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08_Curves&amp;Surfaces(ch8)

# 08_Curves&amp;Surfaces(ch8) - Curves and Surfaces...

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Curves and Surfaces Chapter 8

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Representation of Curves and Surfaces n Key words n surface modeling n parametric surface n continuity, control points n basis functions n Bezier curve n B-spline 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?

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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 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

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Intrinsic and Extrinsic Properties n Intrinsic properties n shape definitions that use intrinsic properties (of the shape) are axis-independent . ) 3 , 7 ( ) , ( ) 3 , 2 ( ) , ( 1 0 ) 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 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.

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Curve & Surface Models n Explicit/implicit representations n Parametric/non-parametric representations n Approximation n polygon mesh : a collection of edges, vertices, and polygons
Nonparametric Explicit 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 Single-valued function 2 2 2 2 x r y x r y - - = - = ) ( x f y =

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Nonparametric Implicit Representation n n Define curves as a solution of equation system n e.g., a circle: 0 ) , , ( = z y x f 0 ) , ( 2 2 2 = - + = r y x y x f
Nonparametric Implicit Representation n Algebraic quadric surfaces : f is a polynomial of degree <= 2 0 2 2 2 2 2 2 ) , , ( 2 2 2 = + + + + + + + + + = k jz hy gx fxz eyz dxy cz by ax z y x f 0 0 0 1 : 0 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

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Nonparametric Implicit Representation n Coefficients determine geometric properties n Can represent closed or multi-valued curves n Easy to classify point-membership n Hard to render (have to solve non-linear equation system) 0 1 : sphere 2 2 2 = - + + z y x
Parametric Curve 30 31 2 32 3 33 20 21 2 22 3 23 10 11 2 12 3 13 ) ( ) ( ) ( a t a t a t a t z a t a t a t a t y a t a t a t a t x + + + = + + + = + + + =

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Parametric Curve (Example) ) sin , (cos ) ( 1 : 1 0 ) 1 ( ) 1 ( ) ( ) ( 2 1 2 1 2 1 2 2 2 1 1 1 q q q = - £ £ + - = + - = = = Q t t P P t ty y t y tx x t x ,y x P ,y x P circle unit A functions blening : , points control , to from line A
Parametric Curve Characteristics n Simple to render n evaluate parameter function n

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08_Curves&amp;Surfaces(ch8) - Curves and Surfaces...

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