# Curves - Curves and Surfaces Chap. 8 Intro. to Computer...

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Curves and Surfaces Chap. 8 Intro. to Computer Graphics, Spring 2009, Y. G. Shin

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Representation of Curves and Surfaces Key words surface modeling parametric surface continuity, control points basis functions Bezier curve B-spline curve
All shapes can be described in terms of points. But, it is impractical to enumerate the points that comprise a shape 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 Intrinsic properties CFG properties B has four sides All four sides of B have equal length All four angles of B are 90 o, ...... Extrinsic properties Coordinate dependent geometry B has two horizontal sides Two vertical sides vertices of B are at x y B ) , ( and ), , )( , ( ), , ( 4 4 3 3 2 2 1 1 y x
intrinsic and extrinsic properties shape definitions that use extrinsic properties (of the shape) are dependent on the coordinate system used. a line: y = 3, 2 ≤ x ≤ 7 Axis dependency 2 7 3

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intrinsic and extrinsic properties 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 Explicit/implicit Parametric/non-parametric Approximation polygon mesh : a collection of edges, vertices, and polygons
Nonparametric explicit representation x = x y = f(x) successive values of y can be obtained by plugging in successive values of x. easy to generate polygons or line segments single-valued function 2 2 2 2 x r y x r y = =

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Nonparametric implicit representation Define curves as solution of equation system E.g., a circle: 0 ) , , ( = z y x f 2 2 2 r y x = +
Nonparametric implicit representation 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 paraboloid z y x corn y x cylinder z y x sphere

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

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Parametric Curve (Example) )) 2 u sin( ), 2 u (cos( ) u ( Q circle unit functions blening : t 1 , t points control : P , P 1 t 0 ty y ) t 1 ( y tx x ) t 1 ( x ) ,y (x P to ) ,y (x P from line 2 1 2 1 2 1 2 2 2 1 1 1 π π = + = + = = =
Parametric Curve Characteristics Simple to render evaluate parameter function Can represent closed or multi-valued curves Curve or surface can be easily translated or rotated.

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## This note was uploaded on 04/29/2010 for the course CSE 4190.411 taught by Professor Shinyeonggil during the Fall '08 term at Seoul National.

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Curves - Curves and Surfaces Chap. 8 Intro. to Computer...

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