1 Dr. Scott Schaefer Smooth Curves

2/109 Smooth Curves Interpolation  Interpolation through Linear Algebra  Lagrange interpolation Bezier curves B-spline curves

3/109 Smooth Curves How do we create smooth curves?

4/109 Smooth Curves How do we create smooth curves? Parametric curves with polynomials

5/109 Smooth Curves Controlling the shape of the curve

6/109 Smooth Curves Controlling the shape of the curve

7/109 Smooth Curves Controlling the shape of the curve

8/109 Smooth Curves Controlling the shape of the curve

9/109 Smooth Curves Controlling the shape of the curve

10/109 Smooth Curves Controlling the shape of the curve

11/109 Smooth Curves Controlling the shape of the curve

12/109 Smooth Curves Controlling the shape of the curve

13/109 Smooth Curves Controlling the shape of the curve

14/109 Smooth Curves Controlling the shape of the curve Power-basis coefficients not intuitive for controlling shape of curve!!!

15/109 Interpolation Find a polynomial that passes through specified values

16/109 Interpolation Find a polynomial that passes through specified values

17/109 Interpolation Find a polynomial that passes through specified values

18/109 Interpolation Find a polynomial that passes through specified values

19/109 Interpolation Find a polynomial that passes through specified values Intuitive control of curve using “control points”!!!

20/109 Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

21/109 Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

22/109 Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

23/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

24/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

25/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

26/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

27/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

28/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

29/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

30/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

31/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

32/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

33/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

34/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

35/109 Lagrange Interpolation Identical to matrix method but uses a geometric construction

36/109 Uniform Lagrange Interpolation Base Case  Linear interpolation between two points Inductive Step  Assume we have points y i, …, y n+1+i  Build interpolating polynomials f(t), g(t) of degree n for y i, …, y n+i and y i+1, …, y n+1+i   h(t) interpolates all points y i, …, y n+1+i and is of degree n+1  Moreover, h(t) is unique

37/109 Lagrange Evaluation Pseudocode Lagrange(pts, i, t) n = length(pts)-2 if n is 0 return pts[0](i+1-t)+pts[1](t-i) f=Lagrange(pts without last element, i, t) g=Lagrange(pts without first element, i+1, t) return ((n+1+i-t)f+(t-i)g)/(n+1)

38/109 Problems with Interpolation

39/109 Problems with Interpolation

40/109 Bezier Curves Polynomial curves that seek to approximate rather than to interpolate

41/109 Bernstein Polynomials Degree 1: (1-t), t Degree 2: (1-t) 2, 2(1-t)t, t 2 Degree 3: (1-t) 3, 3(1-t) 2 t, 3(1-t)t 2, t 3

42/109 Bernstein Polynomials Degree 1: (1-t), t Degree 2: (1-t) 2, 2(1-t)t, t 2 Degree 3: (1-t) 3, 3(1-t) 2 t, 3(1-t)t 2, t 3 Degree 4: (1-t) 4, 4(1-t) 3 t, 6(1-t) 2 t 2, 4(1-t)t 3,t 4

43/109 Bernstein Polynomials Degree 1: (1-t), t Degree 2: (1-t) 2, 2(1-t)t, t 2 Degree 3: (1-t) 3, 3(1-t) 2 t, 3(1-t)t 2, t 3 Degree 4: (1-t) 4, 4(1-t) 3 t, 6(1-t) 2 t 2, 4(1-t)t 3,t 4 Degree 5: (1-t) 5, 5(1-t) 4 t, 10(1-t) 3 t 2, 10(1-t) 2 t 3,5(1-t)t 4,t 5 … Degree n: for

44/109 Bezier Curves

45/109 Bezier Curves

46/109 Bezier Curves

47/109 Bezier Curve Properties Interpolate end-points

48/109 Bezier Curve Properties Interpolate end-points

49/109 Bezier Curve Properties Interpolate end-points

50/109 Bezier Curve Properties Interpolate end-points

51/109 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

52/109 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

53/109 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

54/109 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge Curve lies within the convex hull of the control points

55/109 Convex Hull The smallest convex set containing all points p i

56/109 Convex Hull The smallest convex set containing all points p i

57/109 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge Curve lies within the convex hull of the control points Bezier Lagrange

58/109 Pyramid Algorithms for Bezier Curves Polynomials aren’t pretty Is there an easier way to evaluate the equation of a Bezier curve?

59/109 Pyramid Algorithms for Bezier Curves

60/109 Pyramid Algorithms for Bezier Curves

61/109 Pyramid Algorithms for Bezier Curves

62/109 Bezier Evaluation Pseudocode Bezier(pts, t) if ( length(pts) is 1) return pts[0] newPts={} For ( i = 0, i < length(pts) – 1, i++ ) Add to newPts (1-t)pts[i] + t pts[i+1] return Bezier ( newPts, t )

63/109 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

64/109 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

65/109 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

66/109 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

67/109 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

68/109 Subdividing Bezier Curves Control points for left curve!!!

69/109 Subdividing Bezier Curves Control points for right curve!!!

70/109 Subdividing Bezier Curves

71/109 Adaptive Rendering of Bezier Curves If control polygon is close to a line, draw the control polygon If not, subdivide and recur on subdivided pieces

72/109 Adaptive Rendering of Bezier Curves If control polygon is close to a line, draw the control polygon If not, subdivide and recur on subdivided pieces

73/109 Adaptive Rendering of Bezier Curves If control polygon is close to a line, draw the control polygon If not, subdivide and recur on subdivided pieces

74/109 Adaptive Rendering of Bezier Curves If control polygon is close to a line, draw the control polygon If not, subdivide and recur on subdivided pieces

75/109 Adaptive Rendering of Bezier Curves If control polygon is close to a line, draw the control polygon If not, subdivide and recur on subdivided pieces

76/109 Adaptive Rendering of Bezier Curves If control polygon is close to a line, draw the control polygon If not, subdivide and recur on subdivided pieces

77/109 Applications: Intersection Given two Bezier curves, determine if and where they intersect

78/109 Applications: Intersection Check if convex hulls intersect If not, return no intersection If both convex hulls can be approximated with a straight line, intersect lines and return intersection Otherwise subdivide and recur on subdivided pieces

79/109 Applications: Intersection

80/109 Applications: Intersection

81/109 Applications: Intersection

82/109 Applications: Intersection

83/109 Applications: Intersection

84/109 Applications: Intersection

85/109 Applications: Intersection

86/109 Applications: Intersection

87/109 Applications: Intersection

88/109 Applications: Intersection

89/109 Applications: Intersection

90/109 Application: Font Rendering

91/109 Application: Font Rendering

92/109 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve

93/109 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve

94/109 B-spline Curves Not a single polynomial, but lots of polynomials that meet together smoothly Local control

95/109 B-spline Curves Not a single polynomial, but lots of polynomials that meet together smoothly Local control

96/109 Rendering B-spline Curves Lane-Reisenfeld subdivision algorithm Linearly subdivide the curve by inserting the midpoint on each edge Perform averaging by replacing each edge by its midpoint d times Subdivide the curve repeatedly

97/109 Rendering B-spline Curves

98/109 Rendering B-spline Curves

99/109 Rendering B-spline Curves

100/109 Rendering B-spline Curves

101/109 Rendering B-spline Curves

102/109 Rendering B-spline Curves

103/109 Rendering B-spline Curves

104/109 Rendering B-spline Curves

105/109 Rendering B-spline Curves

106/109 B-spline Properties Curve lies within convex hull of control points Variation Diminishing: Curve wiggles no more than control polygon Influence of one control point is bounded Degree of curve increases by one with each averaging step Smoothness increases by one with each averaging step

107/109 B-spline Curve Example

108/109 B-spline Curve Example

109/109 B-spline Curve Example