NURBS Drawing Week 5, Lecture 10
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1 CS 430/536 Computer Graphics I NURBS Drawing Week 5, Lecture 10 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University Outline Conic Sections via NURBS Knot insertion algorithm The de Boor s algorithm for B-Splines for NURBS Oslo Algorithm Barycentric Coordinates Discussion of homework #3 1 Conic Sections via NURBS Conic Sections via NURBS Obtained via projection of the 3D parabola onto a plane Note: 3D Case: rational curve is a 4D object D Case: rational curve is a 3D object assign w to each control point 3 Define the curve with three control points Weights of first/last control points are 1 For center control point w<1 gives an ellipse w>1 gives a hyperbola w=1 gives a parabola Knot vector is {0.0, 0.0,0.0, 1.0, 1.0, 1.0} 4 Conic Sections via NURBS: A Circular Arc The two sides of the control polygon are of equal length The chord connecting the first and last control points meets each leg at an angle q equal to half the angular extent of the desired arc (for instance, 30 for a 60 arc The weight of the inner control point is equal to the cosine of q Knot vector is {0.0, 0.0, 0.0, 1.0, 1.0, 1.0} 5 Conic Sections via NURBS: A Circle What if we need an arc of >180? Idea: Use multiple 90 or 10 arcs B0 = B6 = - stitch them together with knots Example: 3 arcs of 10 { 0.866,0.5,1} B = {- 1.73, 1,0.5} B = { 0, 1,1 } B = { 1.73, 1,0.5} 1 - B 5 = { 0,,0.5} - 0,0,0,1,1,,,3,3,3 B = 4 { 0.866,0.5,1} 3-6 1
2 Conic Sections via NURBS Knot Insertion Example: 4 arcs of 90 { } B 3 = -1,1, { } B 1 = 1,1, Issue: More control points mean more control How do we add more points and keep same curve? { } B 5 = -1,-1, { } B 7 = 1,-1, 7 8 Knot Insertion Basic Approach Decide where we d like to tweak the curve Add a new knot Find affected d-1 control points Replace it with d new control points Example: New knot at u=.6 9 Knot Insertion Given: n+1 control points (P 0, P 1,..., P n, a knot vector of m+1 knots U = { u 0, u1,..., u m } and a degree d B- spline curve C(u. Insert a new knot t into the knot vector without changing the shape of the curve. If t lies in knot span [u k, u k+1, only the basis functions for (P k, P k-d are non-zero. Find d new control points Q k on edge P k-1 P k, Q k-1 on edge P k- P k-1,..., and Q k-d+1 on edge P k-d P k-d+1 All other control points are unchanged. Note that d-1 control points of the original control polyline are removed and replaced with d new control points. 10 See Knot Insertion Algorithm Create new control point Q j = (1 a j P j-1 + a j P j Where a is defined as a j = t - u j u j +d - u j Properties of Knot Insertion Increasing the multiplicity of a knot decreases the number of non-zero basis functions at this knot At a knot of multiplicity d, there will be only one non-zero basis function Corresponding point on the curve p(u is affected by exactly one control point p i In fact p(u is p i! See
3 Generalization of de Casteljau's algorithm It provides a fast and numerically stable way for finding a point on a B-spline curve Observation: if a knot u is inserted d times to a B-spline, then p(u is the point on the curve. Idea: We just simply insert u d times and the last point is p(u! See
4 De Boor s Algorithm If u lies in [u k,u k+1 and u!= u k, let h = d If u = u k and u k is a knot of multiplicity s, let h = d s Copy the affected control points p k-s, p k-s-1,..., p k-d+1, p k-d to a new array and rename them as p k-s,0, p k-s-1,0,..., p k-d+1,0 for r := 1 to h do for i := k-d+r to k-s do { Let a i,r = (u - u i / ( u i+d-r+1 - u i Let p i,r = (1 - a i,r p i-1,r-1 + a i,r p i,r-1 } p k-s,d-s is the point p(u. De Boor s Algorithm (cont for u := 0 to u max do { for r := 1 to h do for i := k-p+r to k-s do { Let a i,r = (u - u i / ( u i+p-r+1 - u i Let p i,r = (1 - a i,r p i-1,r-1 + a i,r p i,r-1 } p k-s,p-s is the point p(u. } 19 1 Example of de Boor s Algorithm Degree 3 B-spline curve (i.e., d = 3 Defined by seven control points p 0,..., p 6 And knot vector: u = 0.4 a 4,1 = (u - u 4 / ( u u 4 = 0. a 3,1 = (u - u 3 / ( u u 3 = 8/15 = 0.53 a,1 = (u - u / ( u +3 - u = 0.8 p 4,1 = (1 - a 4,1 p 3,0 + a 4,1 p 4,0 = 0.8p 3,0 + 0.p 4,0 p 3,1 = (1 - a 3,1 p,0 + a 3,1 p 3,0 = 0.47p, p 3,0 p,1 = (1 - a,1 p 1,0 + a,1 p,0 = 0.p 1, p,0 a 4, = (u - u 4 / (u u 4 = 0.3 a 3, = (u - u 3 / (u u 3 = 0.8 p 4, = (1 - a 4, p 3,1 + a 4, p 4,1 = 0.7p 3, p 4,1 p 3, = (1 - a 3, p,1 + a 3, p 3,1 = 0.p, p 3,1 a 4,3 = (u - u 4 / (u u 4 = 0.6 p 4,3 = (1 - a 4,3 p 3, + a 4,3 p 4, = 0.4p 3, + 0.6p 4, u 0 = u 1 = u = u 3 u 4 u 5 u 6 u 7 = u 8 = u 9 = u Similar but Different De Casteljau's: Dividing points are computed with a pair of numbers (1 - u and u that never change Can be used for curve subdivision Uses all control points De Boor's These pairs of numbers are different and depend on the column number and control point number Intermediate control points are not sufficient d+1 affected control points are involved in the computation 3 De Boor s: Curves Oslo Algorithm A subdivision algorithm for B-splines, the basic idea: Take the curve with m+1 control points P 0 to P m Insert a knot in any point (0.5 maybe? As a result you will have new points P k and P k Take curves with m+1 control points P 0 P k, P k P m-1 and P 1 P k, P k P m Apply procedure recursively on each part 5 Animated by Max Drexel University 6 4
5 Oslo Algorithm 7 Barycentric Coordinates By Ceva's Theorem: For any point K inside the triangle ABC Consider the existence of masses w A, w B, and w C, placed at the vertices of the triangle Their center of gravity (barycenter will coincide with the point K. A, w A August Ferdinand Moebius ( defined (187 w A, w B, and w C as the barycentric coordinates of K K A B + w C C B, w B K 8 C, w C Recorded from: Properties of Barycentric Coordinates Not unique Can be generalized to negative masses Can be made unique by setting w A + w C = 1 w A = 0 for points on BC w B = 0 for points on AC w C = 0 on AB B, w B A, w A K C, w C Area Ta: (b-p x (c-p / Calculating the Weights Calculating Weights (cont. Given vertices A, B, C and Centroid K What are the weights, w A, w B, w C? x K x A x B + w C x C Solve for w A and w B w A = (x B (y C - y K - (x C - x K (y B (x A (y B - (x B (y A y K y A y B + w C y C Substitute w C = 1 - w A - w B x K x A x B + (1- w A - w B x C y K y A y B + (1- w A - w B y C w B = (x A (y C - y K - (x C - x K (y A (x B (y A - (x A (y B w C = 1 - w A - w B
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