3D Geometry and Transformations 11 고려대학교 컴퓨터학과 김 창 헌

Contents Translation Scaling Rotation Other transformations Transformation of coordinate systems

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

3D Translation Translation of a Point y x z

3D Scaling Uniform scaling y x z

Relative Scaling Scaling with a selected fixed position x z y Original position Translate Scaling Inverse Translate

3D Rotation 1. Coordinate-Axes Rotations 좌표축을 기준으로 회전 2. General Three-Dimensional Rotations 좌표축에 평행한 회전축 기준 회전 임의의 회전축(직선) 기준 회전

Coordinate-Axis Rotations y X축 중심 회전 Z 축 중심 회전 x z x축 중심 회전 y z y x z축 중심 회전 Y축 중심 회전 x z y축 중심 회전

Order of rotations affects the final position of an object

Rotation about an Principal axis  좌표축과 평행한 회전축 중심 회전 물체를 좌표축과 평행하게 이동 (회전축이동) 회전 물체를 원위치로 이동 (회전축 원위치)

Rotation about an arbitrary axis  Basic Idea 1. 원점을 지나도록 회전축을 평행이동 2. 좌표축과 일치하도록 회전축을 회전 3. 축에 대한 회전 4. 회전축을 원래 방향으로 역회전 5. 회전축을 원위치로 평행 이동 y T (x2,y2,z2) R (x1,y1,z1) R-1 x T-1 z

Rotation about an arbitrary axis Step 1. Translation (x2,y2,z2) (x1,y1,z1) x z y

Rotation about an arbitrary axis Step 2. Establish [ TR ]x x axis y (0,b,c) (a,b,c) Projected Point   x z Rotated Point

Rotation about an arbitrary axis Step 3. Rotate about y axis by  y (a,b,c) l Projected Point d x  (a,0,d) Rotated Point z

Rotation about an arbitrary axis Step 4. Rotate about z axis by the desired angle  y l x  z

Rotation about an arbitrary axis Step 5. Apply the reverse translation to place the axis back in its initial position x z y l

Rotation about an arbitrary axis Ex) Find the new coordinates of a unit cube rotated about an axis defined by its endpoints A(2,1,0) and B(3,3,1). Step1. Translate point A to the origin y B’(1,2,1) A’(0,0,0) x A Unit Cube z

Rotation about an arbitrary axis Step 2. Rotate axis A’B’ about the x axis by and angle , until it lies on the xz plane. y Projected point (0,2,1) B’(1,2,1) l  x z B”(1,0,5)

Rotation about an arbitrary axis Step 3. Rotate axis A’B’’ about the y axis by and angle , until it coincides with the z axis. y l  x (0,0,6) B”(1,0,  6) z

Rotation about an arbitrary axis Step 4. Rotate the cube 90° about the z axis Finally, the concatenated rotation matrix about the arbitrary axis AB becomes,

Rotation about an arbitrary axis

Rotation about an arbitrary axis Multiplying [TR]AB by the point matrix of the original cube

3D Reflections & Shears Reflection relative to the xy plane z-axis shear y y z z x x

Transformation of Coordinate System Front-Wheel Tractor System World Coordinate Coordinate Coordinate

Transformation of Coordinate System Use of Multiple Coordinate System zworld