🗒️ Spline (6)

1. 2022.05.01 [Computer Animation] Quaternion Catmull-Rom Spline

   Quaternion Catmull-Rom Spline Quaternion(사원수) Operations Unit Quaternion Algebra Unit Quaternion space is closed under multiplication and inverse, but not closed under addition and subtraction Rotation Quaternion Exp and Log Scale Quaternion Catmull-Rom Spline Quaternion Bezier Quaternion Bezier Curve Linear Combination of quaternion? De Casteljau by Slerp by De Casteljau Algorithm Multi-linear ..
2. 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^{..
3. 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..
4. 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_..
5. 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..
6. 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..