2d/3d shape manipulation, 3d printing

Discrete Differential GeometryPlanar Curves

2D/3D Shape Manipulation,3D Printing

March 13, 2013

Slides from Olga Sorkine, Eitan Grinspun

Olga Sorkine-Hornung #

Differential Geometry – Motivation

● Describe and analyze geometric characteristics of shapes e.g. how smooth?

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Differential Geometry – Motivation

● Describe and analyze geometric characteristics of shapes e.g. how smooth? how shapes deform

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Differential Geometry Basics

● Geometry of manifolds● Things that can be discovered by

local observation: point + neighborhood

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manifold point

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manifold point

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manifold pointcontinuous 1-1 mapping

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non-manifold point

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non-manifold pointx

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continuous 1-1 mapping

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continuous 1-1 mapping

u

v

If a sufficiently smooth mapping can be constructed, we can look at its first and second derivativesTangents, normals, curvaturesDistances, curve angles, topology

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Planar Curves

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Curves

● 2D:

● must be continuous

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● Equal pace of the parameter along the curve

● len (p(t1), p(t2)) = |t1 – t2|

Arc Length Parameterization

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Secant

• A line through two points on the curve.

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Secant

● A line through two points on the curve.

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Tangent

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● The limiting secant as the two points come together.


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

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Tangent, normal, radius of curvature

p

r

Osculating circle“best fitting circle”

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Circle of Curvature

● Consider the circle passing through three points on the curve…

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Circle of Curvature

• …the limiting circle as three points come together.

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Radius of Curvature, r

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Radius of Curvature, r = 1/

Curvature

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Signed Curvature

+

–

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• Clockwise vs counterclockwisetraversal along curve.

#

Gauss map

● Point on curve maps to point on unit circle.


Curvature = change in normal direction

curve Gauss map curve Gauss map

● Absolute curvature (assuming arc length t)

● Parameter-free view: via the Gauss map

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Curvature Normal

• Assume t is arc-length parameter)(ˆ)( tt np

[Kobbelt and Schröder]

p(t)

)(ˆ tn

p(t)

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Curvature Normal – Examples

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Turning Number, k

• Number of orbits in Gaussian image.

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• For a closed curve, the integral of curvature is an integer multiple of 2.

• Question: How to find curvatureof circle using this formula?

Turning Number Theorem

+2

–2

+4

0

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Discrete Planar Curves

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Discrete Planar Curves

● Piecewise linear curves● Not smooth at vertices● Can’t take derivatives

● Generalize notions fromthe smooth world forthe discrete case!

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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

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Tangents, Normals

● For vertices, we have many options

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● Can choose to average the adjacent edge normals

Tangents, Normals

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Tangents, Normals

● Weight by edge lengths

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Inscribed Polygon, p

• Connection between discrete and smooth

• Finite number of verticeseach lying on the curve,connected by straight edges.

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p1

p2

p3

p4

The Length of a Discrete Curve

• Sum of edge lengths

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The Length of a Continuous Curve

• Length of longest of all inscribed polygons.

March 13, 2013 40sup = “supremum”. Equivalent to maximum if maximum exists.


• …or take limit over a refinement sequence

h = max edge length

The Length of a Continuous Curve

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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

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normal changes at vertices – record the turning angle!

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normal changes at vertices – record the turning angle!

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same as the turning anglebetween the edges


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● Zero along the edges● Turning angle at the vertices

= the change in normal direction

1, 2 > 0, 3 < 0

1 2

3


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Total Signed Curvature

• Sum of turning angles 1 2

3

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#

Discrete Gauss Map

● Edges map to points, vertices map to arcs.

#

Discrete Gauss Map

● Turning number well-defined for discrete curves.


Discrete Turning Number Theorem

● For a closed curve, the total signed curvature is an integer multiple of 2. proof: sum of exterior angles

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Discrete Curvature – Integrated Quantity!

● Integrated over a local area associated with a vertex

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1 2

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1 2A1

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1 2A1

A2

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1 2A1

A2

The vertex areas Ai form a covering of the curve.They are pairwise disjoint (except endpoints).

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Structure Preservation

• Arbitrary discrete curve– total signed curvature obeys

discrete turning number theorem– even coarse mesh (curve)– which continuous theorems to preserve?

• that depends on the application…

discrete analogueof continuous theorem

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Convergence

• Consider refinement sequence– length of inscribed polygon approaches

length of smooth curve – in general, discrete measure approaches

continuous analogue– which refinement sequence?

• depends on discrete operator• pathological sequences may exist

– in what sense does the operator converge? (point-wise, L2; linear, quadratic)

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Recap

ConvergenceStructure-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.

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Thank You

March 13, 2013

2d/3d shape manipulation, 3d printing

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