Jack Pinches

INFO410 & INFO350 S2 2015

INFORMATIONSCIENCE

Geometric Modelling 2

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Previously in Geometric Modelling…

Geometry Information Topological Information Geometric Model Techniques:

Wireframe Modelling Surface Modelling Solid Modelling (CSG Example)

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How to model curves and free-form surfaces?

Primitives: Lines Triangles

http://hessan.annahid.com/game-development-days/day-6/

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Outline

Previous material Parametric Curves

Bézier curves B – Splines NURBS

Free-Form Surface Modelling

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Modelling Curves

Manufacturing Car Aircraft Boats

Medical Imaging Molecular Modelling Computational fluid dynamics Physical simulation

in applied mechanics

http://www.cis.upenn.edu/~jean/tabcont.pdf

Oceanography Shape reconstruction Weather analysis Computer animation Architecture Art

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Modelling Curves: Parametric Curves

Bezier B – Spline NURBS Beta – Spline Cubic Splines

Parametric Curves: parametric equations of a curve express the coordinates as functions of a variable – called a parameter. t often denotes the parameter.

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Modelling Curves: Bézier Curves

1960s research: Paul de Casteljau and Pierre Bézier Curve defined by:

P for 0 ≤ t ≤ 1

Where are the Bernstein base functions, of degree n

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Modelling Curves: Bézier Curves

Bézier curve is tangent to the first and last sections of the control polygon

Curve order is equal to number of vertices of control polygon

Curve is contained in the convex hull of control points

https://en.wikipedia.org/wiki/B%C3%A9zier_curve#/media/File:Bezier_curve.svg

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Modelling Curves: Bézier Curves

Problem: Lack of local control

P

For each , will contribute.

Example:25 control points, move P2, entire curve is modified.

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Modelling Curves: Bézier Curves

Piecewise Bézier Curve: Join small curves More local control Called “B-Splines”

http://graphics.cs.ucdavis.edu/~joy/GeometricModelingLectures/Unit-2/Why/Why.mp4

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Modelling Curves: B – Spline

Basis Spline 1946: Curve fitting for experimental data 1963: Used in CAD systems by J. Ferguson (Boeing)

𝑃 (𝑡)=∑𝑖=0

𝑛

𝑃 𝑖𝑁 𝑖 ,𝑘(𝑡) are points of the control polygonare the B-Spline base functionsan order k, and a set of knots {, }

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Modelling Curves: B – Spline

Multiple Bézier curves Low degree to reduce complexity Similar properties Increased local control Curve doesn’t have to go to

end points

https://en.wikipedia.org/wiki/B-spline#/media/File:B-spline_curve.svg

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Modelling Curves: NURBS

Non - Uniform Rational B-spline Commonly used in CAD, CAM, CAE 3D modelling and animation software Control points: directly connected, connected by a link

https://en.wikipedia.org/wiki/File:Spline01.gif

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Modelling Curves: NURBS Curve Definition

NURBS Curve defined by four things: Degree Control points Knot vector Evaluation rule

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Modelling Curves: NURBS Curve Definition

Degree: Positive whole number, often 1-5 Linear, quadratic, cubic, quantic Order of curve: degree + 1

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Modelling Curves: NURBS Curve Definition

Control Points: At least degree + 1 number of points Change curve shape Points are weighted Shape formed: Control Polygon

https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline#/media/File:NURBstatic.svg

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Modelling Curves: NURBS Curve Definition

Knot Vector: The knot vector is a sequence of parameter values that determines where and how the control points affect the NURBS curve.

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Modelling Curves: NURBS Curve Definition

Knots (Knot Vector): List of (degree+N-1) numbers, N is number of control points Several conditions

Example:3 degree NURBS curve with 11 control points0,0,0,1,2,2,2,3,7,7,9,9,9 0,0,0,1,2,2,2,2,7,7,9,9,9

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Modelling Curves: NURBS Curve Definition

Evaluation Rule: Mathematical formula, takes a number and assigns a point

“You can think of the evaluation rule as a black box that eats a parameter and produces a point location. The degree, knots, and control points determine how the black box works.”

https://www.rhino3d.com/nurbs

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Modelling Curves:

The development of curve modelling is progressive. The new methods were often designed to over come weaknesses present in other methods.

B-Splines created to achieve more control of curve shape than Bézier is an example. (Slide 12)

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Freeform Surface Modelling

https://en.wikipedia.org/wiki/Freeform_surface_modelling#/media/File:Freeform1.gif

Modelling in CAD: Create curves, mesh surface Create surface, manipulate control points

Model car bodies, boat hulls, aircraft Most CAD software uses NURBS

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Freeform Surface Modelling

Continuity between surfaces Require continuous rate

of change High quality, higher degree Context dependent

http://www.doschdesign.com/images2/Red-D3D-Cars2011-AutoCAD.jpg

Potential Exam Questions

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Exam Questions

How do Bézier curves and B-Splines differ? Including reasons why B-Splines was developed.

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ResourcesGeneral: http://

www.mar.ist.utl.pt/mventura/Projecto-Navios-I/EN/SD-1.5.2-Geometric%20Modeling.pdf http://fagweb.hin.no/ala/geomod/ERBSbok_main.pdf NURBS: http://www.mactech.com/articles/develop/issue_25/schneider.html https://www.rhino3d.com/nurbs Bezier and B-Splines: http://graphics.cs.ucdavis.edu/~joy/GeometricModelingLectures/ Free-Form Surface Modelling: http://www.mif.vu.lt/cs2/courses/surfaces.pdf Title page image: https://commons.wikimedia.org/wiki/File:Bicubic_Patches.png