1. 5. Curves and Curve Modeling
ME 521
Computer Aided Design
5. Curves and Curve Modeling
Assoc.Dr. Ahmet Zafer Şenalp
Mechanical Engineering Department
Gebze Technical University

2. Mechanical Engineering Department, GTU
Purpose
5. Curves and Curve Modeling
Curves are the basics for surfaces
Before learning surfaces curves have to be known
When asked to modify a particular entity on a CAD system, knowledge of the entities can increase your productivity
Understand how the math presentation of various curve entities relates to a user interface
Understand what is impossible and which way can be more efficient when creating or modifying an entity
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

3. Mechanical Engineering Department, GTU
Purpose
5. Curves and Curve Modeling
Curves are the basics for surfaces
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

4. Why Not Simply Use a Point Matrix to Represent a Curve?
5. Curves and Curve Modeling
Storage issue and limited resolution
Computation and transformation
Difficulties in calculating the intersections or curves and physical properties of objects
Difficulties in design (e.g. control shapes of an existing object)
Poor surface finish of manufactured parts
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

5. Advantages of Analytical Representation for Geometric Entities
5. Curves and Curve Modeling
A few parameters to store
Designers know the effect of data points on curve behavior, control, continuity, and curvature
Facilitate calculations of intersections, object properties, etc.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

6. Mechanical Engineering Department, GTU
Curve Definitions
5. Curves and Curve Modeling
Explicit form :
Implicit form :
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

7. Drawbacks of Conventional Representations
5. Curves and Curve Modeling
Conventional explicit and implicit forms have several drawbacks.
They represent unbounded geometry
They may be multi-valued
Difficult to evaluate points along the curve
Depends on coordinate system
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

8. Parametric Representation
5. Curves and Curve Modeling
Curves are defined as a function of a single parameter:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

9. Parametric Representation
5. Curves and Curve Modeling u u
Surface, P=P(u,v)
Curve, P=P(u)
P(u)=[x(u),y(u),z(u)]T v
P(u, v)=[x(u, v), y(u, v), z(u, v)]T
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

10. Parametric Representation
5. Curves and Curve Modeling
Changing curve equation into parametric form:
Let’s use “t” parameter ;
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

11. Parametric Explicit Form- Implicit Form Conversion Example :
5. Curves and Curve Modeling
Planar 2. degree curve:
How to obtain implicit form?
t is extracted as:
Replacing t in y equation;
Rearranging the above equation;
Rearranging again;
We obtain implicit form.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

12. Parametric Explicit Form- Implicit Form Conversion Example :
5. Curves and Curve Modeling
Planar 2. degree curve :
plot
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

13. Mechanical Engineering Department, GTU
Curve Classification
5. Curves and Curve Modeling
Curve Classification:
Analytic Curves
Synthetic curves
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

14. Mechanical Engineering Department, GTU
Analytic Curves
5. Curves and Curve Modeling
These curves have an analytic equation
point
line
arc
circle
fillet
Chamfer
Conics (ellipse, parabola,and hyperbola))
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

15. Forming Geometry with Analytic Curves
5. Curves and Curve Modeling
line
arc
circle
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

16. Mechanical Engineering Department, GTU
Analytic Curves Line
5. Curves and Curve Modeling
Line definition in cartesian coordinate system:
Here;
m: slope of the line
b: point that intersects y axis
x: independent varaible of y function.
Parametric form;
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

17. Analytic Curves Line Example: implicit-explicit form change
5. Curves and Curve Modeling
Line equation:
implicit form
explicit form
Changing to parametric form. In this case
Let Replacing this value into y equation.
is obtained.
As a result;
Parametric line equation is obtained.
To turn back to implicit or explicit nonparametric form t is replaced in x and y equalities
From here the form at the beginning is obtained.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

18. Analytic Curves Circle
5. Curves and Curve Modeling
Circle definition in Cartesian coordinate system:
Here;
a,b: x,y coordinates of center point
r: circle radius
Parametric form
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

19. Analytic Curves Ellipse
5. Curves and Curve Modeling
Ellipse definition in Cartesian coordinate system:
Here;
h,k: x,y coordinates of center point
a: radius of major axis
b: radius of minör exis
Parametric form
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

20. Analytic Curves Parabola
5. Curves and Curve Modeling
Parabola definition in Cartesian coordinate system:
Usual form;
y = ax2 + bx + c
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

21. Analytic Curves Hyperbola
5. Curves and Curve Modeling
Hyperbola definition in Cartesian coordinate system:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

22. Mechanical Engineering Department, GTU
Synthetic Curves
5. Curves and Curve Modeling
As the name implies these are artificial  curves
Lagrange interpolation curves
Hermite interpolation curves
Bezier
B-Spline
NURBS
etc.
Analytic curves are usually not sufficient to meet geometric design requirements of mechanical parts.
Many products need free-form, or synthetic curved surfaces
These curves use a series of control points either interploated or aproximated
It is the definition method for complex curves.
It should be controllable by the designer.
Calculation and storage should be easy.
At the same time called as free form curves.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

23. Mechanical Engineering Department, GTU
Synthetic Curves
5. Curves and Curve Modeling
open curve
closed curve
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

24. Mechanical Engineering Department, GTU
Synthetic Curves
5. Curves and Curve Modeling
interpolated
control points
approximated
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

25. Mechanical Engineering Department, GTU
Composite Curves
5. Curves and Curve Modeling
Curves can be represented by connected segments to form a composite curve
There must be continuity at the mid-points 1 2 3 4
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

26. Mechanical Engineering Department, GTU
Degrees of Continuity
5. Curves and Curve Modeling
Position continuity
Slope continuity
1st derivative
Curvature continuity
2nd derivative
Higher derivatives as necessary
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

27. Mechanical Engineering Department, GTU
Position Continuity
5. Curves and Curve Modeling
Mid-points are connected 2 1 3
Connected (C0 continuity)
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

28. Mechanical Engineering Department, GTU
Slope Continuity
5. Curves and Curve Modeling
Both curves have the same 1. derivative value at the connection point. At the same time position continuity is also attained. 1 2
Continuous tangent
Tangent continuity (C1 continuity)
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

29. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
Curvature Continuity
Both curves have the same 2.derivative value at the connection point.
At the same time position and slope continuity is also attained. 1 2
Continuous curvature
Curvature continuity (C2 continuity)
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

30. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
Composite Curves
A cubic spline has C2 continuity at intermediate points
Cubic splines do not allow local control 1 2 3 4
Cubic polynomials
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

31. Mechanical Engineering Department, GTU
Linear Interpolation
5. Curves and Curve Modeling
General Linear Interpolation: One of the simplest method is linear interpolation.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

32. Parametric Cubic Polynomial Curves
5. Curves and Curve Modeling
Cubic polynomials are the lowest-order polynomials that can represent a non-planar curve
The curve can be defined by 4 boundary conditions
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

33. Mechanical Engineering Department, GTU
Cubic Polynomials
5. Curves and Curve Modeling
Lagrange interpolation - 4 points
Hermite interpolation - 2 points, 2 slopes p0 p1
P1’
P0’
Hermite p0 p3 p2 p1
Lagrange
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

34. Lagrange Interpolation
5. Curves and Curve Modeling
2 xi terms should not be the same, For N+1 data points ; (x0,y0),...,(xN,yN) için Lagrange interpolation form is in the form of linear combination:
Below polynomial is called Lagrange base polynomial;
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

35. Lagrange Interpolation
5. Curves and Curve Modeling
This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x) (in black), which is the sum of the scaled basis polynomials y0ℓ0(x), y1ℓ1(x), y2ℓ2(x) and y3ℓ3(x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

36. Lagrange Interpolation Example:
5. Curves and Curve Modeling
A 3. degree L(x) function has the following x and corresponding y values;
The polynomial corresponding to the above values can be determined by Lagrange interpolation method:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

37. Lagrange Interpolation Example:
5. Curves and Curve Modeling
L(x)= -0,7083x4+7,4167x3-22,2917x2+13,5833x+8
obtained.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

38. Cubic Hermite Interpolation
5. Curves and Curve Modeling
Also known as cubic splines.
Enables up to C1 continuity.
General form of Cubic Hermite interpolation:
Position vector at the starting point
Position vector at the end point
Tangent vector at the starting point
Tangent vector at the end point
There are no algebraic coefficints but there are geometric coefficints
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

39. Cubic Hermite Interpolation
5. Curves and Curve Modeling
Cubic Hermite Interpolation
Hermite base functions
Hermite form is obtained by the linear summation of this 4 function at each interval.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

40. Cubic Hermite Interpolation
5. Curves and Curve Modeling
Cubic Hermite Interpolation
The effect of tangent vector to the curve shape
Geometrik katsayı
matrisi
Geometric coefficient matrix controls the shape of the curve.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

41. Cubic Hermite Interpolation
5. Curves and Curve Modeling
Cubic Hermite Interpolation P0
P0’ P2 T2
Hermite curve set with same end points (P0 ve P1), Tangent vectors P0’ and P1’ have the same directions but P0’ have different magnitude P1’ is constant
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

42. Cubic Hermite Interpolation
5. Curves and Curve Modeling
Cubic Hermite Interpolation
All tangent vector magnitudes are equal but the direction of left tangent vector changes.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

43. Cubic Hermite Interpolation
5. Curves and Curve Modeling
Cubic Hermite interpolation form:
Can also be written as:
There are no algebraic coefficints but there are geometric coefficints
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

44. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
Approximated Curves
Bezier
B-Spline
NURBS
etc.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

45. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
Bezier Curves
P. Bezier of the French automobile company of Renault first introduced the Bezier curve (1962).
Bezier curves were developed to allow more convenient manipulation of curves
A system for designing sculptured surfaces of automobile bodies (based on the Bezier curve)
A Bezier curve is a polynomial curve approximating a control polygon
Quadratic and cubic Bézier curves are most common
Higher degree curves are more expensive to evaluate.
When more complex shapes are needed, low order Bézier curves are patched together.
Bézier curves are easily programmable. Bezier curves are widely used in computer graphics.
Enables up to C1 continuity.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

46. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
Bezier Curves
Control polygon
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

47. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
Bezier Curves
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

48. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
Bezier Curves
General Bezier curve form which is controlled by n+1 Pi control points;
where the polynomials
are known as Bernstein basis polynomials of degree n,
defining t0 = 1 and (1 - t)0 = 1.
: binomial coefficient.
Degree of polynomial is one less than the control points used.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

49. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
The points Pi are called control points for the Bézier curve
The polygon formed by connecting the Bézier points with lines, starting with P0 and finishing with Pn, is called the Bézier polygon (or control polygon). The convex hull of the Bézier polygon contains the Bézier curve.
The curve begins at P0 and ends at Pn; this is the so-called endpoint interpolation property.
The curve is a straight line if and only if all the control points are collinear.
The start (end) of the curve is tangent to the first (last) section of the Bézier polygon.
A curve can be split at any point into 2 subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

50. Bezier Curves Linear Curves
5. Curves and Curve Modeling
Bezier Curves Linear Curves
t= [0,1] form of a linear Bézier curve turns out to be linear interpollation form. Curve passes through points P0 ve P1. Animation of a linear Bézier curve, t in [0,1]. The t in the function for a linear Bézier curve can be thought of as describing how far B(t) is from P0 to P1. For example when t=0.25, B(t) is one quarter of the way from point P0 to P1. As t varies from 0 to 1, B(t) describes a curved line from P0 to P1.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

51. Bezier Curves Quadratic Curves
5. Curves and Curve Modeling
Bezier Curves Quadratic Curves
Curve passes through P0 , P1 & P2 points.
For quadratic Bézier curves one can construct intermediate points Q0 and Q1 such that as t varies from 0 to 1:
Point Q0 varies from P0 to P1 and describes a linear Bézier curve.
Point Q1 varies from P1 to P2 and describes a linear Bézier curve.
Point B(t) varies from Q0 to Q1 and describes a quadratic Bézier curve.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

52. Bezier Curves Higher Order Curves
5. Curves and Curve Modeling
Bezier Curves Higher Order Curves
For higher-order curves one needs correspondingly more intermediate points. Cubic Bezier Curve Curve passes through P0 , P1, P2 & P3 points. For cubic curves one can construct intermediate points Q0, Q1 & Q2 that describe linear Bézier curves, and points R0 & R1 that describe quadratic Bézier curves
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

53. Bezier Curves Bernstein Polynomials
5. Curves and Curve Modeling
Bezier Curves Bernstein Polynomials
Most of the graphics packages confine Bézier curve with only 4 control points. Hence n = 3 . t
f(t) 1
BB1
BB4
BB2
BB3
Bernstein polinomials
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

54. Bezier Curves Higher Order Curves
5. Curves and Curve Modeling
Bezier Curves Higher Order Curves
Fourth Order Bezier Curve Curve passes through P0 , P1, P2, P3 & P4 points. For fourth-order curves one can construct intermediate points Q0, Q1, Q2 & Q3 that describe linear Bézier curves, points R0, R1 & R2 that describe quadratic Bézier curves, and points S0 & S1 that describe cubic Bézier curves:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

55. Bezier Curves Polinomial Form
5. Curves and Curve Modeling
Bezier Curves Polinomial Form
Sometimes it is desirable to express the Bézier curve as a polynomial instead of a sum of less straightforward Bernstein polynomials.
Application of the binomial theorem to the definition of the curve followed by some rearrangement will yield:
and
This could be practical if Cj can be computed prior to many evaluations of B(t); however one should use caution as high order curves may lack numeric stability (de Casteljau's algorithm should be used if this occurs).
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

56. Bezier Curves Example:
5. Curves and Curve Modeling
Bezier Curves Example:
Coordinatess of 4 control poits are given as:
What is the equation of Bezier curve that will be obtained by using above points?
What are the coordinate values on the curve corresponding to t=0,1/4,2/4,3/4,1 ?
Solution:
For 4 points 3. order Bezier form is used:
: Bezier curve equation
Points on B(t) curve
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

57. Bezier Curves Example:
5. Curves and Curve Modeling
Bezier Curves Example:
Equation of Bezier curve:
Control points
Points on B(t) curve
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

58. Bezier Curves Disadvantages
5. Curves and Curve Modeling
Bezier Curves Disadvantages
Difficult to interpolate points
Cannot locally modify a Bezier curve
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

59. Bezier Curves Global Change
5. Curves and Curve Modeling
Bezier Curves Global Change
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

60. Bezier Curves Local Change
5. Curves and Curve Modeling
Bezier Curves Local Change
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

61. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
Bezier Curves Example
2 cubic composite Bézier curve - 6. order Bézier curve
comparisson
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

62. Bezier Curves Modeling Example
5. Curves and Curve Modeling
Bezier Curves Modeling Example
Contains 32 curve
Polygon representation
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

63. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
B-Spline Curves
B-splines are generalizations of Bezier curves
A major advantage is that they allow local control
B-spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition.
A fundamental theorem states that every spline function of a given degree, smoothness, and domain partition, can be represented as a linear combination of B-splines of that same degree and smoothness, and over that same partition.
The term B-spline was coined by Isaac Jacob Schoenberg and is short for basis spline. B-splines can be evaluated in a numerically stable way by the de Boor algorithm.
A B-spline is simply a generalisation of a Bézier curve, and it can avoid the Runge phenomenon without increasing the degree of the B-spline.
The degree of curve obtained is independent of number of control points.
Enables up to C2 continuity.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

64. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
B-Spline Curves
Pi defines B-Spline curve with given n+1 control points:
Here Ni,k(u) is B-Spline functions are proposed by Cox and de Boor in 1972.
k parameter controls B-Spline curve degree (k-1) and generally independent of number of control points.
aksi durumda
ui is called parametric knots or (knot vales) for an open curve B-Spline:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

65. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
B-Spline Curves
if a curve with (k-1) degree and ( n+1) control points is to be developed, (n+k+1) knots then are required.
This inequality shows that;
for linear curve at least 2
for 2. degree curve at least 3
for cubic curve at least 4 control points are necessary.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

66. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
B-Spline Curves
Below figures show B-Spline functions:
Linear function
k=2
2. degree function
k=3
cubic function
k=4
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

67. B-Spline Curves Properties
5. Curves and Curve Modeling
B-Spline Curves Properties
Number of control points is independent than the degree of the polynomial.
Linear
k=2
vertex
Quadratic B-Spline; k=3
Cubic B-Spline; k=4
Fourth Order B-Spline; k=5
n=3
The higher the order of the B-Spline, the less the influence the close
control point
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

68. B-Spline Curves Properties
5. Curves and Curve Modeling
B-Spline Curves Properties
B-spline allows better local control. Shape of the curve
can be adjusted by moving the control points.
Local control: a control point only influences k segments.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

69. B-Spline Curves Example:
5. Curves and Curve Modeling
B-Spline Curves Example:
Cubic Spline; k=4, n=3 8 knots are required.
reminder : To define a (k-1) degree curve with (n+1) control points (n+k+1) knots are required.
Equation results 8 knots
Limits of u parameter:
Bezier curve equality;
reminder : *
B-Spline vector can be calculated together with knot vector;
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

70. B-Spline Curves Example:
5. Curves and Curve Modeling
B-Spline Curves Example:
aksi durumda
else
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

71. B-Spline Curves Example:
5. Curves and Curve Modeling
B-Spline Curves Example:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

72. B-Spline Curves Example:
5. Curves and Curve Modeling
B-Spline Curves Example:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

73. B-Spline Curves Example:
5. Curves and Curve Modeling
B-Spline Curves Example:
Replacing into Ni,4 * equality;
By replacing Ni,3 into the above equality the B-Spline curve equation given below is obtained.
This equation is the same with Bezier curve with the same control points.
Hence cubic B-Spline curve with 4 control points is the same with cubic Bezier curve with the same control points.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

74. Bezier /B-Spline Curves
5. Curves and Curve Modeling
Bezier /B-Spline Curves
Bezier Blending Functions; Bi,n
B-spline Blending Functions; Ni,k
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

75. Bezier /B-Spline Curves
5. Curves and Curve Modeling
Bezier /B-Spline Curves
Point that is moved
This point is moving
This point is not moving
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

76. Uniform B-Spline Curves
5. Curves and Curve Modeling
Uniform B-Spline Curves
When B-spline is uniform B-spline functions with n degrees are just shifted copies of each other.
Knots are equally spaced along the curve.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

77. Rational Curves and NURBS
5. Curves and Curve Modeling
Rational Curves and NURBS
Rational polynomials can represent both analytic and polynomial curves in a uniform way
Curves can be modified by changing the weighting of the control points
A commonly used form is the Non-Uniform Rational B-spline (NURBS)
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

78. Rational Bezier Curves
5. Curves and Curve Modeling
Rational Bezier Curves
The rational Bézier adds adjustable weights to provide closer approximations to arbitrary shapes. The numerator is a weighted Bernstein-form Bézier curve and the denominator is a weighted sum of Bernstein polynomials. Given n + 1 control points Pi, the rational Bézier curve can be described by:
or simply
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

79. Rational B-Spline Curves
5. Curves and Curve Modeling
Rational B-Spline Curves
One rational curve is defined by ratios of 2 polynomials. In rational curve control points are defined in homogenous coordinates.
Then rational B-Spline curve can be obtained in the following form:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

80. Rational B-Spline Curves
5. Curves and Curve Modeling
Rational B-Spline Curves
Ri,k(u) is the rational B-Spline basis functions.
The above equality show that; Ri,k(u) basis functions are the generelized form of Ni,k(u).
When hi=1 is replaced in Ri,k(u) equality shows the same properties with the
nonrational form.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

81. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
NURBS
It is non uniform rational B-Spline formulation. This mathematical model is generally used for constructing curves and surfaces in computer graphics.
NURBS curve is defined by its degree, control points with weights and knot vector.
NURBS curves and surfaces are the generalized form of both B-spline and Bézier curves and surfaces.
Most important difference is the weights in the control points which makes NURBS rational curve.
NURBS curves have only one parametric direction (generally named as s or u). NURBS surfaces have 2 parametric directions.
NURBS curves enables the complete modeling of conic curves.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

82. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
NURBS
General form of a NURBS curve;
k: is the number of control points (Pi)
wi: weigths
The denominator is a normalizing factor that evaluates to one if all weights are one. This can be seen from the partition of unity property of the basis functions. It is customary to write this as
Rin: are known as the rational basis functions.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

83. NURBS Examples 5. Curves and Curve Modeling Uniform knot vector
Nonuniform knot vector
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

84. NURBS Development of NURBS
5. Curves and Curve Modeling
NURBS Development of NURBS
Boeing: Tiger System in 1979
SDRC: Geomod in 1993
University of Utah: Alpha-1 in 1981
Industry Standard: IGES, PHIGS, PDES,Pro/E, etc.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

85. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
NURBS Advantages
Serve as a genuine generalizations of non-rational B-spline forms as well as rational and non-rational Bezier curves and surfaces
Offer a common mathematical form for representing both standard analytic shapes (conics, quadratics, surface of revolution, etc) and free-from curves and surfaces precisely. B-splines can only approximate conic curves.
By evaluating a NURBS curve at various values of the parameter, the curve can be represented in cartesian two- or three-dimensional space. Likewise, by evaluating a NURBS surface at various values of the two parameters, the surface can be represented in cartesian space.
Provide the flexibility to design a large variety of shapes by using control points and weights. increasing the weights has the effect of drawing a curve toward the control point.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

86. Mechanical Engineering Department, GTU
5. Curves and Curve Modeling
NURBS Advantages
Have a powerful tool kit (knot insertion/refinement/removal, degree elevation, splitting, etc.)
They are invariant under affine as well as perspective transformations: operations like rotations and translations can be applied to NURBS curves and surfaces by applying them to their control points.
Reasonably fast and computationally stable. They reduce the memory consumption when storing shapes (compared to simpler methods).
They can be evaluated reasonably quickly by numerically stable and accurate algorithms.
Clear geometric interpretations
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU