CurvesSurfaces - 3D Curves and Surfaces Implicit Form One...

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Unformatted text preview: 3D Curves and Surfaces Implicit Form One or more equations describe the relationship between the co-ordinates of the points of curve. ex: f(x,y) = 0 : Implicit form of any curve lying on the xy plane. Line: ax+by+c = 0 Circle: x 2 +y 2 –1 = 0 Implicit Form (continued) 3D Examples: Plane: Ax + By + Cz + D = 0 Sphere: x 2 +y 2 +z 2-r 2 =0 or (x/r) 2 + (y/r) 2 + (z/r) 2 – 1 = 0 Elipsoid: (x/r x ) 2 + (y/r y ) 2 + (z/r z ) 2 – 1 = 0 Torus: 1 = r θ = u x z ( 29 2 2 2 axial =- +- r z r x Torus: 1 = r θ = u x z 2 2 2 axial 2 2 =- + - + r z r y x y Parametric Form A curve is a vector valued function of one parameter. It maps a straight line segment into Euclidean 3D space. Each co-ordinate of the curve is represented separately as a function of the independent parameter. ( 29 ( 29 ( 29 b u a u f u f u f u z u y u x ≤ ≤ = where ) ( ) ( ) ( 3 2 1 Parametric Functions 1 = r θ = u π 2 where sin cos ) ( ) ( ) ( ≤ ≤ = u u u u z u y u x Implicit vs. Parametric By adding a z co-ordinate, the parametric method easily represent curves in 3-D space. Implicit vs. Parametric By adding a z co-ordinate, the parametric method easily represent curves in 3-D space. : 2D in rep Implicit line. a of Equation : ex = + + c by ax Implicit vs. Parametric By adding a z co-ordinate, the parametric method easily represent curves in 3-D space. ? ? : 3D in rep Implicit : 2D in rep Implicit line. a of Equation : ex = + + c by ax Implicit vs. Parametric By adding a z co-ordinate, the parametric method easily represent curves in 3-D space. + + = = + + u a a u a a u y u x c by ax y y x x 1 1 ) ( ) ( : 2D in rep Parametric ? ? : 3D in rep Implicit : 2D in rep Implicit line. a of Equation : ex Implicit vs. Parametric By adding a z co-ordinate, the parametric method easily represent curves in 3-D space. + + + = + + = = + + u a a u a a u a a u z u y u x u a a u a a u y u x c by ax z z y y x x y y x x 1 1 1 1 1 ) ( ) ( ) ( : 3D in rep Param etric ) ( ) ( : 2D in rep Param etric : 2D in line a of rep Im plicit : ex Parametric Form A surface is a vector valued function of two parameter. It maps a rectangle into Euclidean 3D space....
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This note was uploaded on 08/22/2010 for the course CAP 6701 taught by Professor Staff during the Spring '10 term at University of Central Florida.

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CurvesSurfaces - 3D Curves and Surfaces Implicit Form One...

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