🗒️ 스플라인 (5)

1. 2022.05.01 [Computer Animation] B-Spline

   B-Spline B-Splines Is it possible to achieve both $C^{2}$-continuity and local controllability? B-splines can do! Uniform cubic B-spline basis functions Uniform B-spline basis functions Bell-shaped basis function for each control points Overlapping basis functions Control points correspond to knot points Uniform B-splines We have $(n+1)+2$ unknowns. B-spline Properties Variation Diminishing $C^{..
2. 2022.05.01 [Computer Animation] Natural Cubic Spline

   Natural Cubic Spline Natural Cubic Splines $C^{n-1}$-continuity can be achieved from splines of degree $n$. Cubic Splines can have $C^{2}$-continuity. We have $4n$ unknowns. $n$ cubic segments (4 coefficients for each segment) We have $4n$ unknowns $n$ cubic segments (4 coefficients for each segment) We have $(4n-2)$ equations $2n$ equations for end point interpolation $(n-1)$ equations for tang..
3. 2022.05.01 [Computer Animation] Catmull-Rom Spline

   Catmull-Rom Spline Without tangent vectors? Cubic Splines τ(타우) 값에 따른 그래프의 변화 Catmull-Rom Spline Properties Variation Diminishing the curve in 2D space does not oscillate about any straight line more often than the control point polygon. $C^{1}$-continuity Local Controllability Question Find a catmull-rom cubic spline interpolating the four key points with $\tau = 0.5$ $p(t) = L_{0}(t)p_{0} + L_..
4. 2022.05.01 [Computer Animation] Bezier Curve & Bezier Spline

   Bezier Curve & Bezier Spline Basic Functions A linear space of cubic polynomials Monomial basis $(t^{3}, t^{2}, t^{1}, t^{0}$ $x(t) = a_{3}t^{3} + a_{2}t^{2} + a_{1}t + a_{0}$ The coefficients $a_{i}$ do not give tangible(유형의) geometric meaning. Bezier Control Points Control Points $b_{0}, b_{1}, b_{2}, b_{3}$ Demo http://blogs.sitepointstatic.com/examples/tech/canvas-curves/bezier-curve.html Ca..
5. 2022.05.01 [Computer Animation] Keyframing and Splines

   Keyframing and Splines What is Motion? Motion is a time-varying transformation from body local system to world coordinate system. (in a very narrow sense) Transformation Rigid Transformation(강체 변환) Rotate + Translate 3x3 orthogonal matrix + 3-vector $T : x → Rx + b$ Affine Transformation(어파인 변환) Scale + Shear + Rigid Transformation 3x3 matrix + 3-vector $T: x → Ax + b$ Homogeneous Transformation..