2023. 10. 10. 18:29ㆍRun/Computer Graphics
- 이전까지는 직선으로만 도형을 형성했다.
- Curves, Sufaces는 디자인 측면에서 중요하다.
- Curves: 1D objects in 2D or 3D space
- Surfaces: 2D objects in 3D space
곡선을 나타내는 방법
Polynomials
n: degree / a0, a1, ... an: constants
Drawing Polynomials Using Control Points
- point들을 Control Points 라고 한다.
- 위 그림에는 총 21개의 point가 있다. → degree 20 polynomial (20차 방정식) 사용
(ex. 일차방정식 y = ax + b 를 표현하려면 점 2개가 필요하다.)
- 이 방식을 사용하면 high degree polynomial을 사용할 때 굉장히 형태가 불안정하다.
- Cubic Polynomial (degree 3 polynomial) 을 사용하면 깔끔한 결과가 나온다.
- 여러 cubic polynomail을 합친다.
- Cubic polynomial는 4개의 점을 필요로 한다. → 4개의 점을 가지고 진행
Interpolating Cubic
- Express x, y, z points of a curve as functions of a parameter u.
- Each set of 4 control points is connected by a cubic polynomial.
- 0 ≤ u ≤ 1
- At control points, u = 0, 1/3, 2/3, 1
- Coefficient들은 각 control point의 (x, y, z) 좌표를 통해 구할 수 있다.
- C = MI * P
- p0~p3: used to interpolate the first curve, p3~p6: used to interpolate the second curve, ... and so on.
- Use n points, where (n - 1) is divisble by 3.
Bezier Curve
- 지나치게 curvy한 것을 방지하기 위해 사용한다.
- Curve is tangent to line segments p0p1 and p2p3 (위 사진에서 빨간색이 tangent line)
- Curve is within convex hull of p0, p1, p2, p3. (위 사진에서 파란색이 convex hull)
- Convex polygon: 도형 내 모든 각도가 180도 미만 (↔ Non-convex polygon)
- C = MB * P
- Issue: 각 곡선이 만나는 cusps가 sharp하다.
- Slope(tangent) changes drastically at cusp.
B-Spline
- Tangent of adjacent cubics are forced to be the same.
- 4 points control a curve near 2 points.
- p0, p1, p2, p3 control a curve near p1 and p2.
- p1, p2, p3, p4 control a curve near p2 and p3.
...
- Bezeir & Interpolated Cubic use p0~p3, p3~p6, p6~p9...
- B-Splines use p0~p3, p1~p4, p2~p5, ...
- C = Ms * P
- Issue: Endpoints(양끝 점)을 잇지 않는다.
- B-Spline does not have cusps.
- Cusps 또는 연결된 endpoints가 필요하다면, replicating points를 사용한다.
(ex. p0, p1, p2, ... → p0, p0, p1, p2, ... or p0, p0, p0, p1, p2, ...)
- p0 repeated 2 times: pulls curve toward p0. (cusp near p3)
- p0 repeated 3 times: forces curve to start at p0. (sharp cusp at p3)
- With repeated points, B-splines turn out to be incredibly powerful in modeling curves.
- B-spline with repeated points ≠ Bezier curve
- Both tangent to the line segments, but B-spline is closer to the control points
모든 공식을 요약하면,
p = C^T * u 로 표현할 수 있다.
Modeling Surfaces Using Polynomials
- Need 16 points (4 along u * 4 along v)
- x, y, z를 u와 v를 이용하여 표현한다.
- curve/surface를 n개의 점으로 분할하고, line 또는 triangle을 이용해 점을 연결하는 방식은 비효율적이다.
(∵ need to do polynomial evaluations for several points)
Recursive Subdivison (작성 중)
- 장점: Curve approaches smoothness in few steps (without polynomial calculations)
- Bezier cubic does this curve approximate.
- Bezier curve lies within convex hull, unlike Interpolated Cubic, and curve ends at endpoints, unlike B-Spline.
- Interpolated Cubic and B-Spline are converted to Bezier and modeled using recursive subdivison.
Recursive Subdivision : Interpolated Cubic
- Apply recursive subdivision to control points in PI→B
Recursive Subdivision : B-Spline
- Apply recursive subdivision to control points in PS→B
Recursive Subdivision : Bezier Surfaces
Recursive Subdivision : Sphere from Regular Tetrahedron
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