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

Sep 09, 2022 •266 likes •545 views

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

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
More Transforma?ons: Sof Move
Iso- surfaces (aka Meta-forms) NOT “meat-balls” (not a Rhino feature)
(form•Z) Meta-balls
(form•Z) Meta-balls
(form•Z) Meta-balls
6. Complex Curvature - fini -

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