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Unformatted text preview: Curves and Surfaces Curves and Surfaces Chapter 8 Representation of Curves and Representation of Curves and Surfaces Surfaces n Key words n surface modeling n parametric surface n continuity, control points n basis functions n Bezier curve n Bspline curve n All shapes can be described in terms of points. But, it is impractical to enumerate the points that comprise a shape n We define shape indirectly through expressions that relate certain properties of points that comprise them. Why we need surface models? Why we need surface models? Intrinsic and Extrinsic Properties Intrinsic and Extrinsic Properties n Intrinsic properties n CFG (Coord. Free Geometry) properties n CFG: A style of expressing geometric objects and relations that do not rely on any specific coordinate system n B has four sides n All four sides of B have equal length n All four angles of B are 90 o, ...... ) , ( and ), , )( , ( ), , ( 4 4 3 3 2 2 1 1 y x y x y x y x n Extrinsic properties n Coordinate dependent geometry n B has two horizontal sides to an axis n The vertices of two vertical sides of B are at x B y Intrinsic and Extrinsic Properties Intrinsic and Extrinsic Properties n Extrinsic properties n Shape definitions that use extrinsic properties (of the shape) are dependent on the coordinate system used. n a line: ¬ axis dependence 2 7 3 7 2 , 3 £ £ = x y Intrinsic and Extrinsic Properties Intrinsic and Extrinsic Properties n Intrinsic properties n shape definitions that use intrinsic properties (of the shape) are axisindependent . ) 3 , 7 ( ) , ( ) 3 , 2 ( ) , ( 1 ) 1 ( ) 1 ( 2 2 2 1 1 1 2 1 2 1 = = = = £ £ + = + = y x y x y y x x p p p p p p t tp p t y tp p t x Axis Independence Axis Independence A mathematical representation of a line/curve is axis independent if its shape depends on only the relative position of the points defining its characteristic vectors and is independent of the coordinate system used. Curve & Surface Models Curve & Surface Models n Explicit/implicit representations n Parametric/nonparametric representations n Approximation n polygon mesh : a collection of edges, vertices, and polygons Nonparametric Explicit Nonparametric Explicit Representation Representation n n Successive values of y can be obtained by plugging in successive values of x . n easy to generate polygons or line segments n Singlevalued function 2 2 2 2 x r y x r y = = ) ( x f y = Nonparametric Implicit Nonparametric Implicit Representation Representation n n Define curves as a solution of equation system n e.g., a circle: ) , , ( = z y x f ) , ( 2 2 2 = + = r y x y x f Nonparametric Implicit Nonparametric Implicit Representation Representation n Algebraic quadric surfaces : f is a polynomial of degree <= 2 2 2 2 2 2 2 ) , , ( 2 2 2 = + + + + + + + + + = k jz hy gx fxz eyz dxy cz by ax z y x f 1 : 1 : 2 2 2 2 2 2 2 2 2 2 = + + = + = + = + + z y x z y x y x z y x : paraboloid : cone cylinder sphere Nonparametric Implicit Nonparametric Implicit Representation Representation...
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 Fall '08
 Shinyeonggil
 Computer Graphics

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