changeset 0:11ac175098d1

talk: init
author Jordi Gutiérrez Hermoso <jordigh@octave.org>
date Thu, 23 Aug 2018 18:14:16 -0400
parents
children a8a6e95b85b4
files talk/img/Bezierkurve.png talk/img/CatmullRom.png talk/img/FirstBlends.eps talk/img/Messy.png talk/img/after.eps talk/img/before.eps talk/img/bezier.pdf talk/img/blending-functions.pdf talk/img/blendingfunctions.pdf talk/img/bspline.pdf talk/img/catmullrom.pdf talk/img/computer-graphics-spline.jpg talk/img/computer-graphics-spline.pdf talk/img/convex-combination.pdf talk/img/convex-combination.svg talk/img/fakebspline.eps talk/img/fakebspline.pdf talk/img/fakebsplines.eps talk/img/fakebsplines.pdf talk/img/fakecatmullrom.eps talk/img/fakecatmullrom.pdf talk/img/fpu.pdf talk/img/fpu.png talk/img/fpu.svg talk/img/history.pdf talk/img/morestrumming.eps talk/img/piecewise.pdf talk/img/piecewise.png talk/img/strumming.eps talk/img/strumming.pdf talk/img/svg-spline.pdf talk/img/waydown.eps talk/img/waydown.pdf talk/img/x-spline-q-full.pdf talk/img/x-spline.eps talk/img/x-spline.pdf talk/talk.tex
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+    0 G
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+NP 449 6230 m 500 6230 l
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+medium setlinewidth
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+thin setlinewidth
+NP 588 673 m 588 692 l
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+NP 766 673 m 766 692 l
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+NP 944 654 m 944 692 l
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+NP 1122 673 m 1122 692 l
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+NP 1299 673 m 1299 692 l
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+NP 1655 673 m 1655 692 l
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+NP 1833 654 m 1833 692 l
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+NP 2011 673 m 2011 692 l
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+NP 2188 673 m 2188 692 l
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+NP 2366 673 m 2366 692 l
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+NP 2544 673 m 2544 692 l
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+NP 2722 654 m 2722 692 l
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+NP 2900 673 m 2900 692 l
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+NP 3077 673 m 3077 692 l
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+NP 3255 673 m 3255 692 l
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+NP 3433 673 m 3433 692 l
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+NP 3611 654 m 3611 692 l
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+NP 3788 673 m 3788 692 l
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+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
new file mode 100644
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+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
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+(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
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+(2) dup 0 stringbbox 744 exch sub exch 2 idiv 1227 exch sub exch m show
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+showpage
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new file mode 100644
--- /dev/null
+++ b/talk/img/fakecatmullrom.eps
@@ -0,0 +1,1408 @@
+%!PS-Adobe-3.0 EPSF-2.0
+%%Title: Maple plot
+%%Creator: Maple 
+%%Pages:  1
+%%BoundingBox: 93 136 493 655 
+%%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.
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+NP 1293 1245 m 1315 1222 l
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+NP 1272 1259 m 1293 1245 l
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+NP 1250 1265 m 1272 1259 l
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+NP 1208 1266 m 1250 1265 l
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+NP 1081 1266 m 1165 1266 l
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+NP 508 1266 m 510 1266 l
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+  0.3 G
+NP 4494 1266 m 4497 1266 l
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+NP 4491 1266 m 4494 1266 l
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+NP 4489 1266 m 4491 1266 l
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+NP 4483 1266 m 4489 1266 l
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+NP 4456 1266 m 4467 1266 l
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+NP 4434 1266 m 4456 1266 l
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+NP 4413 1266 m 4434 1266 l
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+NP 4372 1266 m 4413 1266 l
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+NP 4332 1266 m 4372 1266 l
+S
+NP 4247 1266 m 4332 1266 l
+S
+NP 4168 1266 m 4247 1266 l
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+NP 4079 1266 m 4168 1266 l
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+NP 4002 1266 m 4079 1266 l
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+NP 3918 1266 m 4002 1266 l
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+NP 3750 1266 m 3833 1266 l
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+NP 3498 1266 m 3582 1266 l
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+NP 3417 1266 m 3498 1266 l
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+NP 3331 1266 m 3417 1266 l
+S
+NP 3253 1266 m 3331 1266 l
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+NP 3167 1266 m 3253 1266 l
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+NP 3086 1266 m 3167 1266 l
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+NP 2996 1266 m 3086 1266 l
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+NP 2915 1266 m 2996 1266 l
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+NP 2831 1266 m 2915 1266 l
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+NP 2745 1266 m 2831 1266 l
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+NP 2662 1266 m 2745 1266 l
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+NP 2586 1266 m 2662 1266 l
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+NP 2498 1266 m 2586 1266 l
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+NP 2413 1266 m 2498 1266 l
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+NP 2334 1266 m 2413 1266 l
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+NP 2312 1266 m 2334 1266 l
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+NP 2291 1264 m 2312 1266 l
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+NP 2270 1257 m 2291 1264 l
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+NP 2248 1241 m 2270 1257 l
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+NP 2207 1185 m 2248 1241 l
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+NP 2166 1103 m 2207 1185 l
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+NP 1039 816 m 1050 807 l
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+NP 1019 844 m 1039 816 l
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+NP 998 884 m 1019 844 l
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+NP 959 994 m 998 884 l
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+NP 919 1131 m 959 994 l
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+NP 898 1203 m 919 1131 l
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+NP 876 1254 m 898 1203 l
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+NP 855 1266 m 866 1265 l
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+    0 G
+medium setlinewidth
+NP 500 692 m 500 6230 l
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+thin setlinewidth
+NP 474 877 m 500 877 l
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+NP 474 1072 m 500 1072 l
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+NP 474 5928 m 500 5928 l
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+NP 449 6122 m 500 6122 l
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+medium setlinewidth
+NP 500 1266 m 4500 1266 l
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+thin setlinewidth
+NP 645 1247 m 645 1266 l
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+NP 790 1247 m 790 1266 l
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+NP 936 1247 m 936 1266 l
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+NP 1081 1247 m 1081 1266 l
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+NP 1227 1228 m 1227 1266 l
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+NP 1372 1247 m 1372 1266 l
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+NP 1518 1247 m 1518 1266 l
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+NP 1663 1247 m 1663 1266 l
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+NP 1809 1247 m 1809 1266 l
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+NP 1954 1228 m 1954 1266 l
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+NP 2100 1247 m 2100 1266 l
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+NP 2245 1247 m 2245 1266 l
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+NP 2390 1247 m 2390 1266 l
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+NP 2536 1247 m 2536 1266 l
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+NP 2681 1228 m 2681 1266 l
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+NP 2827 1247 m 2827 1266 l
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+NP 2972 1247 m 2972 1266 l
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+NP 3118 1247 m 3118 1266 l
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+NP 3263 1247 m 3263 1266 l
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+NP 3554 1247 m 3554 1266 l
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+NP 3700 1247 m 3700 1266 l
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+NP 3845 1247 m 3845 1266 l
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+NP 3990 1247 m 3990 1266 l
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+NP 4136 1228 m 4136 1266 l
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+NP 4427 1247 m 4427 1266 l
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+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
new file mode 100644
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new file mode 100644
--- /dev/null
+++ b/talk/img/fpu.svg
@@ -0,0 +1,93 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<svg xmlns="http://www.w3.org/2000/svg"
+     xmlns:xlink="http://www.w3.org/1999/xlink"
+     width="576px" height="432px" viewBox="0 0 576 432">
+<title>gl2ps_renderer figure</title>
+<desc>
+Creator: GL2PS 1.3.9, (C) 1999-2015 C. Geuzaine
+For: Octave
+CreationDate: Thu Aug 23 12:47:04 2018
+</desc>
+<defs>
+</defs>
+<polygon fill="#ffffff" points="0,0 576,0 576,432 0,432"/>
+<g shape-rendering="crispEdges">
+<polygon fill="#ffffff" points="0,432 576,432 576,0 0,0"/>
+<clipPath id="cp00576432">
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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}
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