object 3d(2)

Parametric Curves 고려대학교 컴퓨터 그래픽스 연구실

Overview ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Introduction ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Requirements ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Mathematical Splines ,[object Object],[object Object],[object Object],[object Object],[object Object]

Parametric Curves ,[object Object],[object Object],[object Object]

Parametric Polynomial Curves ,[object Object],[object Object],[object Object],[object Object]

Piecewise Parametric Polynomial Curves ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Bezier Curves ,[object Object],[object Object],[object Object],[object Object]

Bezier Curves: Basic Properties ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Bezier Curves: Explicit Formulation ,[object Object],[object Object]

Explicit Formulation (cont.) ,[object Object],[object Object],[object Object]

Bezier Curves: More Properties ,[object Object],[object Object],[object Object]

More Properties (cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Cubic Curves ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Matrix Form of Bezier Curves ,[object Object]

Display: Recursive Subdivision ,[object Object],[object Object]

Display (cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Splines ,[object Object],[object Object],[object Object],[object Object],[object Object]

Advantages of Splines ,[object Object],[object Object],[object Object],[object Object]

Tangent (G 1 ) Continuity ,[object Object],[object Object],[object Object]

Explanation ,[object Object],[object Object],[object Object],Q n (u) Q n (u) Q n (u) Q n+1 (u) Q n+1 (u) Q n+1 (u)

Curvature (C 2 ) Continuity ,[object Object],[object Object]

Operator Calculus ,[object Object],[object Object],[object Object],[object Object]

Taking the Derivative ,[object Object],[object Object],[object Object],[object Object]

Specializing to n=3 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],V 0 V 1 V 2 V 3 Q’(1) Q’(0) W 0 W 1 W 2 W 3

Taking the Second Derivative ,[object Object],[object Object]

Second-Order Continuity ,[object Object],[object Object],[object Object],V 0 V 1 V 2 V 3 Q’’(1) Q’’(0) Q’(0) Q’(1)

C 3 Continuity V 0 V 1 V 2 V 3

C 3 Continuity V 0 V 1 V 2 V 3 W 0 W 1 W 2 W 3

C 3 Continuity ,[object Object],[object Object],[object Object],V 0 V 1 V 2 V 3 W 0 W 1 W 2 W 3

Creating Continuous Splines ,[object Object],[object Object],[object Object],[object Object]

C 2 Interpolating Splines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],joint

In-Depth Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

End Conditions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

C 2 Interpolating Splines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Global vs. Local Control ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

B-Splines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Explanation ,[object Object],[object Object],V 0 V 1 B 0 B 1 B 2 B 3 V 2 V 0 V 1 V 2 V 3

B-Spline Construction n+1 control points polynomials of degree d-1

B-Spline Construction ,[object Object],n+1 control points polynomials of degree d-1

B-Spline Properties ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Algebraic Construction of B-Splines ,[object Object],[object Object],[object Object],[object Object],[object Object]

Algebraic Construction of B-Splines (cont.) ,[object Object]

Drawing B-Splines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Multiple Vertices ,[object Object],[object Object],[object Object],[object Object],[object Object]

Catmull-Rom Splines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Derivation of Catmull-Rom Splines ,[object Object],[object Object],[object Object],[object Object],[object Object],B 1 B 3 B 0 B 2 =V 3 =V 0

Derivation of Catmull-Rom (cont.) ,[object Object],[object Object],B 1 B 3 B 0 B 2 =V 3 =V 0

Matrix Formulation ,[object Object],[object Object]

Properties ,[object Object],[object Object],[object Object],[object Object],[object Object]

object 3d(2)

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