Lecture_Chapter3_b

Higher order parametric curves linear representation

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Unformatted text preview: esentations •  Parametric representation f (u) = (x(u), y (u), z (u)) Coordinates expressed in terms of one independent variable, the parameter. ECS 175 Chapter 3: Object Representation 46 Parametric Representation •  Set of control points •  Piecewise linear representation vs. higher-order parametric curves Linear representation Linear interpolation ECS 175 Chapter 3: Object Representation Bézier curve (cubic) Approximation 47 Bézier Curves •  Relaxing the interpolation condition allows for more stable curve construction (approximation) •  Goals: •  Easy construction •  Suitable for rendering and basic geometric operations •  Control over smoothness •  Stability •  Easy computation of derivatives ECS 175 Chapter 3: Object Representation 48 Bézier Curves •  Geometric construction of a cubic Bézier curve Four control points are weighted to construct a cubic Bézier curve. ECS 175 Chapter 3: Object Representation 49 Bézier Curves •  Bézier curve constructed by repeated linear interpolation f ( u) = n ￿ bin (u)pi curve blends together control po...
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