convex hulls in 2d and 3d · 2013. 1. 28. · convex hulls in 2d and 3d. extreme point. extreme...
TRANSCRIPT
![Page 1: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/1.jpg)
Convex Hulls in 2d and 3d
![Page 2: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/2.jpg)
Convex Hulls in 2d and 3d
![Page 3: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/3.jpg)
![Page 4: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/4.jpg)
Extreme point
![Page 5: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/5.jpg)
Extreme point
Support line
![Page 6: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/6.jpg)
Jarvis march (Gift wrapping)
![Page 7: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/7.jpg)
Jarvis march (Gift wrapping)
The lowest point is extreme
![Page 8: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/8.jpg)
Jarvis march (Gift wrapping)
![Page 9: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/9.jpg)
Jarvis march (Gift wrapping)
![Page 10: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/10.jpg)
Jarvis march (Gift wrapping)
![Page 11: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/11.jpg)
Jarvis march (Gift wrapping)
![Page 12: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/12.jpg)
Jarvis march (Gift wrapping)
Next point is found
![Page 13: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/13.jpg)
Jarvis march (Gift wrapping)
Next point is found
Then the next
![Page 14: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/14.jpg)
Jarvis march (Gift wrapping)
Next point is found
Then the next
Etc...
![Page 15: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/15.jpg)
Jarvis march (Gift wrapping)
Input : S set of points.u = the lowest point of S;min =∞For all w ∈ S \ {u}
if angle(ux, uw) < min then min = angle(ux, uw); v = w;u.next = v;Do
S = S \ {v}For all w ∈ S
min =∞If angle(v.previous v, vw) < min then
min = angle(v.previous v, vw); v.next = w;v = v.next;
While v 6= u
![Page 16: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/16.jpg)
Jarvis march (Gift wrapping)
Complexity ?
Input : S set of points.u = the lowest point of S;min =∞For all w ∈ S \ {u}
if angle(ux, uw) < min then min = angle(ux, uw); v = w;u.next = v;Do
S = S \ {v}For all w ∈ S
min =∞If angle(v.previous v, vw) < min then
min = angle(v.previous v, vw); v.next = w;v = v.next;
While v 6= u
![Page 17: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/17.jpg)
Jarvis march (Gift wrapping)
Complexity ?
O(n)
Input : S set of points.u = the lowest point of S;min =∞For all w ∈ S \ {u}
if angle(ux, uw) < min then min = angle(ux, uw); v = w;u.next = v;Do
S = S \ {v}For all w ∈ S
min =∞If angle(v.previous v, vw) < min then
min = angle(v.previous v, vw); v.next = w;v = v.next;
While v 6= u
![Page 18: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/18.jpg)
O(n)
Jarvis march (Gift wrapping)
Complexity ?
Input : S set of points.u = the lowest point of S;min =∞For all w ∈ S \ {u}
if angle(ux, uw) < min then min = angle(ux, uw); v = w;u.next = v;Do
S = S \ {v}For all w ∈ S
min =∞If angle(v.previous v, vw) < min then
min = angle(v.previous v, vw); v.next = w;v = v.next;
While v 6= u
![Page 19: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/19.jpg)
Jarvis march (Gift wrapping)
Complexity ?
Input : S set of points.u = the lowest point of S;min =∞For all w ∈ S \ {u}
if angle(ux, uw) < min then min = angle(ux, uw); v = w;u.next = v;Do
S = S \ {v}For all w ∈ S
min =∞If angle(v.previous v, vw) < min then
min = angle(v.previous v, vw); v.next = w;v = v.next;
While v 6= u
![Page 20: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/20.jpg)
Jarvis march (Gift wrapping)
Complexity ?
Input : S set of points.u = the lowest point of S;min =∞For all w ∈ S \ {u}
if angle(ux, uw) < min then min = angle(ux, uw); v = w;u.next = v;Do
S = S \ {v}For all w ∈ S
min =∞If angle(v.previous v, vw) < min then
min = angle(v.previous v, vw); v.next = w;v = v.next;
While v 6= u O(n)
![Page 21: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/21.jpg)
Jarvis march (Gift wrapping)
Complexity ?
Input : S set of points.u = the lowest point of S;min =∞For all w ∈ S \ {u}
if angle(ux, uw) < min then min = angle(ux, uw); v = w;u.next = v;Do
S = S \ {v}For all w ∈ S
min =∞If angle(v.previous v, vw) < min then
min = angle(v.previous v, vw); v.next = w;v = v.next;
While v 6= u
O(n2)
![Page 22: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/22.jpg)
Jarvis march (Gift wrapping)
Complexity ?
Input : S set of points.u = the lowest point of S;min =∞For all w ∈ S \ {u}
if angle(ux, uw) < min then min = angle(ux, uw); v = w;u.next = v;Do
S = S \ {v}For all w ∈ S
min =∞If angle(v.previous v, vw) < min then
min = angle(v.previous v, vw); v.next = w;v = v.next;
While v 6= u
O(nh)
![Page 23: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/23.jpg)
Graham scan
![Page 24: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/24.jpg)
Graham scan
One interior point
![Page 25: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/25.jpg)
Graham scan
One interior point
Sorting around this point
![Page 26: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/26.jpg)
Graham scan
One interior point
Sorting around this point
![Page 27: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/27.jpg)
Graham scan
One interior point
Sorting around this point
![Page 28: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/28.jpg)
Graham scan
One interior point
Sorting around this point
![Page 29: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/29.jpg)
Graham scan
One interior point
Sorting around this point
![Page 30: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/30.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 31: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/31.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 32: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/32.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 33: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/33.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 34: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/34.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 35: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/35.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 36: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/36.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 37: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/37.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 38: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/38.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 39: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/39.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 40: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/40.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 41: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/41.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 42: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/42.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 43: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/43.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 44: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/44.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 45: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/45.jpg)
Graham scan
One interior point
Sorting around this point
Scan starting from the lowest point
![Page 46: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/46.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham Scan
![Page 47: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/47.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
![Page 48: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/48.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
![Page 49: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/49.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)
![Page 50: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/50.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)O(1)
![Page 51: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/51.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)O(1)
O(n)
![Page 52: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/52.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)O(1)
O(n)
![Page 53: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/53.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)O(1)
O(n)
![Page 54: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/54.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)O(1)
O(n)
at most n deletions
![Page 55: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/55.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)O(1)
O(n)
at most n deletions
![Page 56: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/56.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)O(1)
O(n)
at most n deletions
at most n times
![Page 57: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/57.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)O(1)
O(n)
O(n)
![Page 58: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/58.jpg)
Input : S a set of n points.origin = barycenter of 3 points of S;sort S around the origin;u = the lowest point of S;v = u;while v.next 6= u
if (v, v.next, v.next.next) turn leftv = v.next;
elsev.next = v.next.next;if v 6= u v = v.previous;
Graham ScanComplexity
O(n log n)
![Page 59: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/59.jpg)
Graham alternative: origin at y = −∞
![Page 60: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/60.jpg)
Graham alternative: origin at y = −∞
Sort in x
![Page 61: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/61.jpg)
Graham alternative: origin at y = −∞
Sort in x
![Page 62: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/62.jpg)
Graham alternative: origin at y = −∞
Sort in x
Upper envelope
![Page 63: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/63.jpg)
Graham alternative: origin at y = −∞
Upper envelope
Add the lower envelope
![Page 64: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/64.jpg)
Deterministic incremental algorithm
![Page 65: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/65.jpg)
Deterministic incremental algorithm
![Page 66: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/66.jpg)
Deterministic incremental algorithm
![Page 67: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/67.jpg)
Deterministic incremental algorithm
![Page 68: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/68.jpg)
Deterministic incremental algorithm
![Page 69: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/69.jpg)
Deterministic incremental algorithm
![Page 70: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/70.jpg)
Deterministic incremental algorithm
![Page 71: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/71.jpg)
Deterministic incremental algorithm
![Page 72: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/72.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
Deterministic incremental algorithm
![Page 73: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/73.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
vu
Deterministic incremental algorithm
![Page 74: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/74.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
vu
Deterministic incremental algorithmu
![Page 75: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/75.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
vu
Deterministic incremental algorithmu
u
![Page 76: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/76.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
vu
Deterministic incremental algorithm
w
![Page 77: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/77.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
vu
Deterministic incremental algorithm
ww
![Page 78: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/78.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
Deterministic incremental algorithm
ww
u
![Page 79: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/79.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
Deterministic incremental algorithmComplexity
![Page 80: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/80.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
Deterministic incremental algorithmComplexity
O(n log n)
![Page 81: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/81.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
Deterministic incremental algorithm
![Page 82: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/82.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
Deterministic incremental algorithmComplexity
Draw an edge in the triangulation
![Page 83: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/83.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
Deterministic incremental algorithmComplexity
Draw an edge in the triangulation
nb of edges ' 3n
![Page 84: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/84.jpg)
Input : S set of n points.sort S in x;initialize a circular list with the 3 leftmost points
such that u is on the right u, u.next, u.next.next turn left;For v the next point in x
w=uwhile (v, u, u.next) turn right
u = u.next;v.next = u; u.previous = v;while (v, w, w.previous) turn left
w = w.previous;v.previous = w; w.next = v;u = v;
Deterministic incremental algorithmComplexity
O(n log n)
![Page 85: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/85.jpg)
Divide & conquer algorithm
![Page 86: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/86.jpg)
Divide & conquer algorithm
![Page 87: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/87.jpg)
Divide & conquer algorithm
![Page 88: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/88.jpg)
Divide & conquer algorithm
Common tangents
![Page 89: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/89.jpg)
Upper tangent
![Page 90: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/90.jpg)
Upper tangent
Topmost points
![Page 91: Convex Hulls in 2d and 3d · 2013. 1. 28. · Convex Hulls in 2d and 3d. Extreme point. Extreme point Support line. Jarvis march (Gift wrapping) Jarvis march (Gift wrapping) The lowest](https://reader034.vdocuments.site/reader034/viewer/2022051822/5fec2e8e3a5d9a7fca07cb71/html5/thumbnails/91.jpg)
Upper tangent
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Upper tangent
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Upper tangent
O(n)
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Divide & conquer algorithm
Complexity
f (n) =
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Divide & conquer algorithm
Complexity
f (n) = A · n + f (n2) + f (n2)
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Divide & conquer algorithm
Complexity
f (n) = A · n + f (n2) + f (n2)
= O(n log n)
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Divide & conquer algorithm
Complexity
f (n) = A · n + f (n2) + f (n2)
= O(n log n)
Balanced partition
Divide and merge in O(n)
(preprocessing in O(n log n)
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CHAN’s ALGORITHM
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CHAN’s ALGORITHMCompute the CH of k subsets of P
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CHAN’s ALGORITHMWrap around the CH of the k subsets
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CHAN’s ALGORITHMCompute the CH of k = n/m subsets of size m:
each in O(m logm), all in k/m×O(m logm) = O(n logm)
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CHAN’s ALGORITHM
Find the si: O(n)
Compute the CH of k = n/m subsets of size m:each in O(m logm), all in k/m×O(m logm) = O(n logm)
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CHAN’s ALGORITHM
One wrapping step:find the bitangents ti on eachsubset:n/m+ O(nb points visited)=n/m+ O(nb points removed)
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CHAN’s ALGORITHM
Find the si: O(n)
Compute the CH of k = n/m subsets of size m:each in O(m logm), all in k/m×O(m logm) = O(n logm)
One wrapping step:find the bitangents ti on eachsubset:n/m + O(nb points visited)=n/m + O(nb points removed)
For h wrapping steps:O(hn/m+ total points removed)
= O(hn/m + n)
Total:O(n logm + hn/m + n) =O(n(1 + h/m + logm))
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CHAN’s ALGORITHMSet a parameter H, call h the actual size of the convex hull of P
Hull(P,m,H)Do H wrapping steps
If the last wrap comes back to the first point then return successElse return incomplete
• Success if H > h• Complexity of Hull(P,H,H) is O(n logH)
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CHAN’s ALGORITHMSet a parameter H, call h the actual size of the convex hull of P
Hull(P,m,H)Do H wrapping steps
If the last wrap comes back to the first point then return successElse return incomplete
• Success if H > h• Complexity of Hull(P,H,H) is O(n logH)
Hull(P)For i = 1, 2, ... doL = Hull(P,H,H) with m = H = min(22i , n)If L 6= incomplete then return L
Complexity:Nb of iterations = O(log log h)Cost of ith iterations = O(n logH) = O(n2i)
Total: O(∑log logn
i=1 n2i) = O(n2log logn+1) =O(n log h)
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Special case: simple polygonal line
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Special case: simple polygonal line
already seen: monotone polyline in Graham’s scan
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Special case: simple polygonal line
Graham does not work
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Special case: simple polygonal line
Graham does not work
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Special case: simple polygonal line
Graham does not work
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Special case: simple polygonal line
Graham does not work
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Special case: simple polygonal line
Graham does not work
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Special case: simple polygonal line
Graham does not work
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Special case: simple polygonal line
Graham does not work
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Special case: simple polygonal line
Graham does not work
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Special case: simple polygonal line
p1
p2
p3
p4
p6
p5
p8p10
p11
p12p13
p14
P Q
P = p1, p2, ..., p14Q contains the subsequence p2, p4, p6, p11
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Special case: simple polygonal line
p1
succ(AM)
pi−1
pi
AM
Incremental algorithm : order given by the polylineAM = highest rank point in CH(Pi−1)
Property: pi interior to CH(Pi−1) iff pi ∈ H+pred ∩H+
succ
pred(AM)
H+succ
H+pred
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Special case: simple polygonal line
p1
succ(AM)
pi−1
pi
AM
Incremental algorithm : order given by the polylineAM = highest rank point in CH(Pi−1)
Property: pi interior to CH(Pi−1) iff pi ∈ H+pred ∩H+
succ
Update CH(Pi) as in the incremental algorithm
Complexity: O(n)
pred(AM)
H+succ
H+pred
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
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3D: Gift wrapping
Complexity: O(nh)
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3D Divide & conquer algorithm
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3D Divide & conquer algorithm
Sort in x and divide
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3D Divide & conquer algorithm
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3D Divide & conquer algorithmFind new edge: construct the CH of the projections,
use the 2d algo for bitangent
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3D Divide & conquer algorithmUse a wrapping algorithm around the new edges
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3D Divide & conquer algorithmUse a wrapping algorithm around the new edges
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3D Divide & conquer algorithmSearch only in the star of the new edge vertices
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3D Divide & conquer algorithmSearch only in the star of the new edge vertices
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3D Divide & conquer algorithmSearch only in the star of the new edge vertices
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3D Divide & conquer algorithmSearch only in the star of the new edge vertices
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3D Divide & conquer algorithmSearch only in the star of the new edge vertices
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3D Divide & conquer algorithmSearch only in the star of the new edge vertices
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3D Divide & conquer algorithmSearch only in the star of the new edge vertices Merge in O(n)
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3D Divide & conquer algorithmSearch only in the star of the new edge vertices Merge in O(n)
Complexity: O(n log n)
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