2D/3D Shape Manipulation, 3D Printing

2D/3D Shape Manipulation,3D Printing Discrete Differential Geometry Planar Curves Slides from Olga Sorkine, EitanGrinspun

Differential Geometry – Motivation • Describe and analyze geometric characteristics of shapes • e.g. how smooth? Olga Sorkine-Hornung

Differential Geometry – Motivation • Describe and analyze geometric characteristics of shapes • e.g. how smooth? • how shapes deform Olga Sorkine-Hornung

Differential Geometry Basics • Geometry of manifolds • Things that can be discovered by local observation: point + neighborhood Olga Sorkine-Hornung

Differential Geometry Basics • Geometry of manifolds • Things that can be discovered by local observation: point + neighborhood manifold point Olga Sorkine-Hornung

Differential Geometry Basics • Geometry of manifolds • Things that can be discovered by local observation: point + neighborhood manifold point continuous 1-1 mapping Olga Sorkine-Hornung

Differential Geometry Basics • Geometry of manifolds • Things that can be discovered by local observation: point + neighborhood manifold point continuous 1-1 mapping non-manifold point Olga Sorkine-Hornung

Differential Geometry Basics • Geometry of manifolds • Things that can be discovered by local observation: point + neighborhood manifold point continuous 1-1 mapping non-manifold point x Olga Sorkine-Hornung

Differential Geometry Basics • Geometry of manifolds • Things that can be discovered by local observation: point + neighborhood Olga Sorkine-Hornung

Differential Geometry Basics • Geometry of manifolds • Things that can be discovered by local observation: point + neighborhood continuous 1-1 mapping Olga Sorkine-Hornung

Differential Geometry Basics • Geometry of manifolds • Things that can be discovered by local observation: point + neighborhood If a sufficiently smooth mapping can be constructed, we can look at its first and second derivatives continuous 1-1 mapping v Tangents, normals, curvatures Distances, curve angles, topology u Olga Sorkine-Hornung

Planar Curves Olga Sorkine-Hornung

Curves • 2D: • must be continuous Olga Sorkine-Hornung

Arc Length Parameterization • Equal pace of the parameter along the curve • len(p(t1), p(t2)) = |t1 – t2| Olga Sorkine-Hornung

Secant • A line through two points on the curve. Olga Sorkine-Hornung

Tangent • The limiting secant as the two points come together. Olga Sorkine-Hornung

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 Olga Sorkine-Hornung

Tangent, normal, radius of curvature Osculating circle“best fitting circle” r p Olga Sorkine-Hornung

Circle of Curvature • Consider the circle passing through three points on the curve… Olga Sorkine-Hornung

Circle of Curvature • …the limiting circle as three points come together. Olga Sorkine-Hornung

Radius of Curvature, r Olga Sorkine-Hornung

Radius of Curvature, r= 1/ Curvature Olga Sorkine-Hornung

Signed Curvature • Clockwise vs counterclockwisetraversal along curve. + – Olga Sorkine-Hornung

Gauss map • Point on curve maps to point on unit circle.

Curvature = change in normal direction • Absolute curvature (assuming arc length t) • Parameter-free view: via the Gauss map curve Gauss map curve Gauss map Olga Sorkine-Hornung

Curvature Normal • Assume t is arc-length parameter p(t) p(t) [Kobbeltand Schröder] Olga Sorkine-Hornung

Curvature Normal – Examples Olga Sorkine-Hornung

Turning Number, k • Number of orbits in Gaussian image. Olga Sorkine-Hornung

Turning Number Theorem +2 • For a closed curve, the integral of curvature is an integer multiple of 2. • Question: How to find curvatureof circle using this formula? –2 +4 0 Olga Sorkine-Hornung

Discrete Planar Curves Olga Sorkine-Hornung

Discrete Planar Curves • Piecewise linear curves • Not smooth at vertices • Can’t take derivatives • Generalize notions fromthe smooth world forthe discrete case! Olga Sorkine-Hornung

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 Olga Sorkine-Hornung

Tangents, Normals • For vertices, we have many options Olga Sorkine-Hornung

Tangents, Normals • Can choose to average the adjacent edge normals Olga Sorkine-Hornung

Tangents, Normals • Weight by edge lengths Olga Sorkine-Hornung

Inscribed Polygon, p • Connection between discrete and smooth • Finite number of verticeseach lying on the curve,connected by straight edges. Olga Sorkine-Hornung

The Length of a Discrete Curve • Sum of edge lengths p3 p2 p4 p1 Olga Sorkine-Hornung

The Length of a Continuous Curve • Length of longest of all inscribed polygons. sup = “supremum”. Equivalent to maximum if maximum exists. Olga Sorkine-Hornung

The Length of a Continuous Curve • …or take limit over a refinement sequence h = max edge length Olga Sorkine-Hornung

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 Olga Sorkine-Hornung

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! Olga Sorkine-Hornung

Curvature of a Discrete Curve • Curvature is the change in normal direction as we travel along the curve same as the turning anglebetween the edges Olga Sorkine-Hornung

Curvature of a Discrete Curve • Zero along the edges • Turning angle at the vertices= the change in normal direction 1 2 1, 2 > 0, 3 < 0 3 Olga Sorkine-Hornung

Total Signed Curvature • Sum of turning angles 1 2 3 Olga Sorkine-Hornung

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 Olga Sorkine-Hornung

2D/3D Shape Manipulation, 3D Printing

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