674 5776 m 2712 5568 l +S +NP 2601 6059 m 2674 5776 l +S +NP 2529 6213 m 2601 6059 l +S +NP 2510 6228 m 2529 6213 l +S +NP 2490 6228 m 2510 6228 l +S +NP 2470 6214 m 2490 6228 l +S +NP 2451 6187 m 2470 6214 l +S +NP 2411 6099 m 2451 6187 l +S +NP 2372 5974 m 2411 6099 l +S +NP 2334 5818 m 2372 5974 l +S +NP 2297 5627 m 2334 5818 l +S +NP 2259 5398 m 2297 5627 l +S +NP 2222 5132 m 2259 5398 l +S +NP 2182 4817 m 2222 5132 l +S +NP 2142 4475 m 2182 4817 l +S +NP 2103 4122 m 2142 4475 l +S +NP 2063 3773 m 2103 4122 l +S +NP 2027 3473 m 2063 3773 l +S +NP 1991 3199 m 2027 3473 l +S +NP 1955 2956 m 1991 3199 l +S +NP 1919 2748 m 1955 2956 l +S +NP 1838 2400 m 1919 2748 l +S +NP 1756 2166 m 1838 2400 l +S +NP 1686 2004 m 1756 2166 l +S +NP 1616 1873 m 1686 2004 l +S +NP 1534 1758 m 1616 1873 l +S +NP 1451 1681 m 1534 1758 l +S +NP 1382 1643 m 1451 1681 l +S +NP 1312 1623 m 1382 1643 l +S +NP 1236 1616 m 1312 1623 l +S +NP 1159 1615 m 1236 1616 l +S +NP 1000 1615 m 1159 1615 l +S +NP 842 1615 m 1000 1615 l +S +NP 702 1615 m 842 1615 l +S +NP 542 1615 m 702 1615 l +S +NP 500 1615 m 542 1615 l +S + 0.3 G +NP 4358 1615 m 4500 1615 l +S +NP 4215 1615 m 4358 1615 l +S +NP 4056 1615 m 4215 1615 l +S +NP 3908 1615 m 4056 1615 l +S +NP 3743 1615 m 3908 1615 l +S +NP 3594 1615 m 3743 1615 l +S +NP 3441 1615 m 3594 1615 l +S +NP 3284 1615 m 3441 1615 l +S +NP 3208 1615 m 3284 1615 l +S +NP 3132 1614 m 3208 1615 l +S +NP 3096 1612 m 3132 1614 l +S +NP 3061 1605 m 3096 1612 l +S +NP 3026 1592 m 3061 1605 l +S +NP 2991 1572 m 3026 1592 l +S +NP 2910 1498 m 2991 1572 l +S +NP 2829 1401 m 2910 1498 l +S +NP 2790 1357 m 2829 1401 l +S +NP 2751 1324 m 2790 1357 l +S +NP 2732 1312 m 2751 1324 l +S +NP 2712 1305 m 2732 1312 l +S +NP 2693 1302 m 2712 1305 l +S +NP 2674 1305 m 2693 1302 l +S +NP 2601 1367 m 2674 1305 l +S +NP 2529 1521 m 2601 1367 l +S +NP 2490 1649 m 2529 1521 l +S +NP 2451 1802 m 2490 1649 l +S +NP 2411 1969 m 2451 1802 l +S +NP 2372 2143 m 2411 1969 l +S +NP 2334 2315 m 2372 2143 l +S +NP 2297 2496 m 2334 2315 l +S +NP 2259 2688 m 2297 2496 l +S +NP 2222 2894 m 2259 2688 l +S +NP 2182 3127 m 2222 2894 l +S +NP 2142 3371 m 2182 3127 l +S +NP 2103 3621 m 2142 3371 l +S +NP 2063 3865 m 2103 3621 l +S +NP 2027 4074 m 2063 3865 l +S +NP 1991 4261 m 2027 4074 l +S +NP 1955 4421 m 1991 4261 l +S +NP 1919 4549 m 1955 4421 l +S +NP 1878 4649 m 1919 4549 l +S +NP 1838 4691 m 1878 4649 l +S +NP 1817 4690 m 1838 4691 l +S +NP 1797 4680 m 1817 4690 l +S +NP 1777 4663 m 1797 4680 l +S +NP 1756 4639 m 1777 4663 l +S +NP 1686 4504 m 1756 4639 l +S +NP 1616 4294 m 1686 4504 l +S +NP 1575 4140 m 1616 4294 l +S +NP 1534 3969 m 1575 4140 l +S +NP 1492 3783 m 1534 3969 l +S +NP 1451 3589 m 1492 3783 l +S +NP 1382 3259 m 1451 3589 l +S +NP 1312 2941 m 1382 3259 l +S +NP 1236 2626 m 1312 2941 l +S +NP 1159 2362 m 1236 2626 l +S +NP 1080 2141 m 1159 2362 l +S +NP 1000 1965 m 1080 2141 l +S +NP 921 1830 m 1000 1965 l +S +NP 842 1733 m 921 1830 l +S +NP 772 1675 m 842 1733 l +S +NP 702 1640 m 772 1675 l +S +NP 662 1628 m 702 1640 l +S +NP 622 1620 m 662 1628 l +S +NP 582 1617 m 622 1620 l +S +NP 542 1615 m 582 1617 l +S +NP 500 1615 m 542 1615 l +S + 0 G +medium setlinewidth +NP 500 692 m 500 6230 l +S +thin setlinewidth +NP 449 692 m 500 692 l +S +NP 474 876 m 500 876 l +S +NP 474 1061 m 500 1061 l +S +NP 474 1246 m 500 1246 l +S +NP 474 1430 m 500 1430 l +S +NP 449 1615 m 500 1615 l +S +NP 474 1799 m 500 1799 l +S +NP 474 1984 m 500 1984 l +S +NP 474 2169 m 500 2169 l +S +NP 474 2353 m 500 2353 l +S +NP 449 2538 m 500 2538 l +S +NP 474 2723 m 500 2723 l +S +NP 474 2907 m 500 2907 l +S +NP 474 3092 m 500 3092 l +S +NP 474 3276 m 500 3276 l +S +NP 449 3461 m 500 3461 l +S +NP 474 3646 m 500 3646 l +S +NP 474 3830 m 500 3830 l +S +NP 474 4015 m 500 4015 l +S +NP 474 4199 m 500 4199 l +S +NP 449 4384 m 500 4384 l +S +NP 474 4569 m 500 4569 l +S +NP 474 4753 m 500 4753 l +S +NP 474 4938 m 500 4938 l +S +NP 474 5123 m 500 5123 l +S +NP 449 5307 m 500 5307 l +S +NP 474 5492 m 500 5492 l +S +NP 474 5676 m 500 5676 l +S +NP 474 5861 m 500 5861 l +S +NP 474 6046 m 500 6046 l +S +NP 449 6230 m 500 6230 l +S +medium setlinewidth +NP 500 1615 m 4500 1615 l +S +thin setlinewidth +NP 500 1577 m 500 1615 l +S +NP 633 1596 m 633 1615 l +S +NP 766 1596 m 766 1615 l +S +NP 900 1596 m 900 1615 l +S +NP 1033 1596 m 1033 1615 l +S +NP 1166 1577 m 1166 1615 l +S +NP 1300 1596 m 1300 1615 l +S +NP 1433 1596 m 1433 1615 l +S +NP 1566 1596 m 1566 1615 l +S +NP 1700 1596 m 1700 1615 l +S +NP 1833 1577 m 1833 1615 l +S +NP 1966 1596 m 1966 1615 l +S +NP 2100 1596 m 2100 1615 l +S +NP 2233 1596 m 2233 1615 l +S +NP 2366 1596 m 2366 1615 l +S +NP 2500 1577 m 2500 1615 l +S +NP 2633 1596 m 2633 1615 l +S +NP 2766 1596 m 2766 1615 l +S +NP 2900 1596 m 2900 1615 l +S +NP 3033 1596 m 3033 1615 l +S +NP 3166 1577 m 3166 1615 l +S +NP 3300 1596 m 3300 1615 l +S +NP 3433 1596 m 3433 1615 l +S +NP 3566 1596 m 3566 1615 l +S +NP 3700 1596 m 3700 1615 l +S +NP 3833 1577 m 3833 1615 l +S +NP 3966 1596 m 3966 1615 l +S +NP 4100 1596 m 4100 1615 l +S +NP 4233 1596 m 4233 1615 l +S +NP 4366 1596 m 4366 1615 l +S +NP 4500 1577 m 4500 1615 l +S +medium setlinewidth +(\2610.2) dup 0 stringbbox 2 idiv 692 exch sub exch 427 exch sub exch m show +(0) dup 0 stringbbox 1615 exch sub exch 427 exch sub exch m show +(0.2) dup 0 stringbbox 2 idiv 2538 exch sub exch 427 exch sub exch m show +(0.4) dup 0 stringbbox 2 idiv 3461 exch sub exch 427 exch sub exch m show +(0.6) dup 0 stringbbox 2 idiv 4384 exch sub exch 427 exch sub exch m show +(0.8) dup 0 stringbbox 2 idiv 5307 exch sub exch 427 exch sub exch m show +(1) dup 0 stringbbox 2 idiv 6230 exch sub exch 427 exch sub exch m show +(3) dup 0 stringbbox 1560 exch sub exch 2 idiv 1166 exch sub exch m show +(4) dup 0 stringbbox 1560 exch sub exch 2 idiv 1833 exch sub exch m show +(5) dup 0 stringbbox 1560 exch sub exch 2 idiv 2500 exch sub exch m show +(6) dup 0 stringbbox 1560 exch sub exch 2 idiv 3166 exch sub exch m show +(7) dup 0 stringbbox 1560 exch sub exch 2 idiv 3833 exch sub exch m show +(8) dup 0 stringbbox 1560 exch sub exch 2 idiv 4500 exch sub exch m show +(t) dup 0 stringbbox 1172 exch sub exch 2 idiv 2500 exch sub exch m show +boundarythick setlinewidth +% The following draws a box around the plot, +% if the variable drawborder is true +drawborder { +/bd boundarythick 2 idiv def +[] 0 setdash +NP bd bd m bd 6923 bd sub l +5000 bd sub 6923 bd sub l +5000 bd sub bd l +bd bd l S +} if % end of if to draw the border + +showpage +grestore +end +%%EOF
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new file mode 100644 --- /dev/null +++ b/talk/talk.tex @@ -0,0 +1,457 @@ +%%% BEGIN BEAMER PREAMBLE %%% +\documentclass{beamer} +\usepackage{bm, fourier, anyfontsize, xcolor} + +\mode<presentation> +{ + \usetheme{boxes} + \usecolortheme{crane} +} +\beamertemplatenavigationsymbolsempty + +\AtBeginSection[] +{ + \begin{frame}<beamer> + \frametitle{Outline} + \tableofcontents[currentsection] + \end{frame} +} + +\newcommand{\seti}{\setcounter{saveenumi}{\value{enumi}}} +\newcommand{\conti}{\setcounter{enumi}{\value{saveenumi}}} + +\resetcounteronoverlays{saveenumi} + +\usepackage{times} +%%% END BEAMER PREAMBLE %%% + + +%%% BEGIN METADATA %%% + +\author{Jordi G. H. $\langle$jordigh@octave.org$\rangle$ } +\institute{Also @JordiGH@mathstodon.xyz on the Fediverse/Mastodon} + +\title{X-Splines} +\subtitle{Flexible and user-friendly} + +%%% END METADATA%%% + +\begin{document} + +\begin{frame} + \titlepage +\end{frame} + +\begin{frame} + \frametitle{Outline} + \tableofcontents +\end{frame} + +\section{Splines in general} + +\begin{frame}{What is a spline?} + \pause + \begin{center} + \pgfimage[height=2.5in]{img/history.pdf} + \end{center} + Historically: a mechanical device for drawing curves +\end{frame} + +\begin{frame}{What is a spline?} + You probably know them from computer graphics or CAD. + \begin{center} + \pgfimage[height=2.5in]{img/svg-spline.pdf} + \end{center} +\end{frame} + +\begin{frame}{What is a spline?} + You probably know them from computer graphics or CAD. + \begin{center} + \pgfimage[height=2.5in]{img/computer-graphics-spline.pdf} + \end{center} +\end{frame} + +\begin{frame}{Guiding example: b-splines} + These splines \emph{approximate}. + \begin{center} + \pgfimage[height=2.5in]{img/bspline.pdf} + \end{center} +\end{frame} + +\begin{frame}{Guiding example: Catmull-Rom splines} + These splines \emph{interpolate}. + \begin{center} + \pgfimage[height=2.5in]{img/catmullrom.pdf} + \end{center} +\end{frame} + +\begin{frame}{Another famous example: Bézier curves} + Closely related to B-splines + \begin{center} + \pgfimage[height=2.5in]{img/bezier.pdf} + \end{center} +\end{frame} + + +\begin{frame}{What makes a spline a spline?} + General formula: + + \pause + For each \emph{node} in 2d (or 3d) space there is a + corresponding \emph{blending function}: + \[ + \begin{matrix} + (x_0, y_0) = P_0 & \longleftrightarrow & F_0(t) \\ + (x_1, y_1) = P_1 & \longleftrightarrow & F_1(t) \\ + & \vdots & \\ + (x_n, y_n) = P_n & \longleftrightarrow & F_n(t) + \end{matrix} + \] + \pause + and we form a \emph{linear combination} + \begin{align*} + C(t) &= \sum_{k=0}^{n} F_k(t) P_k \\ + &= F_0(t)P_0 + F_1(t)P_1 + \cdots + F_n(t)P_n. + \end{align*} + \pause + The parameter \(t\) runs continuously from \(0\) to \(n\). +\end{frame} + +\begin{frame}{Convexity (optional)} + \pause + The blending functions \(F_k(t)\) must satisfy + \pause + \begin{itemize} + \item \(1 = \sum_{k=0}^n F_k(t)\) \pause (technical jargon: + \emph{partition of unity}) + \pause + \item \(0 \leq F_k(t) \) + \end{itemize} + \pause + This implies every point on the spline curve + \[ + C(t) = \sum_{k=0}^{n} F_k(t) P_k + \] + \pause + is a \emph{convex combination} of the spline nodes. + \begin{center} + \pgfimage[height=1.2in]{img/convex-combination.pdf} + \end{center} +\end{frame} + +\begin{frame}{Convexity} + Remember? + \begin{center} + \pgfimage[height=2.5in]{img/svg-spline.pdf} + \end{center} +\end{frame} + +\begin{frame}{Smooth and steady} + Splines must be smooth (not spiky) and steady (not too wobbly) + + \pause + Therefore, blending functions \(F_k(t)\) must look like this: + \begin{center} + \pgfimage[height=2.0in, width=3.0in]{img/blending-functions.pdf} + \end{center} + \pause + \begin{itemize} + \item Bump-shaped (smoothness) + \pause + \item Always go in opposite direction when they cross (steadiness) + \end{itemize} +\end{frame} + +\begin{frame}{Local influence} + Each blending function \(F_k(t)\) must not contribute to nodes + \(P_k\) that are too far away. + + \pause + Therefore, the blending functions are zero outside of a small zone + (in technical jargon: \emph{compact support}). + \begin{center} + \pgfimage[height=2.0in, width=3.0in]{img/blending-functions.pdf} + \end{center} +\end{frame} + +\section{X-splines} + +\begin{frame}{Additional requirements} + To summarize, we require at first: + \begin{itemize} + \pause + \item convexity + \pause + \item smoothness + \pause + \item steadiness + \pause + \item locality + \end{itemize} + \pause + We want in addition + \begin{itemize} + \pause + \item flexibility of shape + \pause + \item intuitive shape parameter + \pause + \item smooth transition between approximation and interpolation + \end{itemize} +\end{frame} + +\begin{frame}{The Plan} + \pause + Will begin with polynomial blending functions \(F_k(t)\), e.g. one such + function for Bézier curves + \[ + f_k(t) = 3t^2(1-t) + \] + \pause + To get convexity, (partition of unity) will normalise each blending + function, i.e. + \[ + F_t(t) := \frac{f_k(t)}{\sum_{k=0}^n f_k(t)} + \] + giving a \emph{rational function} (a fraction of polynomials). +\end{frame} + +\begin{frame}{The Plan} + The functions will be defined piecewise on equally-spaced intervals + of the parameter space, i.e. + \[ + f_k(t) := + \begin{cases} + f\left(\frac{t-k}{2}\right) &\text{if } t \in [k-2, k] \\ + f\left(\frac{k-t}{2}\right) &\text{if } t \in [k, k+2] \\ + 0 &\text{otherwise} + \end{cases} + \] + \pause + \begin{center} + \pgfimage[width=3.5in, height=1.8in]{img/piecewise.pdf} + \end{center} +\end{frame} + +\begin{frame}{The Plan} + Recall: blending functions + \begin{center} + \pgfimage[height=2.0in, width=3.0in]{img/blending-functions.pdf} + \end{center} + \pause + Neighbours of $F_k(t)$ cross at $k$, hence \emph{cross-splines} or x-splines. +\end{frame} + +\begin{frame}{The Plan} + We will impose continuity and smoothness (derivatives) conditions at + the parameter knots. + \pause + \begin{itemize} + \item Values of each segment must be equal + \pause + \item First derivatives (tangent lines) must be equal + \pause + \item Second derivatives (curvatures) must be equal + \end{itemize} +\end{frame} + +\begin{frame}{The basic ingredient} + Let's build the basic x-spline function \(f(u)\). + + \pause + For simplicity, let's work on the interval \([0,1]\) and require + \begin{align*} + f(0) &= 0 & f'(0) &= 0 & f''(0) &= 0\\ + f(1) &= 1 & f'(1) &= 0 & f''(1) &= -2p + \end{align*} + \pause + Six equations, so degree 5 polynomial. Solve to get + \[ + f(u) = (6-p)u^5 + (2p-15)u^4 + (10-p)u^3 + \] +\end{frame} + +\begin{frame}{The basic ingredient} + As free parameter \(p\) varies from \(0\) to \(10\), we have some + flexibility in shape: + \begin{center} + \pgfimage[height=2.5in, width=3.5in]{img/fpu.pdf} + \end{center} + \[ + f(u) = (6-p)u^5 + (2p-15)u^4 + (10-p)u^3 + \] +\end{frame} + +\begin{frame}{The basic x-spline} + Summarising, x-spline formula for \(F_k(t)\) is: + \[ + F_K(t) = \frac{f_k(t)}{\sum_{k=0}^n f_k(t)} + \] + \pause + where + \[ + f_k(t) := + \begin{cases} + f\left(\frac{t-k}{2}, p_k^{\mathrm{Left}}\right) &\text{if } t \in [k-2, k] \\ + f\left(\frac{k-t}{2}, p_k^{\mathrm{Right}}\right) &\text{if } t \in [k, k+2] \\ + 0 &\text{otherwise} + \end{cases} + \] + \pause + and + \[ + f(u, p) = (6-p)u^5 + (2p-15)u^4 + (10-p)u^3. + \] + \pause + Observe that each \(F_k(t)\) has its own \(p_k^{\mathrm{Left}}\) and + \(p_k^{\mathrm{Right}}\)parameters too! +\end{frame} + +\begin{frame}{The basic x-spline} + Setting all \(p = 8\), we get a very good approximation of B-splines. + \begin{center} + \pgfimage[width=3.5in]{img/fakebspline.pdf} + \end{center} +\end{frame} + +\section{X-tended X-splines} + +\begin{frame}{Pushing the blending function upwards} + The parameters \(p_k^{\mathrm{Left}}\) and \(p_k^{\mathrm{Right}}\) + give us some freedom to modify the spline more. + + \pause + Will use them to modify the blending functions \(F_k(t)\) like so. + + \begin{center} + \pgfimage[width=2in, height=2.5in]{img/before.eps} + \pgfimage[width=2in, height=2.5in]{img/after.eps} + \end{center} +\end{frame} + +\begin{frame}{Exercise for the audience} + Calculations and notation gets a bit messy, but the idea is + \begin{enumerate} + \pause + \item Assign a new parameter \(s_k \in [0,1]\) to each blending function + \(F_k(t)\). + \pause + \item When all \(s_k = 1\), we have all \(p = 8\) and a basic + approximating x-spline. + \pause + \item Define new zeroing points for the neighbours of \(F_k(t)\): + \begin{align*} + T_{k-1}^{\mathrm{Right}} &:= k + s_k \\ + T_{k+1}^{\mathrm{Left}} &:= k - s_k + \end{align*} + \pause + \item + To preserve curvature (equality of second derivatives at \(k-1\) + and \(k+1\)), this requires setting special values of \(p\) + \begin{align*} + p_{k-1}^{\mathrm{Right}} &= 2(k - T_{k-1}^{\mathrm{Right}})^2 \\ + p_{k+1}^{\mathrm{Left}} &= 2(k - T_{k+1}^{\mathrm{Left}})^2 + \end{align*} + \end{enumerate} +\end{frame} + +\begin{frame}{Can now interpolate sharply} + End result: as \(s_k\) goes from \(1\) to \(0\) at a particular + node, the spline gets closer to the node until it is interpolated + sharply. + \begin{center} + \pgfimage[width=2in, height=2.5in]{img/strumming.pdf} + \end{center} +\end{frame} + +\begin{frame}{Fonts!} + Good for drawing some basic fonts. + \begin{center} + \pgfimage[width=2in, height=2.5in]{img/x-spline.pdf} + \end{center} +\end{frame} + +\section{General X-splines} + +\begin{frame}{Smooth interpolation} + To have smooth interpolation, we have to drop the convexity requirement. + + \pause + Recall, partition of unity: + \begin{align*} + 1 &= \sum_{k=0}^n F_k(t) \\ + 0 &\leq F_k(t) + \end{align*} + \pause + We'll keep the sum equal to 1, but we'll let the blending functions go negative now. + + \pause We've pushed the tails of the blending functions down to zero + (via \(s_k\) parameters). We'll push those tails further down into + the negative. +\end{frame} + +\begin{frame}{Pushing way down} + Blending functions \(F_k(t)\) will now look like this: + \begin{center} + \pgfimage[width=2in, height=2.5in]{img/after.eps} + \pgfimage[width=2in, height=2.5in]{img/waydown.pdf} + \end{center} +\end{frame} + +\begin{frame}{Polynomial requirements again} + This time use two polynomials, \(g(u)\) for the positive part and + \(h(h)\) for the negative part. + + \pause + If we pick the domain of \(h(u)\) to be from \(-1\) to \(0\) + and that of \(g(u)\) to go from \(0\) to \(1\)... + + \pause + 12 conditions on continuity, tangent line, and curvature: + \begin{align*} + h(-1) &= 0 & h'(-1) &= q & h''(-1) &= 0 \\ + h(0) &= 0 & h'(0) &= q & h''(0) &= 4q \\ + g(0) &= 0 & g'(0) &= q & g''(0) &= 4q \\ + g(1) &= 1 & g'(1) &= 0 & g''(1) &= -2p + \end{align*} + \pause + Degree 5 solutions: + \begin{align*} + g(u) &= qu + 2qu^2 + (10 -12q - p)u^3 \\ + &+ (2p + 14q - 15)u^4 + (6 - 5q -p)u^5 \\ + h(u) &= qu + 2qu^2 - 2qu^4 + qu^5 + \end{align*} +\end{frame} + +\begin{frame}{Recovering Catmull-Rom} + New \(q\) parameter to play with, corresponding to the \(s_k\) parameters: + \[ + q_{k-1}^{\mathrm{Right}} := -\frac{s_k}{2} =: q_{k+1}^{\mathrm{Left}} + \] + + \pause + + When \(q = 1/2\) everywhere (i.e. \(s=-1\) everywhere), we get an + almost exact copy of Catmull-Rom. + \begin{center} + \pgfimage[height=2in]{img/x-spline-q-full.pdf} + \end{center} +\end{frame} + +\begin{frame}{Letting \(s\) vary slowly} + Smoothly goes from sharp interpolation to smooth interpolation: + \begin{center} + \pgfimage[height=2.5in]{img/morestrumming.eps} + \end{center} +\end{frame} + +\begin{frame}{Thanks!} + \begin{center} + jordigh@octave.org or @JordiGH@mathstodon.xyz on Mastodon + + \pause + Remember xfig? + \end{center} +\end{frame} + +\end{document} \ No newline at end of file