1. Dr. S.M. Malaek Assistant: M. Younesi
Curve Modeling
Dr. S.M. Malaek
Assistant: M. Younesi

2. Parametric Polynomials

3. Parametric Polynomials
For interpolating n point we need a polynomial of degree n-1.

4. Parametric Polynomials
Polynomial interpolation has several disadvantages:
If the polynomial degree is high, unwanted wiggles are introduced.
If the polynomial degree is law, it gives too little flexibility.
No local operations.

5. Solution:
Polynomial Splines

6. Spline Curve

7. Spline Curve
In drafting terminology, a spline is a flexible strip used to produce a smooth curve through a designated set of points.
Flexible Strip

8. Spline Curve
Several small weights are distributed along the length of the strip to hold it in position on the drafting table as the curve is drawn.
Weights

9. Spline Curve
The term spline curve originally referred to a curve drawn in this manner.

10. Spline Curve
In modeling, the term spline curve refers to a any composite curve formed with polynomial section satisfying specified continuity conditions at the boundary of the pieces.

11. Control Points
A spline curve is specified by given a set of coordinate positions, called control points.
Control points indicates the general shape of the curve.

12. Control Points
A spline curve is defined, modified, and manipulated with operations on the control points.

13. Convex Hull
The convex polygon boundary that enclose a set control points is called the convex hull.

14. Convex Hull
The convex hull is a rubber band stretched around the positions of the control points so that each control point is either on the perimeter of the hull or inside it.

15. Convex
Remind:
A region is convex if the line segment joining any two points in the region is also within the region.

16. Shape of
the Spline Curve

17. Shape of the Curve Interpolation Splines Approximation Splines
Control points are fitted with piecewise continuous parametric polynomial function in one of two way:
Interpolation Splines
Approximation Splines

18. Interpolation Splines
When polynomial sections are fitted so that the curve passes through each control point, the resulting curve is said to interpolate the set of control points.

19. Interpolation Splines
Interpolation curves are commonly used to digitize drawing or to specify animation paths.

20. Approximation Splines
When polynomial sections are fitted to the general control point path without necessarily passing through any control point, the resulting curve is said to approximate the set if control points.

21. Approximation Splines
Approximation curves are uses as design tools to structure object surfaces.

22. Continuity Conditions
Parametric
Continuity Conditions

23. Parametric Continuity Conditions
To represent a curve as a series of piecewise parametric curves, these curves to fit together reasonably …Continuity!

24. Parametric Continuity Conditions
Each section of a spline is described with a set of parametric coordinate functions of the form:

25. Parametric Continuity Conditions
Let C1(u) and C2(u) , be two parametric Curves.
We set parametric continuity by matching the parametric derivation of adjoining curve sections at their common boundary.

26. Parametric Continuity Conditions
Zero order parametric (C0): Simply the curves meet, C1(1) = C2(0) .
First order parametric (C1): The first parametric derivations for two successive curve sections are equal at their joining point, C´1(1)= C´2(0)

27. Parametric Continuity Conditions
Second order parametric (C2): Both the first and second parametric derivatives of the two curve sections are the same at the intersection, C˝1(1)= C˝2(0)

28. Parametric Continuity Conditions
Simple Example:
The curve consists of two parabolas, domains [-1,0] and [0,1], respectively
f(u) = ( u, -u2, 0 )
g(v) = ( v, v2, 0 )
f'(u) = ( 1, -2u, 0 ) ,f''(u) = ( 0, -2, 0 )
g'(u) = ( 1, 2v, 0 ) ,g''(v) = ( 0, 2, 0 )
Two curves are C1 continuous at the origin.
f''(0) = ( 0, -2, 0 ) is not equal to g''(v) = ( 0, 2, 0 ), they are not C2 continuous at the origin.
Curvature of f(u) = 2/(1 + 4u2)1.5 Curvature of g(v) = 2/(1 + 4v2)1.5
Two curve segments may be C1 continuous and even curvature continuous, but not C2 continuous

29. Parametric Continuity Conditions
Problems with
Parametric Continuity Conditions
Representation

30. Problems with Parametric Continuity
Two line segments:
f(u) = A + u(B - A)
g(v) = B + v(C - B)
f'(u) = B - A
g'(v) = C - B
Thus, f'(u) = B - A is, in general, not equal to g'(v) = C - B and consequently these two line segments are not C1 continuous at the joining point B!
Is it strange?
Re-parameterizing the curve segments may overcome the problem
F(u) = A + u(B - A)/ | B - A |
G(v) = B + v(C - B)/ | C - B |
u is in the range of 0 and | B - A | and v is in the range of 0 and | C – B|

31. Problems with Parametric Continuity
Semi circle:
f(u) = ( -cos(u2 PI/2), sin(u2 PI/2), 0 )
g(v) = ( sin(v2 PI/2), cos(v2 PI/2), 0 )
At (0,1,0) = f(1) = g(0). ‍ ) C0)
f'(u) = ( PI u sin(u2 PI/2), PI u cos(u2 PI/2), 0 )
f''(u) = ( PI2 u2 cos(u2 PI/2), -PI2 u2 sin(u2 PI/2), 0 )
f'(u) × f''(u) = ( 0, 0, -PI3 u3 )
| f'(u) | = PI u
| f'(u) × f''(u) | = PI3 u3
k(u) = 1
g'(v) = ( PI v cos(v2 PI/2), -PI v sin(v2 PI/2), 0 )
g''(v) = ( -PI2 v2 cos(v2 PI/2), -PI2 v2 cos(v2 PI/2), 0 )
g'(v) × g''(v) = ( 0, 0, -PI3 u3 )
| g'(v) | = PI v
| g'(v) × g''(v) | = PI3 v3
k(v) = 1

32. Problems with Parametric Continuity
Semi circle:
Let u2 = p in f(u) and let v2 = q in g(v). The new equations are:
f(p) = ( -cos(p PI/2), sin(p PI/2), 0 )
g(q) = ( sin(q PI/2), cos(q PI/2), 0)
Their derivatives are
f'(p) = ( (PI/2) sin(p PI/2), (PI/2) cos(p PI/2), 0 )
f''(p) = ( (PI/2)2 cos(p PI/2), -(PI/2)2 sin(p PI/2), 0 )
g'(q) = ( (PI/2) cos(q PI/2), -(PI/2) sin(q PI/2), 0 )
g''(q) = ( -(PI/2)2 sin(q PI/2), -(PI/2)2 cos(q PI/2), 0 )
Therefore, after changing variables, both f'(1) and g'(0) are equal to ( PI/2, 0, 0 ) and hence they are C1 continuous. Moreover, both f''(1) and g''(0) are equal to ( 0, -(PI/2)2, 0 ) and hence they are C2 continuous! They are also curvature continuous because they have curvature 1 everywhere (remember they are circles).

33. Continuity Conditions
Geometric
Continuity Conditions

34. Geometric Continuity Conditions
Two curve segments are said Gk geometric continuous at the joining point if and only if there exists two parameterizations, one for each curve segment, such that all i-th derivatives, i less than or equal to k, computed with these new parameterizations agree at the joining point.

35. Geometric Continuity Conditions
Two C0 curve segments are said G1 geometric continuous at the joining point if and only if vectors f'(u) and g'(v) are in the same direction at the joining point. Note that f'(u) and g'(v) are evaluated at the joining point.
The tangent vectors point to the opposite directions, and, as a result, they are not G1 at the joining point

36. Geometric Continuity Conditions
Two C1 curve segments are said G2 geometric continuous at the joining point if and only if vector f''(u) - g''(v) is parallel to the tangent vector at the joining point. Note that f''(u) and g''(v) are evaluated at the joining point.

37. Geometric Continuity Conditions
Example:two parabola segments with joining point at ( 0, 1, 0):
f(u) = ( -1 + u2, 2u - u2, 0 ) g(v) = ( 2u - u2, 1 - u2, 0 )
f'(u) = ( 2u, 2 - 2u, 0 ) f''(u) = ( 2, -2, 0 ) f'(u) × f''(u) = ( 0, 0, -4 ) | f'(u) | = 2SQRT(1 - 2u + 2u2 ) | f'(u) × f''(u) | = 4 k(u) = 1/(2(1 - 2u + 2u2)1.5) g'(v) = ( 2 - 2v, -2v, 0 ) g''(v) = ( -2, -2, 0 ) g'(v) × g''(v) = ( 0, 0, -4 ) | g'(v) | = 2SQRT(1 - 2v + 2v2 ) | g'(v) × g''(v) | = 4 k(v) = 1/(2(1 - 2v + 2v2)1.5)

38. Geometric Continuity Conditions
From f'(1) = g'(0) = ( 2, 0, 0 ), we know that these two curve segments are C1 at the joining point. Since f''(1) = ( 2, -2, 0 ) is not equal to g''(0) = ( -2, -2, 0 ), they are not C2.
Since the curves are C1, checking for G2 is meaningful. Since f''(1) - g''(0) = ( 4, 0, 0 ) is parallel to the tangent vector ( 2, 0, 0 ) at the joining point, these two curve segments are G2 continuous at the joining point ( 0, 1, 0 ).

39. Geometric Continuity Conditions
Geometric Continuity Conditions: Only require parametric derivatives of the two sections to be proportional to each other at their common boundary instead of equal to each other.

40. Geometric Continuity Conditions
Zero order geometric (G0): Same as C0
C1(1) = C2(0)
First order geometric (G1): The parametric first derivatives are proportional at the intersection of two successive sections.
C´1(1)= αC´2(0)
Second order geometric (G2): Both the first and second parametric derivatives of the two curve sections are proportional at their boundary,
C˝1(1)= αC˝2(0)

41. Parametric & Geometric Continuity
A curve generated with geometric continuity condition is similar to one generated with parametric continuity , but with slight differences in curve shape.
With geometric continuity, the curve is pulled toward the section with the greater tangent vector.

42. Spline Interpolation

43. Linear
Spline Interpolation

44. Linear Spline Interpolation
Given a set of control points, linear interpolation spline is obtained by fitting the input points with a piecewise linear polynomial curve that passes through every control points.

45. Linear Spline Interpolation
A linear polynomial:

46. Linear Spline Interpolation
We need to determine the values of a and b in the linear polynomial for each of the n curve section.
Substituting endpoint values 0 and 1 for parameter u:
Linear Spline Interpolation

47. Linear Spline Interpolation
Solving this equation for the polynomial coefficients:
P(u) can be written in terms of the boundary conditions:
Mspline
Mgeometry U

48. Linear Spline Interpolation
The matrix representation for linear spline representation:
Mgeometry is the matrix containing the geometry constraint values (boundary condition) on the spline.
Mspline is the matrix that transform the geometric constraint values to the polynomial coefficients and provides a characterization for the spline curve (Basis Matrix).

49. Linear Spline Interpolation
We can expand the matrix representation:
Blending Functions

50. Linear Spline Interpolation
The Linear spline blending function:

51. Linear Spline Interpolation
The curve is linear combination of the blending functions.
The piecewise linear interpolation is C0 continuous.
For a linear polynomial, 2 constraints are required for each segment.
A polynomial of degree k has k+1 coefficients and thus requires k+1 independent constraints to uniquely determine it.

52. Cubic
Spline Interpolation

53. Cubic Spline Interpolation
Cubic splines are used to:
Set up paths for object motion
A representation for an existing object or drawing
Design object shape
Cubic splines are more flexible for modeling arbitrary curve shapes.

54. Cubic Spline Interpolation
Given a set of control points:
Cubic interpolation spline is obtained by fitting the input points with a piecewise cubic polynomial curve that passes through every control points.

55. Cubic Spline Interpolation
A cubic polynomial:

56. Linear Spline Interpolation
We need to determine the values of a , b, c and d in the linear polynomial for each of the n curve section.
By setting enough boundary conditions at the “joints” between curve sections we can obtain numerical values for all the coefficients.
Linear Spline Interpolation

57. Cubic Spline Interpolation
Methods for setting the boundary conditions for cubic interpolation splines:
Natural Cubic Splines
Hermit Splines
Cardinal Splines
Kochanek-Bartels Splines   

58. Natural
Cubic Splines

59. Natural Cubic Splines
Natural cubic splines are developed for graphics application.
Natural cubic splines have C2 continuity.

60. Natural Cubic Splines We have n curve sections
If we have n+1 control points to fit, then:
We have n curve sections
4n polynomial coefficients to be determined.
3 control points (n=2)
2 curve sections
4×2 polynomial coefficients to be determined.

61. Natural Cubic Splines
At each n-1 interior control points, we have 4 boundary condition:
The two curve sections on either side of a control point must first and second parametric derivatives at that control point, and each curve must pass through that control point.
At P1 interior control:
1. C´1(1)=C'2(0)
2. C˝1(1)=C˝2(0)
3. C1(1)=P1
4. C2(0)=P1

62. Natural Cubic Splines Additional equation: First control point P0
End control points Pn
5. C1(0)=P0
6. C2(1)=P2

63.  Natural Cubic Splines
We still need two more conditions to be able to determine values for all coefficients. 

64. Natural Cubic Splines Method 1:
Set the second derivatives at p0 and pn to 0
7. C˝1(0)=0
8. C˝2(1)=0

65. Natural Cubic Splines Method 2:
Add a control point P-1 and a control point Pn+1.
P-1 P3
7. P-1
8. P3

66. Natural Cubic Splines
Natural cubic splines are a mathematical model for the drafting spline.
Disadvantage:
No “Local Control”: we cannot reconstruct part of the curve without specifying an entirely new set of control points.

67. Hermit Splines

68. Hermit Splines
Given a set of control points:
Hermit splines is an interpolating piecewise cubic polynomial with a specified tangent at each control point.

69. Hermit Splines
Hermit splines can be adjusted locally because each curve section is only dependent on its endpoint constraints (unlike the natural cubic splines).

70. Hermit Splines
Example: Change in Magnitude of T0

71. Hermit Splines
Example: Change in Direction of T0

72. Hermit Splines
P(u) represents a parametric cubic point function for the section between pk and pk+1.
The boundary conditions are:
Dpk and Dpk+1 are the values for the parametric derivations (slope of the curve) at control points Pk and Pk+1

73. We can write the constraints
Hermit Splines
Boundary conditions
The derivative of the point function:
We can write the constraints
in a matrix form:

74. Hermit Splines
Solving this equation for the polynomial coefficients:
MH, the Hermit matrix, is the inverse of the boundary constraint matrix.

75. Blending Functions

76. Hermit Blending Functions
Hermit Splines
Hermit Blending Functions

77. Hermit Splines
Hermit polynomials can be useful for some digitizing applications where it may not be too difficult to specify or approximate the curve slopes.
For most problems in modeling, it is more useful to generate spline curve without requiring input values for curve slopes.
But…

78. Cardinal
Splines

79. Cardinal Splines
Cardinal splines are interpolating piecewise cubics with specified endpoint tangents at the boundary of each curve section.
The difference is that we do not have to give the values for the endpoint tangents.
For a cardinal spline, the value for the slope at a control point is calculated from the coordinates of the two adjacent control points.

80. Cardinal Splines
A cardinal spline section is completely specified with four consecutive control points.
The middle two control points are the section endpoints, and the other two points are used in the calculation of the endpoint slopes.

81. Cardinal Splines The boundary conditions:
Thus, the slopes at control points pk and pk+1 are taken to be proportional to the chords and

82. Cardinal Splines
The boundary conditions:

83. Cardinal Splines Parameter t is called the tension parameter.
Tension parameter controls how loosely or tightly the cardinal spline fits the input control points.

84. Cardinal Splines
When t=0 is called the Catmull-Rom splines, or Overhauser splines.

85. Cardinal Splines

86. Cardinal Splines
The polynomials CARk(u) for k=0,1,2,3 are the cardinal functions.

87. Cardinal Splines
Cardinal functions