Geometric Modelling 2 INFO410 & INFO350 S2 2015 Jack Pinches INFORMATION SCIENCE

Previously in Geometric Modelling… Geometry Information Topological Information Geometric Model Techniques: Wireframe Modelling Surface Modelling Solid Modelling (CSG Example)

How to model curves and free-form surfaces? Primitives: Lines Triangles Primitives are mathematically exact, resolution independent, unaffected by changes in position/scale/orientation. Could use primitives for arbitrarily shaped objects, but at the cost of the above properties. http://hessan.annahid.com/game-development-days/day-6/

Outline Previous material Parametric Curves Bézier curves B – Splines NURBS Free-Form Surface Modelling

Computational fluid dynamics Physical simulation in applied mechanics Modelling Curves Manufacturing Car Aircraft Boats Medical Imaging Molecular Modelling Computational fluid dynamics Physical simulation in applied mechanics Oceanography Shape reconstruction Weather analysis Computer animation Architecture Art http://www.cis.upenn.edu/~jean/tabcont.pdf

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.

Modelling Curves: Bézier Curves 1960s research: Paul de Casteljau and Pierre Bézier Curve defined by: P(𝑡)= 𝑖=0 𝑛 𝑃 𝑖 𝐵 𝑖, 𝑛 (𝑡) for 0 ≤ t ≤ 1 Where 𝐵 𝑛, 𝑖 are the Bernstein base functions, of degree n For each t in put into the equation, every Pi will contribute. e.g. have 25 points, move P1, the entire curve will be modified. Bernestein polynomial known since 1912 Renault and Citeron.

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

Modelling Curves: Bézier Curves Problem: Lack of local control P(𝑡)= 𝑖=0 𝑛 𝑃 𝑖 𝐵 𝑖, 𝑛 (𝑡) For each 𝑡, 𝑃 𝑖 will contribute. Example: 25 control points, move P2, entire curve is modified. For each t in put into the equation, every Pi will contribute. e.g. have 25 points, move P1, the entire curve will be modified. Poses a problem in CAD/animation Bernstein problem: high number of points can be computationally intensive.

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

Modelling Curves: B – Spline Basis Spline 1946: Curve fitting for experimental data 1963: Used in CAD systems by J. Ferguson (Boeing) 𝑃 𝑖 are points of the control polygon 𝑁 𝑖,𝑘 are the B-Spline base functions an order k, and a set of knots { 𝑡 0 , 𝑡 1 ,…, 𝑡 𝑛+𝑘 } 𝑃(𝑡)= 𝑖=0 𝑛 𝑃 𝑖 𝑁 𝑖,𝑘 (𝑡) Knots tell us the parameter values where the curves pieced together are joined.

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

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

Modelling Curves: NURBS Curve Definition NURBS Curve defined by four things: Degree Control points Knot vector Evaluation rule NURBS curves are generalization on Bsplines/Bezier curves.

Modelling Curves: NURBS Curve Definition Degree: Positive whole number, often 1-5 Linear, quadratic, cubic, quantic Order of curve: degree + 1 1 = linear 2 = circles (quadratic) 3-5 are most free form curves Order = degree+1

Modelling Curves: NURBS Curve Definition Control Points: At least degree + 1 number of points Change curve shape Points are weighted Shape formed: Control Polygon Connecting lines to make the points easier to see/their relationship to curve. Desirable property: control points make very localised changes, little/no effect on parts of the curve further away. https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline#/media/File:NURBstatic.svg

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. This slide displays the answer I gave in the Q&A for the presentation.

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 points 0,0,0,1,2,2,2,3,7,7,9,9,9 0,0,0,1,2,2,2,2,7,7,9,9,9 Vector has nothing to do with direction as usually meant by vector. Conditions: can’t repeat a number more times than the degree - Can keep the numbers the same, or get larger as go down list

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

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)

Freeform Surface Modelling Modelling in CAD: Create curves, mesh surface Create surface, manipulate control points Model car bodies, boat hulls, aircraft Most CAD software uses NURBS https://en.wikipedia.org/wiki/Freeform_surface_modelling#/media/File:Freeform1.gif

Freeform Surface Modelling Continuity between surfaces Require continuous rate of change High quality, higher degree Context dependent Continuity: how smoothly they connect to one another. http://www.doschdesign.com/images2/Red-D3D-Cars2011-AutoCAD.jpg

Potential Exam Questions

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

Resources General: 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