geometry images xiang gu harvard university steven j. gortler harvard university hugues hoppe...
TRANSCRIPT
![Page 1: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/1.jpg)
Geometry Images
Xiang Gu
Harvard University
Steven J. Gortler
Harvard university
Hugues Hoppe
Microsoft Research
•Some slides taken from Hugues Hoppe
![Page 2: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/2.jpg)
Background
• Modeling geometry with (triangular) meshes
• regular / iregular meshes
• current tecniques form semi-regular meshes
• typically: cut into disks
parametrize each disk
each disk is regular
all disks are iregular network
![Page 3: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/3.jpg)
Advantages of Regularity
• Improved compression (implicit connectivity)
• Reduce tangent non-uniformity
• Better start for hirarchial resolution
![Page 4: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/4.jpg)
Introduction
• Remesh into a completely regular structure
• geometry as 2D array of points
• more arrays for “surface signals”:– normals– colours– texture
• encode as RGB picture
![Page 5: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/5.jpg)
![Page 6: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/6.jpg)
![Page 7: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/7.jpg)
The General Algorithm
• Cut the mesh
• parametrize into an n*n square
• [x,y,z] <==> [R,G,B]
• encode using image compression algorithms
• other attributes - additional images
• implicit parametrization
![Page 8: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/8.jpg)
Challenges
• Find a “good cut” (form a disk-like surface)
• Avoid “cracks in the cut” along the boundary
• Evenly distributing parametrization
• lossy compression - cut fusing
![Page 9: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/9.jpg)
Creating The Geometry Image
![Page 10: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/10.jpg)
The Cut
• Cut = set of edges
• M = Original mesh M’ = New mesh
![Page 11: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/11.jpg)
mates
Split
• Split (non boundary) edges
• form “open cut” ’’ is a directed loop = edge of M’
![Page 12: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/12.jpg)
Split - cont.
• Valence k ==> k replications
• valence 2 ==> “cut node”
![Page 13: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/13.jpg)
Cut Path
• Path between 2 ordered cut nodes of ’
• mate to cut path
![Page 14: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/14.jpg)
Parametrization
• (piecewise) linear map : [n * n] => [vM’]
• resample mesh at grid
• triangle interpolation
![Page 15: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/15.jpg)
Parametrization
![Page 16: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/16.jpg)
Boundary parametrization
• Map ’ ==> boundary of square
• cut nodes => on grid
• cut mates => same length
![Page 17: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/17.jpg)
Boundary parametrization - cont.
• How is it done?– Allocate proportional length– redistribute
![Page 18: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/18.jpg)
Boundary parametrization - cont.
• Avoid degeneracies:– no full triangle on 1 side (split middle)– break edges over corners (add vertex)
![Page 19: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/19.jpg)
Interior parametrization
• Geometric stretch parametrization– minimize spacing on surface (parameters
distribute evenly)– P.V. Sander, J. Snyder, S.J. Gortler and H.
Hoppe. Texture mapping progressive meshes.
• Also serves as metric for mesh
![Page 20: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/20.jpg)
Cutting
![Page 21: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/21.jpg)
Initial Cut
• Fact:– genus 0 ==> 1 edge cut– genus g ==> 2g generator loops cut (that form
basis to surface’s fundamental group)
![Page 22: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/22.jpg)
Find the cut - retractor algorithm
• Remove a seed triangle
• Repeatedly remove an edge adjacent to exactly 1 triangle and the triangle
• Repeatedly remove ‘dandling’ edges (vertices with degree 1 + the edge)
• do not change boundary (if exists)
![Page 23: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/23.jpg)
Original mesh
![Page 24: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/24.jpg)
Remove seed triangle
![Page 25: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/25.jpg)
![Page 26: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/26.jpg)
Remove edges and triangles
![Page 27: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/27.jpg)
![Page 28: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/28.jpg)
Iterate
![Page 29: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/29.jpg)
![Page 30: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/30.jpg)
![Page 31: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/31.jpg)
![Page 32: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/32.jpg)
![Page 33: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/33.jpg)
![Page 34: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/34.jpg)
![Page 35: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/35.jpg)
![Page 36: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/36.jpg)
![Page 37: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/37.jpg)
![Page 38: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/38.jpg)
![Page 39: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/39.jpg)
![Page 40: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/40.jpg)
![Page 41: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/41.jpg)
![Page 42: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/42.jpg)
![Page 43: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/43.jpg)
![Page 44: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/44.jpg)
![Page 45: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/45.jpg)
![Page 46: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/46.jpg)
![Page 47: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/47.jpg)
![Page 48: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/48.jpg)
![Page 49: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/49.jpg)
![Page 50: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/50.jpg)
![Page 51: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/51.jpg)
![Page 52: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/52.jpg)
![Page 53: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/53.jpg)
![Page 54: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/54.jpg)
![Page 55: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/55.jpg)
![Page 56: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/56.jpg)
![Page 57: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/57.jpg)
Done: no triangles left (no faces)
![Page 58: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/58.jpg)
Remove dandling edges
![Page 59: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/59.jpg)
Iterate
![Page 60: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/60.jpg)
![Page 61: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/61.jpg)
![Page 62: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/62.jpg)
![Page 63: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/63.jpg)
![Page 64: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/64.jpg)
![Page 65: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/65.jpg)
![Page 66: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/66.jpg)
![Page 67: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/67.jpg)
![Page 68: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/68.jpg)
We are left with this:
![Page 69: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/69.jpg)
Initial Cut
• Result of the algorithm:– generator loops (genus > 0)– one vertex (genus 0)
• if 1 vertex : add 2 more (adjacent)
![Page 70: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/70.jpg)
Iterated Cut Augmentation
• Parametrize on unit circle (Floater)– M. Floater. Parametrization and smooth
approximation of surface triangulations.
• Find vertex with highest geometric stretch
• Find shortest path to boundary
• Add path to cut
• Stop when geometric stretch increases
• for genus 0: disregard original cut
![Page 71: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/71.jpg)
![Page 72: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/72.jpg)
![Page 73: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/73.jpg)
![Page 74: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/74.jpg)
![Page 75: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/75.jpg)
![Page 76: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/76.jpg)
![Page 77: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/77.jpg)
![Page 78: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/78.jpg)
![Page 79: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/79.jpg)
![Page 80: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/80.jpg)
Putting it all together
![Page 81: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/81.jpg)
![Page 82: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/82.jpg)
![Page 83: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/83.jpg)
![Page 84: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/84.jpg)
![Page 85: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/85.jpg)
![Page 86: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/86.jpg)
![Page 87: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/87.jpg)
Rendering
![Page 88: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/88.jpg)
![Page 89: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/89.jpg)
![Page 90: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/90.jpg)
![Page 91: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/91.jpg)
![Page 92: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/92.jpg)
![Page 93: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/93.jpg)
![Page 94: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/94.jpg)
Compression
![Page 95: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/95.jpg)
![Page 96: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/96.jpg)
![Page 97: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/97.jpg)
Again this Crack!
![Page 98: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/98.jpg)
Topological Sideband
• Table of nodes in • for each node, record:
– valence (k)– k coordinate pairs (s,t)
• Deduce ’ matching from the table
• “Fuse cut” using matchings
![Page 99: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/99.jpg)
![Page 100: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/100.jpg)
![Page 101: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/101.jpg)
![Page 102: Geometry Images Xiang Gu Harvard University Steven J. Gortler Harvard university Hugues Hoppe Microsoft Research Some slides taken from Hugues Hoppe](https://reader031.vdocuments.site/reader031/viewer/2022032709/56649eb45503460f94bbc620/html5/thumbnails/102.jpg)