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

Info iconThis preview shows pages 1–16. Sign up to view the full content.

View Full Document Right Arrow Icon
Curves and Surfaces Chap. 8 Intro. to Computer Graphics, Spring 2009, Y. G. Shin
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Representation of Curves and Surfaces Key words surface modeling parametric surface continuity, control points basis functions Bezier curve B-spline curve
Background image of page 2
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?
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Curve & Surface Models Explicit/implicit Parametric/non-parametric Approximation polygon mesh : a collection of edges, vertices, and polygons
Background image of page 8
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 = =
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 = +
Background image of page 10
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
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 12
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 + + + = + + + = + + + =
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 = + = + = = =
Background image of page 14
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.
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 16
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 85

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

This preview shows document pages 1 - 16. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online