diff --git a/Code/hull.cpp b/Code/hull.cpp
index 25249b6ccbbc01eb20c197c883d7a27f7d2b8811..00c06bab9e628e69ee25b88ce38cd8d2361d6684 100644
--- a/Code/hull.cpp
+++ b/Code/hull.cpp
@@ -40,14 +40,18 @@ struct vec {
//}}}
+int sign(float x) {
+ if(x > 0) return 1 ;
+ if(x < 0) return -1 ;
+ return 0 ;
+}
+
int orient(const vec& v0, const vec& v1) {
//determinant
float d = v0.x*v1.y - v0.y*v1.x ;
//encode as sign
- if(d > 0) return 1 ;
- if(d < 0) return -1 ;
- return 0 ;
+ return sign(d) ;
}
struct vec_compare {
@@ -90,13 +94,13 @@ void compute_hull(std::vector<vec>& points, std::vector<vec>& hull) {
for(size_t i = 2; i < points.size(); ++i) {
//check whether the last segment creates a wrong turn
const vec& p = points[i] ;
- vec v1 = p - *(hull.end() - 1) ;
+ vec v1 = p - *(hull.end() - 2) ;
vec v2 = *(hull.end() - 1) - *(hull.end() - 2) ;
while(hull.size() > 1 && orient(v1, v2) < 0) {
//in case of wrong turn, rease the last point of the hull and iterate
hull.erase(hull.end() - 1) ;
- v1 = p - *(hull.end() - 1) ;
- v2 = p - *(hull.end() - 2) ;
+ v1 = p - *(hull.end() - 2) ;
+ v2 = *(hull.end() - 1) - *(hull.end() - 2) ;
}
hull.push_back(p) ;
}
diff --git a/Presentation/Figures/hull.svg b/Presentation/Figures/hull.svg
index 7a44f379651decc26660176777ea06ba1ac7524b..e177c819ca41e0b0cb03affa18505a198cda56fa 100644
--- a/Presentation/Figures/hull.svg
+++ b/Presentation/Figures/hull.svg
@@ -27,6 +27,92 @@
</metadata>
<defs
id="defs1894">
+ <marker
+ inkscape:isstock="true"
+ style="overflow:visible;"
+ id="marker5177"
+ refX="0.0"
+ refY="0.0"
+ orient="auto"
+ inkscape:stockid="Arrow2Send">
+ <path
+ transform="scale(0.3) rotate(180) translate(-2.3,0)"
+ d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ id="path5175" />
+ </marker>
+ <marker
+ inkscape:isstock="true"
+ style="overflow:visible;"
+ id="marker4465"
+ refX="0.0"
+ refY="0.0"
+ orient="auto"
+ inkscape:stockid="Arrow2Mend">
+ <path
+ transform="scale(0.6) rotate(180) translate(0,0)"
+ d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ id="path4463" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Mend"
+ orient="auto"
+ refY="0.0"
+ refX="0.0"
+ id="marker4219"
+ style="overflow:visible;"
+ inkscape:isstock="true"
+ inkscape:collect="always">
+ <path
+ id="path4217"
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
+ transform="scale(0.6) rotate(180) translate(0,0)" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Mstart"
+ orient="auto"
+ refY="0.0"
+ refX="0.0"
+ id="Arrow2Mstart"
+ style="overflow:visible"
+ inkscape:isstock="true">
+ <path
+ id="path841"
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
+ transform="scale(0.6) translate(0,0)" />
+ </marker>
+ <marker
+ inkscape:isstock="true"
+ style="overflow:visible;"
+ id="marker3855"
+ refX="0.0"
+ refY="0.0"
+ orient="auto"
+ inkscape:stockid="Arrow2Mend">
+ <path
+ transform="scale(0.6) rotate(180) translate(0,0)"
+ d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ id="path3853" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Mend"
+ orient="auto"
+ refY="0.0"
+ refX="0.0"
+ id="marker3809"
+ style="overflow:visible;"
+ inkscape:isstock="true"
+ inkscape:collect="always">
+ <path
+ id="path844"
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
+ transform="scale(0.6) rotate(180) translate(0,0)" />
+ </marker>
<marker
inkscape:stockid="Arrow2Send"
orient="auto"
@@ -34,7 +120,8 @@
refX="0.0"
id="Arrow2Send"
style="overflow:visible;"
- inkscape:isstock="true">
+ inkscape:isstock="true"
+ inkscape:collect="always">
<path
id="path2197"
style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#f57900;stroke-opacity:1;fill:#f57900;fill-opacity:1"
@@ -55,6 +142,21 @@
d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
transform="scale(0.6) rotate(180) translate(0,0)" />
</marker>
+ <marker
+ inkscape:stockid="Arrow2Mstart"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow2Mstart-2"
+ style="overflow:visible"
+ inkscape:isstock="true">
+ <path
+ inkscape:connector-curvature="0"
+ id="path841-8"
+ style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ transform="scale(0.6)" />
+ </marker>
</defs>
<sodipodi:namedview
pagecolor="#ffffff"
@@ -70,10 +172,11 @@
id="namedview1892"
showgrid="false"
inkscape:zoom="1.3796869"
- inkscape:cx="454.51793"
- inkscape:cy="483.23057"
- inkscape:current-layer="layer6"
- inkscape:snap-object-midpoints="true" />
+ inkscape:cx="224.13977"
+ inkscape:cy="621.88734"
+ inkscape:current-layer="layer7"
+ inkscape:snap-object-midpoints="true"
+ inkscape:snap-global="true" />
<g
inkscape:groupmode="layer"
id="layer4"
@@ -780,7 +883,7 @@
inkscape:groupmode="layer"
id="layer6"
inkscape:label="partial_hull"
- style="display:inline">
+ style="display:none">
<circle
cx="56"
cy="481"
@@ -824,67 +927,68 @@
inkscape:connector-curvature="0" />
</g>
<g
- inkscape:groupmode="layer"
- id="layer7"
- inkscape:label="adding_point_1"
- style="display:inline">
+ style="display:none"
+ inkscape:label="adding_point_1 copy"
+ id="g4215"
+ inkscape:groupmode="layer">
<path
- inkscape:connector-curvature="0"
- id="path2434"
+ style="opacity:1;fill:none;fill-opacity:1;stroke:#000000;stroke-width:2;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
d="M 249,300 256,213 204,180 75,409 56,481"
- style="opacity:1;fill:none;fill-opacity:1;stroke:#000000;stroke-width:2;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1" />
+ id="path4197"
+ inkscape:connector-curvature="0" />
<circle
- id="circle2437"
- style="display:inline;fill:none;stroke:#000000;stroke-width:2"
- r="12"
+ cx="56"
cy="481"
- cx="56" />
- <circle
- id="circle2439"
- style="display:inline;fill:none;stroke:#000000;stroke-width:2"
r="12"
- cy="409"
- cx="75" />
- <circle
- id="circle2441"
style="display:inline;fill:none;stroke:#000000;stroke-width:2"
- r="12"
- cy="180"
- cx="204" />
+ id="circle4199" />
<circle
- id="circle2443"
- style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ cx="75"
+ cy="409"
r="12"
- cy="213"
- cx="256" />
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ id="circle4201" />
<circle
- id="circle2445"
+ cx="204"
+ cy="180"
+ r="12"
style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ id="circle4203" />
+ <circle
+ cx="256"
+ cy="213"
r="12"
- cy="300"
- cx="249" />
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ id="circle4205" />
<circle
- cx="412"
- cy="193"
+ cx="249"
+ cy="300"
r="12"
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ id="circle4207" />
+ <circle
+ id="circle4209"
style="display:inline;fill:#f57900;stroke:#f57900;stroke-width:2"
- id="circle1838-62-9" />
+ r="12"
+ cy="193"
+ cx="412" />
<path
- style="display:inline;opacity:1;fill:#f57900;fill-opacity:1;stroke:#f57900;stroke-width:4;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
+ inkscape:connector-curvature="0"
+ id="path4211"
d="M 249,300 412,193"
- id="path2160"
- inkscape:connector-curvature="0" />
+ style="display:inline;opacity:1;fill:#f57900;fill-opacity:1;stroke:#f57900;stroke-width:4;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1" />
<path
- style="display:inline;opacity:1;fill:none;fill-opacity:1;stroke:#f57900;stroke-width:4;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#Arrow2Send)"
- d="m 286.70433,227.21216 c 0,6.93447 5.56889,12.58358 12.50265,12.68274 6.93376,0.0992 12.66193,-5.38837 12.86024,-12.32 0.19831,-6.93163 -4.18172,-9.66261 -12.13483,-13.0351"
- id="path2162"
+ sodipodi:nodetypes="cssc"
inkscape:connector-curvature="0"
- sodipodi:nodetypes="cssc" />
+ id="path4213"
+ d="m 286.70433,227.21216 c 0,6.93447 5.56889,12.58358 12.50265,12.68274 6.93376,0.0992 12.66193,-5.38837 12.86024,-12.32 0.19831,-6.93163 -4.18172,-9.66261 -12.13483,-13.0351"
+ style="display:inline;opacity:1;fill:none;fill-opacity:1;stroke:#f57900;stroke-width:4;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#Arrow2Send)" />
</g>
<g
inkscape:groupmode="layer"
id="layer8"
- inkscape:label="adding_point_2" />
+ inkscape:label="adding_point_2"
+ style="display:none" />
<g
inkscape:groupmode="layer"
id="layer3"
@@ -1086,6 +1190,101 @@
y2="481"
id="line1888" />
</g>
+ <g
+ inkscape:groupmode="layer"
+ id="layer1"
+ inkscape:label="sort_comparison"
+ style="display:none">
+ <path
+ style="opacity:1;fill:#000000;fill-opacity:1;stroke:#000000;stroke-width:3.9987402;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#marker3855)"
+ d="m 56,481 239.39339,76.13113"
+ id="path3804"
+ inkscape:connector-curvature="0"
+ sodipodi:nodetypes="cc" />
+ <path
+ style="opacity:1;fill:#000000;fill-opacity:1;stroke:#000000;stroke-width:3.99900007;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#marker3809)"
+ d="M 56,481 260.93093,372.66266"
+ id="path3806"
+ inkscape:connector-curvature="0"
+ sodipodi:nodetypes="cc" />
+ <path
+ style="opacity:1;fill:none;fill-opacity:1;stroke:#000000;stroke-width:3.99900007;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-start:url(#Arrow2Mstart)"
+ id="path3893"
+ sodipodi:type="arc"
+ sodipodi:cx="56"
+ sodipodi:cy="481"
+ sodipodi:rx="135.90041"
+ sodipodi:ry="135.90041"
+ sodipodi:start="5.86421"
+ sodipodi:end="0.22218386"
+ sodipodi:open="true"
+ d="m 180.14588,425.71237 a 135.90041,135.90041 0 0 1 8.4139,85.23469" />
+ </g>
+ <g
+ inkscape:groupmode="layer"
+ id="layer7"
+ inkscape:label="sweep_comparison"
+ style="display:inline">
+ <path
+ inkscape:connector-curvature="0"
+ id="path2434"
+ d="M 249,300 256,213 204,180 75,409 56,481"
+ style="opacity:1;fill:none;fill-opacity:1;stroke:#000000;stroke-width:2;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1" />
+ <circle
+ id="circle2437"
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ r="12"
+ cy="481"
+ cx="56" />
+ <circle
+ id="circle2439"
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ r="12"
+ cy="409"
+ cx="75" />
+ <circle
+ id="circle2441"
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ r="12"
+ cy="180"
+ cx="204" />
+ <circle
+ id="circle2443"
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ r="12"
+ cy="213"
+ cx="256" />
+ <circle
+ id="circle2445"
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ r="12"
+ cy="300"
+ cx="249" />
+ <circle
+ cx="412"
+ cy="193"
+ r="12"
+ style="display:inline;fill:none;stroke:#000000;stroke-width:2"
+ id="circle1838-62-9" />
+ <path
+ style="display:inline;opacity:1;fill:none;fill-opacity:1;stroke:#000000;stroke-width:4;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#marker4219)"
+ d="M 256,213 392.52453,194.79379"
+ id="path2160"
+ inkscape:connector-curvature="0"
+ sodipodi:nodetypes="cc" />
+ <path
+ sodipodi:nodetypes="cc"
+ inkscape:connector-curvature="0"
+ id="path4461"
+ d="m 256,213 -5.69769,68.15215"
+ style="display:inline;opacity:1;fill:none;fill-opacity:1;stroke:#000000;stroke-width:4;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#marker4465)" />
+ <path
+ style="display:inline;opacity:1;fill:none;fill-opacity:1;stroke:#000000;stroke-width:3.99900007;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-start:url(#Arrow2Mstart-2)"
+ d="m 298.36687,215.56256 c -0.20385,0.89626 -0.0234,1.65148 -0.0697,2.46886 -1.21336,21.43846 -17.36115,36.33106 -39.02351,37.97982"
+ id="path3893-9"
+ inkscape:connector-curvature="0"
+ sodipodi:nodetypes="csc" />
+ </g>
<g
inkscape:groupmode="layer"
id="layer5"
diff --git a/Presentation/Figures/orient.pdf b/Presentation/Figures/orient.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..58c981c49844eff5719425e51f55f833550ebacf
Binary files /dev/null and b/Presentation/Figures/orient.pdf differ
diff --git a/Presentation/Figures/orient.svg b/Presentation/Figures/orient.svg
new file mode 100644
index 0000000000000000000000000000000000000000..772fefe4cc34bf1db466d4c0534482e8df92d693
--- /dev/null
+++ b/Presentation/Figures/orient.svg
@@ -0,0 +1,396 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+ xmlns:ns1="http://www.iki.fi/pav/software/textext/"
+ xmlns:dc="http://purl.org/dc/elements/1.1/"
+ xmlns:cc="http://creativecommons.org/ns#"
+ xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+ xmlns:svg="http://www.w3.org/2000/svg"
+ xmlns="http://www.w3.org/2000/svg"
+ xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+ xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+ width="114.63708mm"
+ height="57.568756mm"
+ viewBox="0 0 114.63708 57.568756"
+ version="1.1"
+ id="svg8"
+ inkscape:version="0.92.4 5da689c313, 2019-01-14"
+ sodipodi:docname="orient.svg">
+ <defs
+ id="defs2">
+ <marker
+ inkscape:stockid="Arrow2Sstart"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow2Sstart"
+ style="overflow:visible"
+ inkscape:isstock="true">
+ <path
+ id="path847"
+ style="fill:#73d216;fill-opacity:1;fill-rule:evenodd;stroke:#73d216;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ transform="matrix(0.3,0,0,0.3,-0.69,0)"
+ inkscape:connector-curvature="0" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow1Sstart"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="marker1657"
+ style="overflow:visible"
+ inkscape:isstock="true">
+ <path
+ id="path1655"
+ d="M 0,0 5,-5 -12.5,0 5,5 Z"
+ style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+ transform="matrix(0.2,0,0,0.2,1.2,0)"
+ inkscape:connector-curvature="0" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow1Sstart"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow1Sstart"
+ style="overflow:visible"
+ inkscape:isstock="true">
+ <path
+ id="path829"
+ d="M 0,0 5,-5 -12.5,0 5,5 Z"
+ style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+ transform="matrix(0.2,0,0,0.2,1.2,0)"
+ inkscape:connector-curvature="0" />
+ </marker>
+ <marker
+ inkscape:isstock="true"
+ style="overflow:visible"
+ id="marker1115"
+ refX="0"
+ refY="0"
+ orient="auto"
+ inkscape:stockid="Arrow2Mend">
+ <path
+ transform="scale(-0.6)"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+ id="path1113"
+ inkscape:connector-curvature="0" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Mend"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow2Mend"
+ style="overflow:visible"
+ inkscape:isstock="true"
+ inkscape:collect="always">
+ <path
+ id="path844"
+ style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ transform="scale(-0.6)"
+ inkscape:connector-curvature="0" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Mend"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow2Mend-9"
+ style="overflow:visible"
+ inkscape:isstock="true"
+ inkscape:collect="always">
+ <path
+ inkscape:connector-curvature="0"
+ id="path844-1"
+ style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ transform="scale(-0.6)" />
+ </marker>
+ <marker
+ inkscape:isstock="true"
+ style="overflow:visible"
+ id="marker1115-2"
+ refX="0"
+ refY="0"
+ orient="auto"
+ inkscape:stockid="Arrow2Mend">
+ <path
+ inkscape:connector-curvature="0"
+ transform="scale(-0.6)"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+ id="path1113-7" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Sstart"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow2Sstart-0"
+ style="overflow:visible"
+ inkscape:isstock="true">
+ <path
+ inkscape:connector-curvature="0"
+ id="path847-9"
+ style="fill:#ef2929;fill-opacity:1;fill-rule:evenodd;stroke:#ef2929;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ transform="matrix(0.3,0,0,0.3,-0.69,0)" />
+ </marker>
+ </defs>
+ <sodipodi:namedview
+ id="base"
+ pagecolor="#ffffff"
+ bordercolor="#666666"
+ borderopacity="1.0"
+ inkscape:pageopacity="0.0"
+ inkscape:pageshadow="2"
+ inkscape:zoom="1.4"
+ inkscape:cx="236.33847"
+ inkscape:cy="47.737009"
+ inkscape:document-units="mm"
+ inkscape:current-layer="layer1"
+ showgrid="false"
+ inkscape:snap-object-midpoints="true"
+ inkscape:snap-global="true"
+ fit-margin-top="0"
+ fit-margin-left="0"
+ fit-margin-right="0"
+ fit-margin-bottom="0" />
+ <metadata
+ id="metadata5">
+ <rdf:RDF>
+ <cc:Work
+ rdf:about="">
+ <dc:format>image/svg+xml</dc:format>
+ <dc:type
+ rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
+ <dc:title></dc:title>
+ </cc:Work>
+ </rdf:RDF>
+ </metadata>
+ <g
+ inkscape:label="Layer 1"
+ inkscape:groupmode="layer"
+ id="layer1"
+ transform="translate(-48.106519,-105.14661)">
+ <g
+ id="g1917">
+ <path
+ inkscape:connector-curvature="0"
+ id="path815"
+ d="M 48.643053,133.96559 H 89.268021"
+ style="opacity:1;fill:#000000;fill-opacity:1;stroke:#000000;stroke-width:1.05799997;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#Arrow2Mend)" />
+ <path
+ style="opacity:1;fill:#000000;fill-opacity:1;stroke:#000000;stroke-width:1.05799997;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#marker1115)"
+ d="M 48.643053,133.96559 78.397999,106.3064"
+ id="path1111"
+ inkscape:connector-curvature="0" />
+ <path
+ d="m 65.353562,122.04938 a 20.524063,20.524063 0 0 1 3.744204,10.23045"
+ sodipodi:open="true"
+ sodipodi:end="6.2009568"
+ sodipodi:start="5.6637237"
+ sodipodi:ry="20.524063"
+ sodipodi:rx="20.524063"
+ sodipodi:cy="133.96559"
+ sodipodi:cx="48.643051"
+ sodipodi:type="arc"
+ id="path1513"
+ style="opacity:1;fill:none;fill-opacity:1;stroke:#73d216;stroke-width:1.05799997;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-start:url(#Arrow2Sstart)" />
+ <g
+ style="stroke-width:2.2442975"
+ transform="matrix(1.0284933,0,0,1.0284933,-90.069698,3.9616292)"
+ ns1:version="0.10.0"
+ ns1:texconverter="pdflatex"
+ ns1:pdfconverter="pdf2svg"
+ ns1:text="$\\mathbf{v}_0$"
+ ns1:preamble="/home/nivoliev/.config/inkscape/extensions/textext/default_packages.tex"
+ ns1:scale="6.54305795713"
+ ns1:alignment="middle center"
+ ns1:jacobian_sqrt="2.308245"
+ id="g1814">
+ <g
+ style="stroke-width:2.2442975"
+ id="surface1">
+ <g
+ style="fill:#000000;fill-opacity:1;stroke-width:2.2442975"
+ id="g1807">
+ <path
+ inkscape:connector-curvature="0"
+ style="stroke:none;stroke-width:0"
+ d="m 5.046875,-3.75 c 0.0625,-0.140625 0.09375,-0.203125 0.734375,-0.203125 v -0.46875 C 5.53125,-4.40625 5.25,-4.390625 5,-4.390625 c -0.25,0 -0.703125,-0.03125 -0.90625,-0.03125 v 0.46875 c 0.1875,0 0.46875,0.03125 0.46875,0.109375 0,0 0,0.03125 -0.046875,0.109375 l -1.15625,2.5 -1.265625,-2.71875 H 2.625 v -0.46875 c -0.328125,0.015625 -1.21875,0.03125 -1.234375,0.03125 -0.28125,0 -0.71875,-0.03125 -1.125,-0.03125 v 0.46875 h 0.625 l 1.75,3.75 c 0.125,0.25 0.25,0.25 0.375,0.25 0.171875,0 0.28125,-0.03125 0.375,-0.25 z m 0,0"
+ transform="translate(148.712,134.765)"
+ id="path1805" />
+ </g>
+ <g
+ style="fill:#000000;fill-opacity:1;stroke-width:2.2442975"
+ id="g1811">
+ <path
+ inkscape:connector-curvature="0"
+ style="stroke:none;stroke-width:0"
+ d="M 3.59375,-2.21875 C 3.59375,-2.984375 3.5,-3.546875 3.1875,-4.03125 2.96875,-4.34375 2.53125,-4.625 1.984375,-4.625 c -1.625,0 -1.625,1.90625 -1.625,2.40625 0,0.5 0,2.359375 1.625,2.359375 1.609375,0 1.609375,-1.859375 1.609375,-2.359375 z M 1.984375,-0.0625 c -0.328125,0 -0.75,-0.1875 -0.890625,-0.75 C 1,-1.21875 1,-1.796875 1,-2.3125 1,-2.828125 1,-3.359375 1.09375,-3.734375 1.25,-4.28125 1.6875,-4.4375 1.984375,-4.4375 c 0.375,0 0.734375,0.234375 0.859375,0.640625 0.109375,0.375 0.125,0.875 0.125,1.484375 0,0.515625 0,1.03125 -0.09375,1.46875 -0.140625,0.640625 -0.609375,0.78125 -0.890625,0.78125 z m 0,0"
+ transform="translate(154.759,136.259)"
+ id="path1809" />
+ </g>
+ </g>
+ </g>
+ <g
+ transform="matrix(1.028493,0,0,1.028493,-101.47796,-21.149805)"
+ ns1:version="0.10.0"
+ ns1:texconverter="pdflatex"
+ ns1:pdfconverter="pdf2svg"
+ ns1:text="$\\mathbf{v}_1$"
+ ns1:preamble="/home/nivoliev/.config/inkscape/extensions/textext/default_packages.tex"
+ ns1:scale="2.91541464204"
+ ns1:alignment="middle center"
+ ns1:jacobian_sqrt="1.028493"
+ id="g1886">
+ <g
+ id="g1884">
+ <g
+ style="fill:#000000;fill-opacity:1"
+ id="g1878">
+ <path
+ style="stroke:none;stroke-width:0"
+ d="m 5.046875,-3.75 c 0.0625,-0.140625 0.09375,-0.203125 0.734375,-0.203125 v -0.46875 C 5.53125,-4.40625 5.25,-4.390625 5,-4.390625 c -0.25,0 -0.703125,-0.03125 -0.90625,-0.03125 v 0.46875 c 0.1875,0 0.46875,0.03125 0.46875,0.109375 0,0 0,0.03125 -0.046875,0.109375 l -1.15625,2.5 -1.265625,-2.71875 H 2.625 v -0.46875 c -0.328125,0.015625 -1.21875,0.03125 -1.234375,0.03125 -0.28125,0 -0.71875,-0.03125 -1.125,-0.03125 v 0.46875 h 0.625 l 1.75,3.75 c 0.125,0.25 0.25,0.25 0.375,0.25 0.171875,0 0.28125,-0.03125 0.375,-0.25 z m 0,0"
+ transform="translate(148.712,134.765)"
+ id="path1876"
+ inkscape:connector-curvature="0" />
+ </g>
+ <g
+ style="fill:#000000;fill-opacity:1"
+ id="g1882">
+ <path
+ style="stroke:none;stroke-width:0"
+ d="m 2.328125,-4.4375 c 0,-0.1875 0,-0.1875 -0.203125,-0.1875 -0.453125,0.4375 -1.078125,0.4375 -1.359375,0.4375 v 0.25 c 0.15625,0 0.625,0 1,-0.1875 v 3.546875 c 0,0.234375 0,0.328125 -0.6875,0.328125 H 0.8125 V 0 c 0.125,0 0.984375,-0.03125 1.234375,-0.03125 0.21875,0 1.09375,0.03125 1.25,0.03125 V -0.25 H 3.03125 c -0.703125,0 -0.703125,-0.09375 -0.703125,-0.328125 z m 0,0"
+ transform="translate(154.759,136.259)"
+ id="path1880"
+ inkscape:connector-curvature="0" />
+ </g>
+ </g>
+ </g>
+ <text
+ id="text1898"
+ y="127.0603"
+ x="73.131424"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.37443352px;line-height:17.77255249px;font-family:'Linux Biolinum';-inkscape-font-specification:'Linux Biolinum';text-align:center;letter-spacing:0px;word-spacing:0px;text-anchor:middle;fill:#73d216;fill-opacity:1;stroke:none;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ xml:space="preserve"><tspan
+ style="fill:#73d216;stroke-width:0.26458332px"
+ y="127.0603"
+ x="73.131424"
+ id="tspan1896"
+ sodipodi:role="line">+</tspan></text>
+ </g>
+ <path
+ style="opacity:1;fill:#000000;fill-opacity:1;stroke:#000000;stroke-width:1.05799997;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#Arrow2Mend-9)"
+ d="m 120.51912,133.91128 h 40.62496"
+ id="path815-6"
+ inkscape:connector-curvature="0" />
+ <path
+ inkscape:connector-curvature="0"
+ id="path1111-0"
+ d="m 120.51912,133.91128 29.75494,27.65919"
+ style="opacity:1;fill:#000000;fill-opacity:1;stroke:#000000;stroke-width:1.05799997;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#marker1115-2)" />
+ <path
+ style="opacity:1;fill:none;fill-opacity:1;stroke:#ef2929;stroke-width:1.05799997;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-start:url(#Arrow2Sstart-0)"
+ id="path1513-6"
+ sodipodi:type="arc"
+ sodipodi:cx="120.51912"
+ sodipodi:cy="-133.91129"
+ sodipodi:rx="20.524063"
+ sodipodi:ry="20.524063"
+ sodipodi:start="5.6637237"
+ sodipodi:end="6.2009568"
+ sodipodi:open="true"
+ d="m 137.22963,-145.82749 a 20.524063,20.524063 0 0 1 3.7442,10.23044"
+ transform="scale(1,-1)" />
+ <g
+ id="g1814-2"
+ ns1:jacobian_sqrt="2.308245"
+ ns1:alignment="middle center"
+ ns1:scale="6.54305795713"
+ ns1:preamble="/home/nivoliev/.config/inkscape/extensions/textext/default_packages.tex"
+ ns1:text="$\\mathbf{v}_0$"
+ ns1:pdfconverter="pdf2svg"
+ ns1:texconverter="pdflatex"
+ ns1:version="0.10.0"
+ transform="matrix(1.0284933,0,0,1.0284933,-19.516552,-11.589708)"
+ style="stroke-width:2.2442975">
+ <g
+ id="surface1-6"
+ style="stroke-width:2.2442975">
+ <g
+ id="g1807-1"
+ style="fill:#000000;fill-opacity:1;stroke-width:2.2442975">
+ <path
+ id="path1805-8"
+ transform="translate(148.712,134.765)"
+ d="m 5.046875,-3.75 c 0.0625,-0.140625 0.09375,-0.203125 0.734375,-0.203125 v -0.46875 C 5.53125,-4.40625 5.25,-4.390625 5,-4.390625 c -0.25,0 -0.703125,-0.03125 -0.90625,-0.03125 v 0.46875 c 0.1875,0 0.46875,0.03125 0.46875,0.109375 0,0 0,0.03125 -0.046875,0.109375 l -1.15625,2.5 -1.265625,-2.71875 H 2.625 v -0.46875 c -0.328125,0.015625 -1.21875,0.03125 -1.234375,0.03125 -0.28125,0 -0.71875,-0.03125 -1.125,-0.03125 v 0.46875 h 0.625 l 1.75,3.75 c 0.125,0.25 0.25,0.25 0.375,0.25 0.171875,0 0.28125,-0.03125 0.375,-0.25 z m 0,0"
+ style="stroke:none;stroke-width:0"
+ inkscape:connector-curvature="0" />
+ </g>
+ <g
+ id="g1811-7"
+ style="fill:#000000;fill-opacity:1;stroke-width:2.2442975">
+ <path
+ id="path1809-9"
+ transform="translate(154.759,136.259)"
+ d="M 3.59375,-2.21875 C 3.59375,-2.984375 3.5,-3.546875 3.1875,-4.03125 2.96875,-4.34375 2.53125,-4.625 1.984375,-4.625 c -1.625,0 -1.625,1.90625 -1.625,2.40625 0,0.5 0,2.359375 1.625,2.359375 1.609375,0 1.609375,-1.859375 1.609375,-2.359375 z M 1.984375,-0.0625 c -0.328125,0 -0.75,-0.1875 -0.890625,-0.75 C 1,-1.21875 1,-1.796875 1,-2.3125 1,-2.828125 1,-3.359375 1.09375,-3.734375 1.25,-4.28125 1.6875,-4.4375 1.984375,-4.4375 c 0.375,0 0.734375,0.234375 0.859375,0.640625 0.109375,0.375 0.125,0.875 0.125,1.484375 0,0.515625 0,1.03125 -0.09375,1.46875 -0.140625,0.640625 -0.609375,0.78125 -0.890625,0.78125 z m 0,0"
+ style="stroke:none;stroke-width:0"
+ inkscape:connector-curvature="0" />
+ </g>
+ </g>
+ </g>
+ <g
+ id="g1886-2"
+ ns1:jacobian_sqrt="1.028493"
+ ns1:alignment="middle center"
+ ns1:scale="2.91541464204"
+ ns1:preamble="/home/nivoliev/.config/inkscape/extensions/textext/default_packages.tex"
+ ns1:text="$\\mathbf{v}_1$"
+ ns1:pdfconverter="pdf2svg"
+ ns1:texconverter="pdflatex"
+ ns1:version="0.10.0"
+ transform="matrix(1.028493,0,0,1.028493,-30.35785,15.081596)">
+ <g
+ id="g1884-0">
+ <g
+ id="g1878-2"
+ style="fill:#000000;fill-opacity:1">
+ <path
+ id="path1876-3"
+ transform="translate(148.712,134.765)"
+ d="m 5.046875,-3.75 c 0.0625,-0.140625 0.09375,-0.203125 0.734375,-0.203125 v -0.46875 C 5.53125,-4.40625 5.25,-4.390625 5,-4.390625 c -0.25,0 -0.703125,-0.03125 -0.90625,-0.03125 v 0.46875 c 0.1875,0 0.46875,0.03125 0.46875,0.109375 0,0 0,0.03125 -0.046875,0.109375 l -1.15625,2.5 -1.265625,-2.71875 H 2.625 v -0.46875 c -0.328125,0.015625 -1.21875,0.03125 -1.234375,0.03125 -0.28125,0 -0.71875,-0.03125 -1.125,-0.03125 v 0.46875 h 0.625 l 1.75,3.75 c 0.125,0.25 0.25,0.25 0.375,0.25 0.171875,0 0.28125,-0.03125 0.375,-0.25 z m 0,0"
+ style="stroke:none;stroke-width:0"
+ inkscape:connector-curvature="0" />
+ </g>
+ <g
+ id="g1882-7"
+ style="fill:#000000;fill-opacity:1">
+ <path
+ id="path1880-5"
+ transform="translate(154.759,136.259)"
+ d="m 2.328125,-4.4375 c 0,-0.1875 0,-0.1875 -0.203125,-0.1875 -0.453125,0.4375 -1.078125,0.4375 -1.359375,0.4375 v 0.25 c 0.15625,0 0.625,0 1,-0.1875 v 3.546875 c 0,0.234375 0,0.328125 -0.6875,0.328125 H 0.8125 V 0 c 0.125,0 0.984375,-0.03125 1.234375,-0.03125 0.21875,0 1.09375,0.03125 1.25,0.03125 V -0.25 H 3.03125 c -0.703125,0 -0.703125,-0.09375 -0.703125,-0.328125 z m 0,0"
+ style="stroke:none;stroke-width:0"
+ inkscape:connector-curvature="0" />
+ </g>
+ </g>
+ </g>
+ <text
+ xml:space="preserve"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.37443352px;line-height:17.77255249px;font-family:'Linux Biolinum';-inkscape-font-specification:'Linux Biolinum';text-align:center;letter-spacing:0px;word-spacing:0px;text-anchor:middle;fill:#ef2929;fill-opacity:1;stroke:#ef2929;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ x="144.44052"
+ y="144.58188"
+ id="text1898-9"><tspan
+ sodipodi:role="line"
+ id="tspan1896-2"
+ x="144.44052"
+ y="144.58188"
+ style="fill:#ef2929;stroke:#ef2929;stroke-width:0.26458332px">-</tspan></text>
+ </g>
+</svg>
diff --git a/Presentation/Figures/sort_comparison.pdf b/Presentation/Figures/sort_comparison.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..e29ce993b2fe28701ee649b9f9fb194db850186e
Binary files /dev/null and b/Presentation/Figures/sort_comparison.pdf differ
diff --git a/Presentation/Figures/sweep_comparison.pdf b/Presentation/Figures/sweep_comparison.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..cfae1bbbcab5446616f0834efdecc54806b9dee4
Binary files /dev/null and b/Presentation/Figures/sweep_comparison.pdf differ
diff --git a/Presentation/presentation.tex b/Presentation/presentation.tex
index d70438fd7851f8cb0f6fa224068996a0c819ef45..a943089726b64c2cb7c1f289646e7174bd588e76 100644
--- a/Presentation/presentation.tex
+++ b/Presentation/presentation.tex
@@ -25,29 +25,29 @@
\frame {
\frametitle{Enveloppe convexe}
\centering
- \includegraphics[width=0.6\li]{Figures/hull.pdf}
+ \includegraphics[width=0.6\li]{Figures/hull}
}
\frame {
\frametitle{1 -- trier par angle}
\centering
- \includegraphics[width=0.6\li]{Figures/hull_sort.pdf}
+ \includegraphics[width=0.6\li]{Figures/hull_sort}
}
\frame {
\frametitle{2 -- balayage}
\centering
- \only<1>{\includegraphics[width=0.6\li]{Figures/partial_hull.pdf}}%
- \only<2>{\includegraphics[width=0.6\li]{Figures/increment_hull_1.pdf}}%
- \only<3>{\includegraphics[width=0.6\li]{Figures/increment_hull_2.pdf}}%
- \only<4>{\includegraphics[width=0.6\li]{Figures/increment_hull_3.pdf}}%
- \only<5>{\includegraphics[width=0.6\li]{Figures/hull.pdf}}%
+ \only<1>{\includegraphics[width=0.6\li]{Figures/partial_hull}}%
+ \only<2>{\includegraphics[width=0.6\li]{Figures/increment_hull_1}}%
+ \only<3>{\includegraphics[width=0.6\li]{Figures/increment_hull_2}}%
+ \only<4>{\includegraphics[width=0.6\li]{Figures/increment_hull_3}}%
+ \only<5>{\includegraphics[width=0.6\li]{Figures/hull}}%
}
\frame {
\frametitle{Robustesse : alignements de points}
\centering
- \includegraphics[width=0.6\li]{Figures/hull_aligned.pdf}
+ \includegraphics[width=0.6\li]{Figures/hull_aligned}
}
\frame {
@@ -80,4 +80,149 @@
\end{ucblalgo}
}
+\section{Perturbation}
+
+\frame{
+ \frametitle{Prédicat : orientation}
+ \centering
+ \includegraphics[width=0.5\li]{Figures/orient}
+
+ \vspace{8mm}
+ \begin{myblock}{0.8\li}
+ \centering
+ $
+ \det{
+ \left(
+ \begin{array}{cc}
+ x_0 & x_1 \\
+ y_0 & y_1 \\
+ \end{array}
+ \right)
+ }
+ $ : sinus de l'angle si vecteurs normalisés.
+
+ \hl{positif, négatif ou nul}
+ \end{myblock}
+}
+
+\frame{
+ \frametitle{Utilisation pour l'enveloppe convexe}
+ \centering
+ \only<1>{\includegraphics[width=0.6\li]{Figures/sort_comparison}}%
+ \only<2>{\includegraphics[width=0.6\li]{Figures/sweep_comparison}}
+}
+
+\newcommand{\eps}{\ensuremath{\varepsilon}}
+
+\frame{
+ \frametitle{Perturbation}
+
+ \begin{align*}
+ \det{
+ \left(
+ \begin{array}{cc}
+ x_0 + \eps_{0,0} & x_1 + \eps_{1,0} \\
+ y_0 + \eps_{0,1} & y_1 + \eps_{1,1} \\
+ \end{array}
+ \right)
+ }
+ &=
+ \det{
+ \left(
+ \begin{array}{cc}
+ x_0 & x_1 \\
+ y_0 & y_1 \\
+ \end{array}
+ \right)
+ } \\
+ &
+ + \eps_{0,0} y_1
+ + \eps_{1,1} x_0
+ - \eps_{0,1} x_1
+ - \eps_{1,0} y_0
+ \\
+ &
+ + \eps_{0,0}\eps_{1,1}
+ - \eps_{0,1}\eps_{1,0}
+ \end{align*}
+
+ \vspace{5mm}
+ \begin{myblock}{0.8\li}
+ \begin{enumerate}
+ \item systématiquement plus de zéro ?
+ \item facilité d'évaluation ?
+ \end{enumerate}
+ \end{myblock}
+}
+
+\frame{
+ \frametitle{Rester dans le pétrin}
+
+ Posons que $ \forall(i,j), \eps_{i,j} = \eps > 0$
+
+ \begin{equation*}
+ \eps y_1
+ + \eps x_0
+ - \eps x_1
+ - \eps y_0
+ + \eps\eps
+ - \eps\eps
+ \end{equation*}
+
+ \vspace{5mm}
+ Si $(x_0,y_0) = (0,1)$ et $(x_1,y_1) = (0,2)$
+
+ \begin{equation*}
+ 2\eps
+ - \eps
+ + \eps\eps
+ - \eps\eps
+ = \eps
+ \quad \Rightarrow \mbox{positif !}
+ \end{equation*}
+
+ \visible<2>{
+ \vspace{5mm}
+ Si $(x_0,y_0) = (1,1)$ et $(x_1,y_1) = (2,2)$
+
+ \begin{equation*}
+ 2\eps
+ + \eps
+ - 2\eps
+ - \eps
+ + \eps\eps
+ - \eps\eps
+ = 0
+ \quad \Rightarrow \mbox{arf.}
+ \end{equation*}
+ }
+}
+
+\frame{
+ \frametitle{Sortir du pétrin}
+
+ Avec la perturbation $ \begin{pmatrix}\eps^1 & \eps^2 \\ \eps^2 & \eps^2\end{pmatrix}$
+
+ \begin{align*}
+ & \eps^1 y_1
+ + \eps^2 x_0
+ - \eps^2 x_1
+ - \eps^2 y_0
+ + \eps^1\eps^2
+ - \eps^2\eps^2 \\
+ &= \eps y_1 + \eps^2(x_0 - x_1 - y_0) + \eps^3 - \eps^4
+ \end{align*}
+
+ \vspace{8mm}
+ \visible<2->{
+ \begin{myblock}{0.8\li}
+ \begin{itemize}
+ \item<1-> si $y_1$ est non nul, renvoyer son signe ;
+ \item<3-> si $x_0 - x_1 - y_0$ est non nul, renvoyer son signe ;
+ \item<4-> sinon renvoyer positif.
+ \end{itemize}
+ \end{myblock}
+ }
+}
+
\end{document}