Curve

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Curve.pdf

 

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목차 ............................................................................................................................................................................................... 2

 

들어 가며 ........................................................................................................................................................................................ 4

1. Combination ...................................................................................................................................................................... 6

1.1. Linear combination ...................................................................................................................................... 6

1.2. Linear independence .................................................................................................................................. 7

1.3. Cartesian coordinate system .............................................................................................................. 8
1.4. Convex combination .................................................................................................................................... 9

2. Polynomial & interpolation .............................................................................................................................. 13

2.1. Linear interpolation .................................................................................................................................. 16
2.2. Polynomial interpolation ..................................................................................................................... 17
2.3. Method of undetermined coefficients ...................................................................................... 18
2.4. Lagrange interpolation .......................................................................................................................... 19
2.5. Divided difference ...................................................................................................................................... 21
2.5.1. Linear interpolation .................................................................................................................... 21
2.5.2. Quadratic interpolation ........................................................................................................... 22
2.5.3. Divided difference & General intepolation............................................................ 23
2.6. Newton interpolation .............................................................................................................................. 25
2.7. Problem of polynomial interpolation ........................................................................................ 26
2.8. Spline interpolation .................................................................................................................................. 28
2.8.1. Cubic spline interpolation ..................................................................................................... 29
2.8.2. Solving natural cubic spline ............................................................................................... 31

3. Parametric equation & continuity ............................................................................................................. 33

3.1. Parametric equation ................................................................................................................................ 34
3.2. Parametric continuity ............................................................................................................................. 37
3.3.Geometric Continuity ................................................................................................................................ 40

4. Bé zier curve ................................................................................................................................................................... 42

4.1. Linear Bé zier curve ................................................................................................................................... 43
4.2. Quadratic Bé zier curve .......................................................................................................................... 43
4.3. Cubic Bé zier curve ..................................................................................................................................... 45
4.4. Composite Bezier curve ........................................................................................................................ 48

5. Spline Curve ................................................................................................................................................................... 50

5.1. Linear spline curve .................................................................................................................................... 50
5.2. Quadratic spline curves ........................................................................................................................ 51
5.3. Spline curves of higher degrees ................................................................................................... 53

6. Hermite spline .............................................................................................................................................................. 56

6.1. Finite difference ........................................................................................................................................... 58
6.2. Cardinal spline ............................................................................................................................................... 58
6.3. Catmul-Rom spline .................................................................................................................................... 58
6.4. Kochanek-Bartels spline ...................................................................................................................... 59
6.5. Monotone cubic interpolation ......................................................................................................... 60

7. B-Spline( Basis spline ) ....................................................................................................................................... 61

8. Nonuniform rational B-Spline( NURBS ) ............................................................................................. 65

8.1. Nonuniform B-spline ................................................................................................................................ 65
8.1.1. Uniform knot vector ................................................................................................................... 65
8.1.2. Open uniform knot vector .................................................................................................... 66
8.1.3. Non-uniform knot vector ....................................................................................................... 67
8.2. Rational curve ................................................................................................................................................ 68
8.3. Non-Uniform Rational B-Spline ..................................................................................................... 71
8.4. Construction of the basis functions ........................................................................................... 72
8.5. General form of a NURBS curve .................................................................................................... 76
8.6. General form of a NURBS surface ............................................................................................... 76

9. Subdivision Surface ................................................................................................................................................ 76