transformation 3d

3D Geometric Transformation 고려대학교 컴퓨터 그래픽스 연구실

Contents ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Transformation in 3D ,[object Object],3  3 : Scaling, Reflection, Shearing, Rotation 3  1 : Translation 1  3 : Homogeneous representation 1  1 : Uniform global Scaling

3D Translation ,[object Object],x z y

3D Scaling ,[object Object],x z y

Relative Scaling ,[object Object],x x x x z z z z y y y y Original position Translate Scaling Inverse Translate

3D Rotation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Coordinate-Axes Rotations ,[object Object],[object Object],[object Object],z y x z y x z y x

Order of Rotations ,[object Object],[object Object],[object Object]

General 3D Rotations ,[object Object],[object Object],[object Object],[object Object]

General 3D Rotations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],(x 2 ,y 2 ,z 2 ) (x 1 ,y 1 ,z 1 ) x z y R -1 T -1 R T

General 3D Rotations ,[object Object],(x 2 ,y 2 ,z 2 ) (x 1 ,y 1 ,z 1 ) x z y

General 3D Rotations ,[object Object],(a,b,c) (0,b,c) Projected Point   Rotated Point x y z

Arbitrary Axis Rotation ,[object Object],(a,b,c) (a,0,d)  l d x y Projected Point z Rotated Point

Arbitrary Axis Rotation ,[object Object], l y x z

Arbitrary Axis Rotation ,[object Object],x y l l z

Example Find the new coordinates of a unit cube 90º-rotated about an axis defined by its endpoints A(2,1,0) and B(3,3,1). A Unit Cube

Example ,[object Object],A’(0,0,0) x z y B’(1,2,1)

Example ,[object Object],x z y l B’(1,2,1)  Projected point (0,2,1) B”(1,0,  5)

Example ,[object Object],x z y l  B”(1,0,  5) (0,0,  6)

Example ,[object Object],[object Object]

Rotations with Quaternions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Rotations with Quaternions ,[object Object],[object Object],[object Object],[object Object]

Example ,[object Object],[object Object],[object Object]

Other Transformations ,[object Object],[object Object],x z y x z y

Coordinate Transformations ,[object Object],[object Object],[object Object],Local Coordinate System

Coordinate Transformations ,[object Object],[object Object],[object Object],Word Coordinate System

Coordinate Transformations ,[object Object],[object Object],[object Object]

Coordinate Transformations ,[object Object],[object Object],[object Object],[object Object],u ' y u ' z u ' x x y z (0,0,0) y’ z’ x’ u ' y u ' z u ' x ( x 0 , y 0 , z 0 ) x y z (0,0,0)

Coordinate Transformations ,[object Object],y’ z’ x’ u ' y u ' z u ' x u ' y u ' z u ' x u ' y u ' z u ' x x y z (0,0,0) x y z (0,0,0) x y z (0,0,0)

transformation 3d

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transformation 3d