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3D 프로그래머를 위한 커브 입문서 : 직선에서 곡면까지 1St Edition

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3D 프로그래머를 위한 커브 입문서 : 직선에서 곡면까지 1St Edition CURVE 3D 프로그래머를 위한 커브 입문서 : 직선에서 곡면까지 1st Edition lifeisforu [email protected] http://lifeisforu.tistory.com 1 목차 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 2 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 3 들어 가며 대부분의 3D 프로그래머들은 커브를 다뤄 본 적이 있을 것입니다. 그런데 안타깝게도 커브의 개 념을 제대로 이해하고 사용하는 사람들은 많지 않습니다. 저도 그런 부류 중의 하나입니다. 대부 분 그냥 라이브러리에 있는 기능을 사용하든지 구글링을 해서 찾은 코드를 사용합니다. 그러다 보니 다음과 같은 요청에 대응할 수가 없습니다. "점을 찍으면 자동으로 path 가 생성되게 해 주세요." "Path 를 원하는 대로 제어할 수 있게 해 주세요." "Path 를 따라 움직이는 오브젝트가 등속도 운동을 하게 해 주세요." "Path 가 부드럽게 연결되게 해 주세요." "이럴 땐 어떤 커브를 사용해야 하나요?" 위와 같은 요청에 응답하기 위해서 코드를 보면 뭔가 곱하고 더하고 난리도 아닙니다. 그리고 모 르는 용어가 자꾸 나오니 이해하기도 어렵습니다. 또한 값을 좀만 바꿔도 원하지 않는 매듭 모양 이 나오거나 깨집니다. 코드를 이거 건드려 보고 저거 건드려 보고, 값을 이리 바꾸고 저리 바꾸고, 삽질을 하다가 보면 시간은 흘러가고 작업 마감일은 다가 옵니다. 그렇다고 제대로 공부하기에는 시간도 부족한 것 같고 참고 자료도 마땅치 않습니다. 특히나 저 같은 문과 출신들에게는 그것이 엄청난 벽으로 다 가 옵니다. 왜냐하면 어디서부터 시작해서 어디에서 끝내야 할지 모르기 때문입니다. 사실 커브 및 서피스에 대한 내용은 수학적인 카테고리로 볼 때 수치해석( numerical analysis ) 과 관계가 깊습니다. 인문계 출신들은 그런 분야의 존재 자체를 모르는 사람도 많을 것입니다. 저만 해도 그런 과목이 있다는 것은 들어는 봤지만, 실제 무슨 내용인지는 이 문서를 정리하면서 수치해석의 내용이 무엇인지 알게 되었습니다. 그러므로 이쪽에 대해 전문가 수준으로 알고자 한 다면 수치해석 책 하나 보시는 것이 깔끔할 것이라 생각합니다. 물론 비전공자가 보기에는 친절 하지 않을 수도 있습니다. 그런데 그저 개념만 파악하고자 하거나 간단한 참고서가 필요하다는 생각을 하실 수 있습니다. 이 글은 ( 저를 포함한 ) 그런 사람들에게 샛길이라도 제공하기 위해서 작성되었습니다. 그러므 로 독자들이 이런 분야에 대한 지식을 많이 가지고 있지 않다는 전제를 깔고 글을 작성하도록 하 겠습니다. 이 글의 내용은 직접 작성한 부분도 있지만 거의 대부분은 참고자료를 번역하거나 정리한 것입니 다. 그리고 정확한 내용을 담고 있지 않을 수 있음을 죄송하게 생각합니다. 제가 수학과 영어에 전문가가 아니기 때문에 잘못된 지식들도 포함되어 있을 수 있습니다. 그러므로 이해가 안 간다 4 싶으면 적당히 다른 참고 문헌들을 참조하시기 바랍니다. 궁금한 사항이 있으시거나 잘못된 내용을 제보해 주시려고 한다면, 저에게 메일 ( [email protected] )을 보내 주시면 됩니다. 다시 한 번 말씀드리지만 전문적인 영역에 접근하는 책은 아니므로
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