Alex Alon
How do you connect smoothly between two points?
Cubic Bezier Splines
Smooth Connection Functions can be computed analytically using fundamental notions of theoretical physics
How do you connect smoothly between two points?
Small curvature radius(high curvature)
Large curvature radius(small curvature)
A cost function sums up the curvature along the curve
ds
The parameter determines the importance of having small curvature =1/r
The cost function can be determined for any given function y(x) describing the curve:
The Smooth Connection Function mimimizes the cost function for a given
and a set of boundary conditions
= 0 [ ]Euler-Lagrange
E. Noether
Examples for Smooth Connection functions
So how do Smooth Connection Functions relate to Shapes?
Many shapes can be approximated
Playing Join-the-dots
http://www.meridiangames.net/game.asp?GID=292
Contours can be defined as interpolation curves through a list of points
Different interpolation schemes lead to different shapes
B-spline interpolationExample for three points: P0 , P1 , P2
For each segment control points are chosen as Pi ± di
By construction this matches the slopes at each point.
From C2 continuity (equal curvatures) it follows that
P1 - 2(P1 - d1) + (P0 + d0) = (P2 - d2) - 2(P1 + d1) + P1 , d0 + 4d1 + d2 = P2 - P0 .
In general case we get banded 3-diagonals linear equations 4d1 + d2 = P2 - P0 - d0 d1 + 4d2 + d3 = P3 - P1 ... ... di + 4di+1 + di+2 = Pi+2 - Pi ... ... dn-2 + 4dn-1 = Pn - Pn-2 - dn
Which can be solved for di.
How to use SCFs for joining the dots?
P0 P1 P2
S01
S12
S = S01(α)+ S12(α)
The total cost is the sum of the cost from each segment.
αα
New optimization problem
𝑆 Σ (α )=∑𝑖=1
𝑛
𝑆𝑖 (α 𝑖 , α 𝑖+1)
The slope at each point affects the cost of the two neighboring segments
P0 P1 P2
S1
S2
P3S3
+ ...
Finding that minimizes corresponds to the best interpolation!
Connecting points by SCFsPro vs Con
Main advantage: • Single tunable parameter controls behavior of
all segment curves (while B-splines have only one solution)
Main disadvantage: • Minimization procedure is computationally
costly (while B-splines require only solving linear equations)
0 20 400.99
0.995
1
1.005
N=3
=1.3
0 20 400.9
0.95
1
=3
0 20 400.4
0.6
0.8
1=10
0 20 400.9
0.95
1
N=4
0 20 400.7
0.8
0.9
1
0 20 400
0.5
1
0 50 1000.7
0.8
0.9
1
N=10
Iteration0 50 1000
0.5
1
Iteration0 50 1000
0.5
1
Iteration
Connecting points by SCFsDoes it work?
Using ‘Gaussian adaptation’ for optimization reduces the cost function
Connecting points by SCFsDoes it work?
0 20 400.99
0.995
1
1.005
N=3
=1.3
0 20 400.9
0.95
1
=3
0 20 400.4
0.6
0.8
1=10
0 20 400.9
0.95
1
N=4
0 20 400.7
0.8
0.9
1
0 20 400
0.5
1
0 50 1000.7
0.8
0.9
1
N=10
Iteration0 50 1000
0.5
1
Iteration0 50 1000
0.5
1
Iteration
The curves evolve differently for each !
Connecting 10 points by SCFs
0 20 400.99
0.995
1
1.005N=3
=1.3
0 20 400.9
0.95
1=3
0 20 400.4
0.6
0.8
1=10
0 20 400.9
0.95
1
N=4
0 20 400.7
0.8
0.9
1
0 20 400
0.5
1
0 50 1000.7
0.8
0.9
1
N=10
Iteration0 50 1000
0.5
1
Iteration0 50 1000
0.5
1
Iteration
And the Octopus?
Not much adaptation with 95 points on the curve …
But there are some subtle differences
Higher promotes round shapes!
Less points introduce more freedom for adaptation
Less points introduce more freedom for adaptation
… and sometimes too much freedom!
Outlook
New ways to define shapes:• Optimal trajectories in transport• Curves in Computer Aided Geometrical Design
New way to analyze shapes• Assigning a typical ν to a shape or a part of it
(e.g. for handwritten letters)
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