© University of Wisconsin, CS559 Fall 2004 Last Time Parametric Curves (Splines) Hermite curves Bezier curves Brief mentions of rational curves, transforming between representation, subdivision for Bezier Homework 7 due Thursday 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Today Parametric surface patches Bsplines 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Parametric Surfaces Define points on the surface in terms of two parameters Simplest case: bilinear interpolation x(s,1) s P1,1 P0,1 x(s,t) t P0,0 s x(s,0) P1,0 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Tensor Product Surface Patches Defined over a rectangular domain Valid parameter values come from within a rectangular region in parameter space: 0s<1, 0t<1 Use a rectangular grid of control points to specify the surface 4 points in the bi-linear case on the previous slide, more in other cases Surface takes the form: For some functions Fi,s and Fj,t 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Bezier Patches As with Bezier curves, Bin(s) and Bjm(t) are the Bernstein polynomials of degree n and m respectively Most frequently, use n=m=3: cubic Bezier patch Need 4x4=16 control points, Pi,j 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Bezier Patches (2) Edge curves are Bezier curves Any curve of constant s or t is a Bezier curve One way to think about it: Each row of 4 control points defines a Bezier curve in s Evaluating each of these curves at the same s provides 4 virtual control points The virtual control points define a Bezier curve in t Evaluating this curve at t gives the point x(s,t) x(s,t) 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Properties of Bezier Patches Which vertices, if any, does the patch interpolate? Why? What can you say about the tangent plane at each corner? Why? Does the patch lie within the convex hull of its control vertices? 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Bezier Patch Matrix Form Note that the 3 matrices stay the same if the control points do not change The middle product can be pre-computed, leaving only: 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Bezier Patch Meshes A patch mesh is just many patches joined together along their edges Patches meet along complete edges Each patch must be a quadrilateral OK Not OK Not OK OK 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Bezier Mesh Continuity Just like curves, the control points must satisfy constraints to ensure parametric continuity How do we ensure C0 continuity along an edge? How do we ensure C1 continuity along an edge? How do we ensure C2 continuity along an edge? For geometric continuity, constraints are less rigid What can you say about the vertices around a corner if there must be C1 continuity at the corner point? 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Rendering Bezier Patches Option 1: Evaluate at fixed set of parameter values and join up with triangles Can’t use quadrilaterals because points may not be co-planar Ideal situation for triangle strips Advantage: Simple, and OpenGL has commands to do it for you Disadvantage: No easy way to control quality of appearance Option 2: Subdivision Allows control of error in the triangle approximation Extend 1D curve subdivision to do surfaces Similar to other subdivision schemes we’ve seen 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 A Potential Problem One (good) way to subdivide, is: If a control mesh is flat enough – draw it Else, subdivide into 4 sub-patches and recurse on each Problem: Neighboring patches may not be subdivided to the same level Cracks can appear because join edges have different control meshes This can be fixed by adding extra edges Crack 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Computing Normal Vectors The partial derivative in the s direction is one tangent vector The partial derivative in the t direction is another Take their cross product, and normalize, to get the surface normal vector 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Bezier Curve/Surface Problems To make a long continuous curve with Bezier segments requires using many segments Same for large surface Maintaining continuity requires constraints on the control point positions The user cannot arbitrarily move control vertices and automatically maintain continuity The constraints must be explicitly maintained It is not intuitive to have control points that are not free 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 B-splines B-splines automatically take care of continuity, with exactly one control vertex per curve segment Many types of B-splines: degree may be different (linear, quadratic, cubic,…) and they may be uniform or non-uniform We will only look closely at uniform B-splines With uniform B-splines, continuity is always one degree lower than the degree of each curve piece Linear B-splines have C0 continuity, cubic have C2, etc 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Uniform Cubic B-spline on [0,1) Four control points are required to define the curve for 0t<1 (t is the parameter) Not surprising for a cubic curve with 4 degrees of freedom The equation looks just like a Bezier curve, but with different basis functions Also called blending functions - they describe how to blend the control points to make the curve 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Basis Functions on [0,1) Does the curve interpolate its endpoints? Does it lie inside its convex hull? B1,4 B2,4 B0,4 B3,4 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Uniform Cubic B-spline on [0,1) The blending functions sum to one, and are positive everywhere The curve lies inside its convex hull The curve does not interpolate its endpoints Requires hacks or non-uniform B-splines There is also a matrix form for the curve: 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Demo 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Uniform B-spline at Arbitrary t The interval from an integer parameter value v to v+1 is essentially the same as the interval from 0 to 1 The parameter value is offset by v A different set of control points is needed To evaluate a uniform cubic B-spline at an arbitrary parameter value t: Find the greatest integer less than or equal to t: v = floor(t) Evaluate: Valid parameter range: 0t<n-3, where n is the number of control points 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Loops To create a loop, use control points from the start of the curve when computing values at the end of the curve: Any parameter value is now valid Although for numerical reasons it is sensible to keep it within a small multiple of n 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Demo 12/7/2004 © University of Wisconsin, CS559 Fall 2004

B-splines and Interpolation, Continuity Uniform B-splines do not interpolate control points, unless: You repeat a control point three times But then all derivatives also vanish (=0) at that point To do interpolation with non-zero derivatives you must use non-uniform B-splines with repeated knots To align tangents, use double control vertices Then tangent aligns similar to Bezier curve Uniform B-splines are automatically C2 All the blending functions are C2, so sum of blending functions is C2 Provides an alternate way to define blending functions To reduce continuity, must use non-uniform B-splines with repeated knots 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Rendering B-splines Same basic options as for Bezier curves Evaluate at a set of parameter values and join with lines Hard to know where to evaluate, and how pts to use Use a subdivision rule to break the curve into small pieces, and then join control points What is the subdivision rule for B-splines? Instead of subdivision, view splitting as refinement: Inserting additional control points, and knots, between the existing points Useful not just for rendering - also a user interface tool Defined for uniform and non-uniform B-splines by the Oslo algorithm 12/7/2004 © University of Wisconsin, CS559 Fall 2004

Non-Uniform B-Splines Uniform B-splines are a special case of B-splines Each blending function is the same A blending functions starts at t=-3, t=-2, t=-1,… Each blending function is non-zero for 4 units of the parameter (if cubic) Non-uniform B-splines Blending functions not all the same Blending functions can star or stop anywhere Fine control of continuity, without vanishing derivatives A deep investigation of B-splines is beyond the scope of this class 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 Rational Curves Each point is the ratio of two curves Just like homogeneous coordinates: NURBS: x(t), y(t), z(t) and w(t) are non-uniform B-splines Advantages: Perspective invariant, so can be evaluated in screen space Can perfectly represent conic sections: circles, ellipses, etc Piecewise cubic curves cannot do this 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 B-Spline Surfaces Defined just like Bezier surfaces: Continuity is automatically obtained everywhere BUT, the control points must be in a rectangular grid OK Not OK 12/7/2004 © University of Wisconsin, CS559 Fall 2004

© University of Wisconsin, CS559 Fall 2004 How to Choose a Spline Hermite curves are good for single segments where you know the parametric derivative or want easy control of it Bezier curves are good for single segments or patches where a user controls the points B-splines are good for large continuous curves and surfaces NURBS are the most general, and are good when that generality is useful, or when conic sections must be accurately represented (CAD) 12/7/2004 © University of Wisconsin, CS559 Fall 2004