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

▶
6 complex curvature
SMART_READER_LITE
LIVE PREVIEW

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

Sep 09, 2022 266 likes •549 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

slide-1
SLIDE 1
  • 6. Complex Curvature

(mostly) Indirect Control of Shape

slide-2
SLIDE 2

Non Uniform Ra?onal Basis1 Splines

aka … NURBS

1 A varia?on on a Bezier curve

slide-3
SLIDE 3

Parametric representa?ons

slide-4
SLIDE 4

Parametric representa?ons

  • Approximate line with polynomial equa?on

y = anxn + a(n-1)x(n-1) + … + a1x + c

  • Parameterize in terms of a parameter “t”

y = antn + a(n-1)t(n-1) + … + a1t + c over t=(0, 1)

  • Polynomial degree (largest exponent)

determines kind of curve you can represent.

Degree 1 Degree 2 Degree 3

slide-5
SLIDE 5

Parametric representa?ons

  • Control points determine parameters
  • Complex curves can be pieced together
  • Three levels of “con?nuity” between pieces

– C0: Posi:onal – C1: Slope of tangent – C2: Radius of curvature

The Golden Spiral Just how con:nuous is it?

slide-6
SLIDE 6

A 4-curve Bezier playground:

hTp://quicksilver.be.washington.edu/java/bezierPlayground/

slide-7
SLIDE 7

Control Point Vocabulary

Edit points are points the line passes through, either preserving con?nuity of slope (knots)

  • r 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.

slide-8
SLIDE 8

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 (C0 con?nuity only) if

desired

slide-9
SLIDE 9

Con?nuity (MakePeriodic)

slide-10
SLIDE 10

NURB Anatomy

slide-11
SLIDE 11

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)
slide-12
SLIDE 12

Transforma2ons of NURBS

Twist Twist and Bend Bend Taper

slide-13
SLIDE 13

(flat shading shows lots of polys!)

slide-14
SLIDE 14

Curves to isocurves (& meshes!)

  • 1. Original

curves

  • 2. Lofed surface (+control

point & weight edits)

  • 2C. Iso-curves

extracted from NURBS object 2Ca.Piped isocurves

  • 2B. Mesh from NURBS object
  • 2A. Mesh from

NURBS control polygon

slide-15
SLIDE 15

A Gazebo Roof

slide-16
SLIDE 16

Mesh Manipula?on

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

slide-17
SLIDE 17

Mesh > Box (divide & conquer!)

slide-18
SLIDE 18

Control Points On [f10]

slide-19
SLIDE 19

Manipula2ng many points

slide-20
SLIDE 20

More Transforma?ons: scale

slide-21
SLIDE 21

More Transforma?ons: Sof Move

slide-22
SLIDE 22

More Transforma?ons: Sof Move

slide-23
SLIDE 23

Iso-surfaces (aka Meta-forms)

NOT “meat-balls” (not a Rhino feature)

slide-24
SLIDE 24

(form•Z) Meta-balls

slide-25
SLIDE 25

(form•Z) Meta-balls

slide-26
SLIDE 26

(form•Z) Meta-balls

slide-27
SLIDE 27
  • 6. Complex Curvature
  • fini -