6. Complex Curvature (mostly) Indirect Control of Shape Non Uniform - PowerPoint PPT Presentation

6. Complex Curvature (mostly) Indirect Control of Shape

Non Uniform Ra?onal Basis 1 Splines aka … NURBS 1 A varia?on on a Bezier curve

Parametric representa?ons

Parametric representa?ons • Approximate line with polynomial equa?on y = a n x n + a (n-1) x (n-1) + … + a 1 x + c • Parameterize in terms of a parameter “t” y = a n t n + a (n-1) t (n-1) + … + a 1 t + c over t=(0, 1) • Polynomial degree (largest exponent) determines kind of curve you can represent. Degree 1 Degree 2 Degree 3

Parametric representa?ons • Control points determine parameters • Complex curves can be pieced together • Three levels of “con?nuity” between pieces – C 0 : Posi:onal – C 1 : Slope of tangent – C 2 : Radius of curvature The Golden Spiral Just how con:nuous is it?

A 4-curve Bezier playground : hTp://quicksilver.be.washington.edu/java/bezierPlayground/

Control Point Vocabulary Edit points are points the line passes through, either preserving con?nuity of slope (knots) or with an op?onal change of direc?on (kinks) Control points are the off-curve points that guide or control the curve. “Weights” are numbers describing the “pull” of any one control point on the curve.

Parametric representa?ons • End-points, tangency and closed curves (“seams” & “deformable” rebuilds) • Higher-degree polynomials can exactly match lower-order polynomials, but not vice versa. • Control points “pull” curve towards their loca?on with a “weight” that is editable. • “kinks” allow corners (C 0 con?nuity only) if desired

Con?nuity ( MakePeriodic )

NURB Anatomy

Edi?ng NURB Curves & Surfaces • Rebuild (going nuclear w/ new controls) – Generate new control point grid – Adjust NURBS degree (1..11, but best if <= 3) • ChangeDegree (changing NURBS degree) • MakePeriodic (comple?ng the circle) • InsertKink (changing con?nuity requirements) • InsertControlPoint (changing control points) • Weight (changing control point influence)

Transforma2ons of NURBS Taper Bend Twist Twist and Bend

(flat shading shows lots of polys!)

Curves to isocurves (& meshes!) 2. Lofed surface (+control 2A. Mesh from point & weight edits) NURBS control polygon 2C. Iso-curves 1. Original extracted from curves NURBS object 2B. Mesh from NURBS object 2Ca.Piped isocurves

A Gazebo Roof

Mesh Manipula?on Control-point-edi?ng, Transforma?on, Cage Edi?ng

Mesh > Box (divide & conquer!)

Control Points On [f10]

Manipula2ng many points

More Transforma?ons: scale

More Transforma?ons: Sof Move

Iso- surfaces (aka Meta-forms) NOT “meat-balls” (not a Rhino feature)

(form•Z) Meta-balls

6. Complex Curvature - fini -

6. Complex Curvature (mostly) Indirect Control of Shape Non Uniform - PowerPoint PPT Presentation

6. Complex Curvature (mostly) Indirect Control of Shape Non Uniform Ra?onal Basis 1 Splines aka NURBS 1 A varia?on on a Bezier curve Parametric representa?ons Parametric representa?ons Approximate line with polynomial equa?on y = a n x n

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6. Complex Curvature (mostly) Indirect Control of Shape Non Uniform - PowerPoint PPT Presentation

6. Complex Curvature (mostly) Indirect Control of Shape Non Uniform Ra?onal Basis 1 Splines aka NURBS 1 A varia?on on a Bezier curve Parametric representa?ons Parametric representa?ons Approximate line with polynomial equa?on y = a n x n

Complex structures and zero-curvature equations for -models Dmitri Bykov Max-Planck-Institut

The Total Curvature and Betti Numbers of Complex Projective Manifolds Convex, Discrete and

Holomorphic Sectional Curvature of Projectivized Vector Bundles over Compact Complex Manifolds

Ghost effect by curvature Providence, November 2011 M&amp;MOCS MATHEMATICS AND MECHANICS OF

Prescribing Gaussian curvature on compact surfaces and geodesic curvature on its boundary David

The conformal curvature flow Xingwang Xu Department of Mathematics, National University of

RIT n. a point on a curve at which the curvature which the curvature Inflection Inflection

JUST THE MATHS SLIDES NUMBER 11.3 DIFFERENTIATION APPLICATIONS 3 (Curvature) by

Oliviers Ricci curvature and applications Sunhyuk Lim Ohio State University lim.991@osu.edu

Thin Shells Plates are naturally flat y &amp; Curvature Based &amp; Curvature-Based Energy

Accelerating Curvature Estimate in 3D Seismic Data Using GPGPU Joner Duarte

Optimal transportation, curvature flows, and related a priori estimates Alexander Kolesnikov

Curvature line parametrized surfaces and orthogonal coordinate systems Discretization with

Drying of complex fluids: Deformations and fractures L. Pauchard FAST Singularities inversion

What do we know about intrinsic metric curvature of affine hypersurfaces? Udo Simon Leuven

On Singularity Formation Under Mean Curvature Flow I.M.Sigal Toronto Joint work with Wenbin

Keyframe animation Process of keyframing Keyframe interpolation Hermite and

Bzier Curves CPSC 453 Fall 2018 Sonny Chan Todays Outline Quadratic Bzier curves

BEZIER SURFACES 1 OUTLINE Quadratic Bezier Surfaces Cubic Bezier Surfaces 2 DE

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ARCS circular segments r c ARCS circular segments angles are calculated clockwise

Drawing Metro Maps using B ezier Curves Martin Fink Lehrstuhl f ur Informatik I

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Assignment #2 Which is something you may wish to do So You Want to Write a Key Framing System

Ghost effect by curvature Providence, November 2011 M&MOCS MATHEMATICS AND MECHANICS OF

Thin Shells Plates are naturally flat y & Curvature Based & Curvature-Based Energy