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Time - Space Sandra Bies Marc van Kreveld maps from triangulations

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Showing travel times between places in a country by train

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50 km

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60 min From Amsterdam: Rotterdam: 63 min.; s-Hertogenbosch: 59 min.; Arnhem: 70 min.

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Can we deform the map so that any time-contour from Amsterdam becomes a circle?

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6090120150

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The idea of using triangulations Triangulations to define map deformations (Saalfeld, SoCG 1987) Triangulations for contiguous-area cartograms (Edelsbrunner and Waupotitsch, SoCG 1995) Spring embedder: Kocmoud and House, 1998

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everything inside a triangle moves according to a linear interpolation of the movement of the vertices Barycentric coordinates

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Triangulations for Time-Space Maps Question 1: Can we extend the deformation naturally outside the convex hull of the point set?

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Triangulations for Time-Space Maps Question 1: Can we extend the deformation naturally outside the convex hull of the point set? Yes, we can make a bounding box with vertices that are stationary. The deformation dies out towards the outside.

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Triangulations for Time-Space Maps Question 2: Is there a triangulation that is good for the original point set and the moved point set?

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Triangulations for Time-Space Maps Question 2: Is there a triangulation that is good for the original point set and the moved point set? Yes, we can use a radial triangulation. No moving point will cause a triangle to collapse.

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Triangulations for Time-Space Maps Question 2: Is there a triangulation that is good for the original point set and the moved point set? Yes, we can use a radial triangulation. No moving point will cause a triangle to collapse.

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Triangulations for Time-Space Maps Question 3: Does this give a nice deformation?

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Triangulations for Time-Space Maps Question 3: Does this give a nice deformation? No, it has many artifacts. Static radial

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Triangulations for Time-Space Maps Question 4: Is there a different triangulation that does not have collapses and gives a nice deformation?

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Triangulations for Time-Space Maps Question 4: Is there a different triangulation that does not have collapses and gives a nice deformation? Usually. For a given input we can flip toward a triangulation that is maximally Delaunay but does not have collapses.

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Triangulations for Time-Space Maps Question 4: Is there a different triangulation that does not have collapses and gives a nice deformation? Usually. For a given input we can flip toward a triangulation that is maximally Delaunay but does not have collapses.

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Triangulations for Time-Space Maps Question 5: Does this give a nice deformation?

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Triangulations for Time-Space Maps Question 5: Does this give a nice deformation? Much nicer. But there are still artifacts. Static hybrid

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Triangulations for Time-Space Maps Question 6: Can we somehow change the triangulation when moving the points from geographic to time location?

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Triangulations for Time-Space Maps Question 6: Can we somehow change the triangulation when moving the points from geographic to time location? This is a great idea! We maintain the Delaunay triangulation for moving points, and flip when the in-circle predicate is violated.

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Triangulations for Time-Space Maps Question 7: Does this give a nice deformation?

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Triangulations for Time-Space Maps Question 7: Does this give a nice deformation? Dynamic Delaunay Usually very nice. Occasionally there may be visually unpleasant parts.

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t=0 t=1 t=0 t=1 t=0 t=1 t=0; t=1

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t=0 t=1 t=0 t=1 t=0; t=1

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Comparison Static radialStatic hybridDynamic Delaunay compute deformation ( n cities ) O(n log n)O(n 2 log n)O(n 3 log n) (*) compute deformed map ( m vertices ) O(m log n) O(m n 3 log n) (*) distance deformation 58.8 km27.3 km7.0 km angle deformation 37.0 o 26.3 o 19.6 o 25 km 25 + X km 60 o 60 o ± Y

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Conclusions Time-space maps Triangulations make deformations Maintaining the Delaunay triangulation during point movement gives good deformations

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1cs542g-term1-2007 Notes. 2 Meshing goals Robust: doesn’t fail on reasonable geometry Efficient: as few triangles as possible Easy to refine later.

1cs542g-term1-2007 Notes. 2 Meshing goals Robust: doesn’t fail on reasonable geometry Efficient: as few triangles as possible Easy to refine later.

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