jack pinches info410 & info350 s2 2015 information science geometric modelling 2
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Jack Pinches
INFO410 & INFO350 S2 2015
INFORMATIONSCIENCE
Geometric Modelling 2
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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 2INFORMATION SCIENCE
Previously in Geometric Modelling…
Geometry Information Topological Information Geometric Model Techniques:
Wireframe Modelling Surface Modelling Solid Modelling (CSG Example)
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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 3INFORMATION SCIENCE
How to model curves and free-form surfaces?
Primitives: Lines Triangles
http://hessan.annahid.com/game-development-days/day-6/
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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 4INFORMATION SCIENCE
Outline
Previous material Parametric Curves
Bézier curves B – Splines NURBS
Free-Form Surface Modelling
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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 5INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 6INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 7INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 8INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 9INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 10INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 11INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 12INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 13INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 14INFORMATION SCIENCE
Modelling Curves: NURBS Curve Definition
NURBS Curve defined by four things: Degree Control points Knot vector Evaluation rule
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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 15INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 16INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 17INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 18INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 19INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 20INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 21INFORMATION SCIENCE
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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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 22INFORMATION SCIENCE
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
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Potential Exam Questions
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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 24INFORMATION SCIENCE
Exam Questions
How do Bézier curves and B-Splines differ? Including reasons why B-Splines was developed.
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INFO410 S2 2015 GEOMETRIC MODELLING 2 SLIDE 25INFORMATION SCIENCE
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