Time - Space

Sandra Bies Marc van Kreveld

maps from triangulations

Showing travel times between places in a country by train

50 km

60 min

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

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

6090120150

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

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

Barycentric coordinates

Triangulations for Time-Space Maps

Question 1: Can we extend the deformation naturally outside the convex hull of the point set?

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.

Triangulations for Time-Space Maps

Question 2: Is there a triangulation that is “good” for the original point set and the moved point set?

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.

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.

Triangulations for Time-Space Maps

Question 3: Does this give a nice deformation?

Triangulations for Time-Space Maps

Question 3: Does this give a nice deformation?

No, it has many artifacts.

Static radial

Triangulations for Time-Space Maps

Question 4: Is there a different triangulation that does not have collapses and gives a nice deformation?

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.

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.

Triangulations for Time-Space Maps

Question 5: Does this give a nice deformation?

Triangulations for Time-Space Maps

Question 5: Does this give a nice deformation?

Much nicer. But there are still artifacts.

Static hybrid

Triangulations for Time-Space Maps

Question 6: Can we somehow change the triangulation when moving the points from geographic to time location?

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.

Triangulations for Time-Space Maps

Question 7: Does this give a nice deformation?

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.

t=0

t=1

t=0t=1

t=0

t=0t=1

t=1

t=0; t=1

t=0

t=0

t=0t=1

t=1

t=1

t=0t=1

t=0; t=1

ComparisonStatic radial Static hybrid Dynamic Delaunay

compute deformation ( n cities )

O(n log n) O(n2 log n) O(n3 log n) (*)

compute deformed map ( m vertices )

O(m log n) O(m log n) O(m n3 log n) (*)

distance deformation 58.8 km 27.3 km 7.0 kmangle deformation 37.0 o 26.3 o 19.6 o

25 km 25 + X km

60o 60o ± Y

Conclusions

• Time-space maps• Triangulations make deformations• Maintaining the Delaunay triangulation during

point movement gives good deformations