Planar Curves 2D3D Shape Manipulation 3D Printing March 13 2013 Slides from Olga Sorkine Eitan Grinspun Differential Geometry Motivation Describe and analyze geometric characteristics of shapes ID: 351743
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Slide1
Discrete Differential GeometryPlanar Curves
2D/3D Shape Manipulation,3D Printing
March 13, 2013
Slides from Olga
Sorkine
,
Eitan
GrinspunSlide2
Differential Geometry – Motivation
Describe and analyze geometric characteristics of shapese.g. how smooth?
March 13, 2013
Olga Sorkine-Hornung
2Slide3
Differential Geometry – Motivation
Describe and analyze geometric characteristics of shapese.g. how smooth?
how shapes deform
March 13, 2013
Olga Sorkine-Hornung
3Slide4
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
March 13, 2013
Olga Sorkine-Hornung
4Slide5
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
manifold point
March 13, 2013
Olga Sorkine-Hornung
5Slide6
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
manifold point
March 13, 2013
Olga Sorkine-Hornung
6Slide7
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
manifold point
continuous 1-1 mapping
March 13, 2013
Olga Sorkine-Hornung
7Slide8
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
manifold point
continuous 1-1 mapping
non-manifold point
March 13, 2013
Olga Sorkine-Hornung
8Slide9
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
manifold point
continuous 1-1 mapping
non-manifold point
x
March 13, 2013
Olga Sorkine-Hornung
9Slide10
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
March 13, 2013
Olga Sorkine-Hornung
10Slide11
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
continuous 1-1 mapping
March 13, 2013
Olga Sorkine-Hornung
11Slide12
Differential Geometry Basics
Geometry of manifoldsThings that can be discovered by local observation: point + neighborhood
continuous 1-1 mapping
u
v
If a sufficiently smooth mapping can be constructed, we can look at its first and second derivatives
Tangents,
normals
, curvatures
Distances, curve angles, topology
March 13, 2013
Olga Sorkine-Hornung
12Slide13
Planar Curves
March 13, 2013
Olga Sorkine-Hornung
13Slide14
Curves
2D: must be continuous
March 13, 2013
Olga Sorkine-Hornung
14Slide15
Equal pace of the parameter along the curve
len
(p
(
t
1
),
p
(
t2
)) = |
t
1
–
t
2
|
Arc Length Parameterization
March 13, 2013
Olga Sorkine-Hornung
15Slide16
Secant
A line through two points on the curve.
March 13, 2013
Olga Sorkine-Hornung
16Slide17
Secant
A line through two points on the curve.
March 13, 2013
Olga Sorkine-Hornung
17Slide18
Tangent
March 13, 2013
Olga Sorkine-Hornung
18
The limiting secant as the two points come together.Slide19
Secant and Tangent – Parametric Form
Secant:
p
(
t
) –
p
(
s
)Tangent:
p
(
t
) = (
x
(
t
),
y
(
t
), …)T
If
t is arc-length:
||
p
(
t
)|| = 1
March 13, 2013
Olga Sorkine-Hornung
19Slide20
Tangent, normal, radius of curvature
p
r
Osculating circle
“best fitting circle”
March 13, 2013
Olga Sorkine-Hornung
20Slide21
Circle of
Curvature
Consider the circle passing through three points on the curve…
March 13, 2013
Olga Sorkine-Hornung
21Slide22
Circle of Curvature
…the limiting circle as three points come together.
March 13, 2013
Olga Sorkine-Hornung
22Slide23
Radius of Curvature,
r
March 13, 2013
Olga Sorkine-Hornung
23Slide24
Radius of Curvature,
r =
1/
Curvature
March 13, 2013
Olga Sorkine-Hornung
24Slide25
Signed Curvature
+
–
March 13, 2013
Olga Sorkine-Hornung
25
Clockwise
vs
counterclockwise
traversal
along curve.Slide26
Gauss map
Point on curve maps to point on unit circle.Slide27
Curvature = change in normal direction
curve
Gauss map
curve
Gauss map
Absolute curvature (assuming arc length
t
)
Parameter-free view: via the Gauss map
March 13, 2013
Olga Sorkine-Hornung
27Slide28
Curvature Normal
Assume t is arc-length parameter
[
Kobbelt
and
Schr
öder
]
p
(
t
)
p
(
t
)
March 13, 2013
Olga Sorkine-Hornung
28Slide29
Curvature Normal – Examples
March 13, 2013
Olga Sorkine-Hornung
29Slide30
Turning Number,
k
Number of orbits in Gaussian image.
March 13, 2013
Olga Sorkine-Hornung
30Slide31
For a closed curve,
the integral of curvature is an integer multiple of 2
.Question: How to find curvature
of circle using this formula?
Turning
Number Theorem
+2
–2
+4
0
March 13, 2013
Olga Sorkine-Hornung
31Slide32
Discrete Planar Curves
March 13, 2013
Olga Sorkine-Hornung
32Slide33
Discrete Planar Curves
Piecewise linear curvesNot smooth at verticesCan’t take derivativesGeneralize notions from
the smooth world forthe discrete case!
March 13, 2013
Olga Sorkine-Hornung
33Slide34
Tangents, Normals
For any point on the edge, the tangent is simply the unit vector along the edge and the normal is the perpendicular vector
March 13, 2013
Olga Sorkine-Hornung
34Slide35
Tangents, Normals
For vertices, we have many options
March 13, 2013
Olga Sorkine-Hornung
35Slide36
Can choose to average the adjacent edge normals
Tangents,
Normals
March 13, 2013
Olga Sorkine-Hornung
36Slide37
Tangents, Normals
Weight by edge lengths
March 13, 2013
Olga Sorkine-Hornung
37Slide38
Inscribed Polygon,
p
Connection
between discrete and smooth
Finite
number of vertices
each lying on the curve,connected by straight edges.
March 13, 2013
Olga Sorkine-Hornung
38Slide39
p
1
p
2
p
3
p
4
The
Length
of a
Discrete Curve
Sum of edge
lengths
March 13, 2013
Olga Sorkine-Hornung
39Slide40
The
Length of a Continuous Curve
Length of longest of all inscribed polygons.
March 13, 2013
Olga Sorkine-Hornung
40
sup = “
supremum
”. Equivalent to maximum if maximum exists.Slide41
…or take limit over
a refinement sequence
h
= max edge length
The
Length
of a
Continuous Curve
March 13, 2013
Olga Sorkine-Hornung
41Slide42
Curvature of a Discrete Curve
Curvature is the change in normal direction as we travel along the curve
no change along each edge –
curvature is zero along edges
March 13, 2013
Olga Sorkine-Hornung
42Slide43
Curvature of a Discrete Curve
Curvature is the change in normal direction as we travel along the curve
normal changes at vertices –
record the turning angle!
March 13, 2013
Olga Sorkine-Hornung
43Slide44
Curvature of a Discrete Curve
Curvature is the change in normal direction as we travel along the curve
normal changes at vertices –
record the turning angle!
March 13, 2013
Olga Sorkine-Hornung
44Slide45
Curvature is the change in normal direction as we travel along the curve
same as the turning angle
between the edges
Curvature of a Discrete Curve
March 13, 2013
Olga Sorkine-Hornung
45Slide46
Zero along the edgesTurning angle at the vertices
= the change in normal direction
1, 2 > 0,
3
< 0
1
2
3
Curvature of a Discrete Curve
March 13, 2013
Olga Sorkine-Hornung
46Slide47
Total Signed Curvature
Sum of turning angles
1
2
3
March 13, 2013
Olga Sorkine-Hornung
47Slide48
Discrete
Gauss
Map
Edges map to points, vertices map to arcs.Slide49
Discrete
Gauss
Map
Turning number well-defined for discrete curves.Slide50
Discrete Turning Number Theorem
For a closed curve, the total signed curvature is an integer multiple of
2
.
proof: sum of exterior angles
March 13, 2013
Olga Sorkine-Hornung
50Slide51
Discrete Curvature – Integrated Quantity!
Integrated over a local area associated with a vertex
March 13, 2013
1
2
Olga Sorkine-Hornung
51Slide52
Discrete Curvature – Integrated Quantity!
Integrated over a local area associated with a vertex
March 13, 2013
1
2
A
1
Olga Sorkine-Hornung
52Slide53
Discrete Curvature – Integrated Quantity!
Integrated over a local area associated with a vertex
March 13, 2013
1
2
A
1
A
2
Olga Sorkine-Hornung
53Slide54
Discrete Curvature – Integrated Quantity!
Integrated over a local area associated with a vertex
March 13, 2013
1
2
A
1
A
2
The vertex areas
A
i
form a covering of the curve.
They are pairwise disjoint (except endpoints).
Olga Sorkine-Hornung
54Slide55
Structure Preservation
Arbitrary discrete curvetotal signed curvature obeysdiscrete turning number theoremeven coarse
mesh (curve)which continuous theorems to preserve?that depends on the
application…
discrete analogue
of continuous theorem
March 13, 2013
Olga Sorkine-Hornung
55Slide56
Convergence
Consider refinement sequencelength of inscribed polygon approaches length of smooth curve in general, discrete measure approaches continuous analoguewhich refinement sequence?
depends on discrete operatorpathological sequences may existin what sense does the operator converge?
(
point-wise, L
2
; linear
, quadratic)
March 13, 2013
Olga Sorkine-Hornung
56Slide57
Recap
Convergence
Structure-
preservation
In the limit of a refinement sequence,
discrete measures of length and curvature
agree
with continuous measures.
For an arbitrary (even coarse) discrete curve,
the discrete measure of curvature
obeys
the discrete turning number theorem.
March 13, 2013
Olga Sorkine-Hornung
57Slide58
Thank You
March 13, 2013