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College of Computer and Information Science, Northeastern UniversityApril 12, CS G140 Graduate Computer Graphics Prof. Harriet Fell Spring 2007 Lecture 6 – February 26, 2007

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College of Computer and Information Science, Northeastern UniversityApril 12, Today’s Topics Bezier Curves and Splines Parametric Bicubic Surfaces Quadrics

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College of Computer and Information Science, Northeastern UniversityApril 12, Curves A curve is the continuous image of an interval in n-space. Implicit f(x, y) = 0 x 2 + y 2 – R 2 = 0 Parametric (x(t), y(t)) = P(t) P(t) = tA + (1-t)B A B Generative proc (x, y)

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College of Computer and Information Science, Northeastern UniversityApril 12, Curve Fitting We want a curve that passes through control points. How do we create a good curve? What makes a good curve?

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College of Computer and Information Science, Northeastern UniversityApril 12, Axis Independence If we rotate the set of control points, we should get the rotated curve.

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Local Control

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College of Computer and Information Science, Northeastern UniversityApril 12, Variation Diminishing Never crosses a straight line more than the polygon crosses it.

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College of Computer and Information Science, Northeastern UniversityApril 12, Continuity C 0 continuity C 1 continuity C 2 continuity G 2 continuity Not C 2 continuity

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College of Computer and Information Science, Northeastern UniversityApril 12, How do we Fit Curves? Lagrange Interpolating Polynomial from mathworld The Lagrange interpolating polynomial is the polynomial of degree n-1 that passes through the n points, (x 1, y 1 ), (x 2, y 2 ), …, (x n, y n ), and is given by

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College of Computer and Information Science, Northeastern UniversityApril 12, Example 1

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College of Computer and Information Science, Northeastern UniversityApril 12, Polynomial Fit P(x) = -.5x(x-2)(x-3)(x-4)

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College of Computer and Information Science, Northeastern UniversityApril 12, Piecewise Fit P a (x) = x (x ) 0 x 1.5 P b (x) = 5.4 x (x ) 1.5 x 2 P c (x) = 0 2 x 4

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College of Computer and Information Science, Northeastern UniversityApril 12, Spline Curves

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College of Computer and Information Science, Northeastern UniversityApril 12, Splines and Spline Ducks Marine Drafting Weights

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College of Computer and Information Science, Northeastern UniversityApril 12, Drawing Spline Today (esc)

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College of Computer and Information Science, Northeastern UniversityApril 12, Hermite Cubics p q Dp Dq P(t) = at 3 + bt 2 +ct +d P(0) = p P(1) = q P'(0) = Dp P'(1) = Dq

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College of Computer and Information Science, Northeastern UniversityApril 12, Hermite Coefficients P(t) = at 3 + bt 2 +ct +d P(0) = p P(1) = q P'(0) = Dp P'(1) = Dq For each coordinate, we have 4 linear equations in 4 unknowns

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College of Computer and Information Science, Northeastern UniversityApril 12, Boundary Constraint Matrix

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College of Computer and Information Science, Northeastern UniversityApril 12, Hermite Matrix MHMH GHGH

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College of Computer and Information Science, Northeastern UniversityApril 12, Hermite Blending Functions

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College of Computer and Information Science, Northeastern UniversityApril 12, Splines of Hermite Cubics a C 1 spline of Hermite curves a G 1 but not C 1 spline of Hermite curves The vectors shown are 1/3 the length of the tangent vectors.

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College of Computer and Information Science, Northeastern UniversityApril 12, Computing the Tangent Vectors Catmull-Rom Spline p1 p1 p2 p2 p3 p3 p4 p4 p 5 P(0) = p 3 P(1) = p 4 P'(0) = ½ (p 4 - p 2 ) P'(1) = ½ (p 5 - p 3 )

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College of Computer and Information Science, Northeastern UniversityApril 12, Cardinal Spline The Catmull-Rom spline P(0) = p 3 P(1) = p 4 P'(0) = ½ (p 4 - p 2 ) P'(1) = ½ (p 5 - p 3 ) is a special case of the Cardinal spline P(0) = p 3 P(1) = p 4 P'(0) = (1 - t)(p 4 - p 2 ) P'(1) = (1 - t)(p 5 - p 3 ) 0 ≤ t ≤ 1 is the tension.

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College of Computer and Information Science, Northeastern UniversityApril 12, Drawing Hermite Cubics How many points should we draw? Will the points be evenly distributed if we use a constant increment on t ? We actually draw Bezier cubics.

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College of Computer and Information Science, Northeastern UniversityApril 12, General Bezier Curves

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College of Computer and Information Science, Northeastern UniversityApril 12, Low Order Bezier Curves p0p0 n = 0 b 0,0 (t) = 1 B(t) = p 0 b 0,0 (t) = p 0 0 ≤ t ≤ 1 p0p0 n = 1 b 0,1 (t) = 1 - t b 1,1 (t) = t B(t) = (1 - t) p 0 + t p 1 0 ≤ t ≤ 1 p1p1 p0p0 n = 2b 0,2 (t) = (1 - t) 2 b 1,2 (t) = 2t (1 - t) b 2,2 (t) = t 2 B(t) = (1 - t) 2 p 0 + 2t (1 - t)p 1 + t 2 p 2 0 ≤ t ≤ 1 p1p1 p2p2

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College of Computer and Information Science, Northeastern UniversityApril 12, Bezier Curves Bezier Arch p q r s n = 3b 0,3 (t) = (1 - t) 3 b 1,3 (t) = 3t (1 - t) 2 b 2,3 (t) = 3t 2 (1 - t) b 2,3 (t) = t 3 B(t) = (1 - t) 3 p + 3t (1 - t) 2 q + 3t 2 (1 - t)r + t 3 s 0 ≤ t ≤ 1

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College of Computer and Information Science, Northeastern UniversityApril 12, Bezier Matrix MBMB B(t) = (1 - t) 3 p + 3t (1 - t) 2 q + 3t 2 (1 - t)r + t 3 s 0 ≤ t ≤ 1 B(t) = a t 3 + bt 2 + ct + d 0 ≤ t ≤ 1 GBGB

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College of Computer and Information Science, Northeastern UniversityApril 12, Geometry Vector

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College of Computer and Information Science, Northeastern UniversityApril 12, Properties of Bezier Curves

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College of Computer and Information Science, Northeastern UniversityApril 12, Geometry of Bezier Arches p q r s Pick a t between 0 and 1 and go t of the way along each edge. B(t)B(t) Join the endpoints and do it again.

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College of Computer and Information Science, Northeastern UniversityApril 12, Geometry of Bezier Arches p q r s We only use t = 1/2. pqrs = B(1/2) pq qr rs pqr qrs

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drawArch(P, Q, R, S){ if (ArchSize(P, Q, R, S) <=.5 ) Dot(P); else{ PQ = (P + Q)/2; QR = (Q + R)/2; RS = (R + S)/2; PQR = (PQ + QR)/2; QRS = (QR + RS)/2; PQRS = (PQR + QRS)/2 drawArch(P, PQ, PQR, PQRS); drawArch(PQRS, QRS, RS, S); }

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College of Computer and Information Science, Northeastern UniversityApril 12, Putting it All Together Bezier Archer and Catmull-Rom Splines

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College of Computer and Information Science, Northeastern UniversityApril 12, Time for a Break

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College of Computer and Information Science, Northeastern UniversityApril 12, Surface Patch P(u,v) 0 u, v 1 A patch is the continuous image of a square in n-space.

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College of Computer and Information Science, Northeastern UniversityApril 12, Bezier Patch Geometry

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College of Computer and Information Science, Northeastern UniversityApril 12, Bezier Patch P 00 P 10 P 01 P 02 P 03 P 11 P 12 P 13 P 20 P 30 P 21 P 22 P 23 P 31 P 32 P 33

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College of Computer and Information Science, Northeastern UniversityApril 12, Bezier Patch Computation

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College of Computer and Information Science, Northeastern UniversityApril 12, Bezier Patch P 00 P 10 P 01 P 02 P 03 P 11 P 12 P 13 P 20 P 30 P 21 P 22 P 23 P 31 P 32 P 33 Q(0,v) Q(1,v) Q(u,0) Q(u,1)

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College of Computer and Information Science, Northeastern UniversityApril 12, Bezier Patch P 00 P 10 P 01 P 02 P 03 P 11 P 12 P 13 P 20 P 30 P 21 P 22 P 23 P 31 P 32 P 33 Q u (0,0) = 3(P 10 – P 00 ) Q v (0,0) = 3(P 01 – P 00 ) Q uv (0,0) = 9(P 00 – P 01 – P 10 + P 11 )

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College of Computer and Information Science, Northeastern UniversityApril 12, Properties of Bezier Surfaces A Bézier patch transforms in the same way as its control points under all affine transformations All u = constant and v = constant lines in the patch are Bézier curves. A Bézier patch lies completely within the convex hull of its control points. The corner points in the patch are the four corner control points. A Bézier surface does not in general pass through its other control points.

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College of Computer and Information Science, Northeastern UniversityApril 12, Rendering Bezier Patches with a mesh Chris Bently - Rendering Bezier Patches 3. Use these derived points as the control points for new Bezier curves running in the v direction 1. Consider each row of control points as defining 4 separate Bezier curves: Q 0 (u) … Q 3 (u) 2. For some value of u, say 0.1, for each Bezier curve, calculate Q 0 (u) … Q 3 (u). 4. Generate edges and polygons from grid of surface points.

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College of Computer and Information Science, Northeastern UniversityApril 12, Subdividing Bezier Patch P 00 P 10 P 01 P 02 P 03 P 11 P 12 P 13 P 20 P 30 P 21 P 22 P 23 P 31 P 32 P 33 Subdivide until the control points are coplanar.

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College of Computer and Information Science, Northeastern UniversityApril 12, Blending Bezier Patches

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Teapot Data double teapot_data[][] = { { , 0.00, 30.00, , , 30.00, , , 30.00, 0.00, , 30.00, , 0.00, 12.00, , , 12.00, , , 12.00, 0.00, , 12.00, , 0.00, 3.00, , , 3.00, , , 3.00, 0.00, , 3.00, , 0.00, 0.00, , , 0.00, , , 0.00, 0.00, , 0.00, }, …

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College of Computer and Information Science, Northeastern UniversityApril 12, Bezier Patch Continuity If these sets of control points are colinear, the surface will have G 1 continuity.

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College of Computer and Information Science, Northeastern UniversityApril 12, Quadric Surfaces ellipsoidelliptic cylinder

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College of Computer and Information Science, Northeastern UniversityApril 12, Quadric Surfaces 1-sheet hyperboloid 2-sheet hyperboloid

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College of Computer and Information Science, Northeastern UniversityApril 12, Quadric Surfaces cones

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College of Computer and Information Science, Northeastern UniversityApril 12, elliptic parabaloid hyperbolic parabaloid

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College of Computer and Information Science, Northeastern UniversityApril 12, Quadric Surfaces hyperbolic cylinder parabolic cylinder

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College of Computer and Information Science, Northeastern UniversityApril 12, General Quadrics in General Position

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College of Computer and Information Science, Northeastern UniversityApril 12, General Quadric Equation Ray Equations

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College of Computer and Information Science, Northeastern UniversityApril 12, Ray Quadric Intersection Quadratic Coefficients A = a*xd*xd + b*yd*yd + c*zd*zd + 2[d*xd*yd + e*yd*zd + f*xd*zd B = 2*[a*x0*xd + b*y0*yd + c*z0*zd + d*(x0*yd + xd*y0 ) + e*(y0*zd + yd*z0 ) + f*(x0*zd + xd*z0 ) + g*xd + h*yd + j*zd] C = a*x0*x0 + b*y0*y0 + c*z0*z0 + 2*[d*x0*y0 + e*y0*z0 + f*x0*z0 + g*x0 + h*y0 + j*z0] + k

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College of Computer and Information Science, Northeastern UniversityApril 12, Quadric Normals Normalize N and change its sign if necessary.

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College of Computer and Information Science, Northeastern UniversityApril 12, MyCylinders

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College of Computer and Information Science, Northeastern UniversityApril 12, Student Images

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College of Computer and Information Science, Northeastern UniversityApril 12, Student Images

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