1. Coordinate Reference Frames ♦Vector Space Vectors have magnitude and direction Vectors have no posotion ♦Affine Space Vector Space + Points (location) Possible Geometric Transform P Truncated plane (No Origin) : Vector Space P 기준의 새 좌표계 설정 : Affine Space

2. Coordinate Reference Frames ♦Affine Space(2) Represent Vector : W W = a1v1 + a2v2 + a3v3 Represent Point : P P = P0 + b1v1 + b2v2 + b3v3 벡터와 점의 구분을 위해서 1X4 행렬의 사용 e3e3 e1e1 e2e2 e3e3 e1e1 e2e2 Basis vectors located at the origin

3. Points & Vector ♦ Points - 좌표계에서의 한점을 차지, 위치표시 ♦Vector (2D) - 두 position 간의 경로차 -Magnitude 와 Direction 으로도 표기 V P2P2 P1P1 x1x1 x2x2 y1y1 y2y2

4. Vector ♦Vector (3D) ♦ Vector Addition and Scalar Multiplication    V x z y

5. Between two Vector ♦Dot Product –Inner Product 라고도 함 – 두 벡터의 사잇각 –V1 V1 = 0 두 벡터가 직각임을 알수가 있다 |V 2 |cos   V2V2 V1V1 Commutative Distributive

6. Between two Vector ♦Cross Product – 두 벡터와 직교하는 또 다른 벡터를 얻을 수 있다 –3D Model shading Relation – 연산의 순서 중요 V1V1 V2V2 V1  V2V1  V2  ※ u x,u y,u z 를 각 축의 단위 vector 라 하면, Properties AntiCommutative Not Assotiative Distributive

7. Geometric Transformations ♦Geometric Transform 기존 물체 속성의 변경 Translate, Rotate, Scale ♦Purpose View 의 조절 물체 (Model) 의 조작 및 조정

8. Position Standard ♦World Coordinates(Global Coordinate) –Only One ♦Modeling Coordinates(Local Coordinate) –Each Object

9. Transformations - Translate ♦Translate

10. Transformations - Rotate ♦Rotate(1) –Origin x = r cos , y = r sin  x’ = r cos (  +  ) = r cos  cos  - r sin  sin  y’ = r sin (  +  ) = r cos  sin  + r sin  cos   x’= x cos  - y sin , y’ = x sin  + y cos  Z axis roteate X axis rotate Y axis rotate (x,y) r  (x’,y’)  r

11. Transformations - Rotate ♦Rotate(2) –Arbitrary Point Translate Fixed Point General Rotate Translate Fixde Point P ’ = T^RTP

12. Transformations - Rotate ♦Rotate(3) –Arbitrary Axis Translation : Translate Arbitrary Axis (x 2,y 2,z 2 ) (x 1,y 1,z 1 ) x z y

13. Transformations - Rotate Establish [ T R ]  x x axis (a,b,c) (0,b,c) Projected Point   Rotated Point x y z

14. Transformations - Rotate Rotate about y axis by  (a,b,c) (a,0,d)  l d x y Projected Point z Rotated Point

15. Transformations - Rotate Rotate about z axis by the desired angle   l y x z

16. Transformations - Rotate Apply the reverse transformation to place the axis back in its initial position x l l z

17. Transformations - Scale ♦Scale –Uniform Scaling X’ = X * Sx, Y’ = Y * Sy Z’ = Z * Sz x z y

18. Transformations - Scale –Fixed Point xxx x z z zz y yy y Original positionTranslateScalingInverse Translate

19. Transformations - Shear ♦Shear X X Y (x’,y’) A Z Y (x,y) x’ = x + y cotA, y’ = y, z’ = z

20. OpenGL Function ◊glPushMatrix – glTranslatef, glRotatef 등의 기록 ◊glPopMatrix – 저장된 glTranslatef, glRotatef 등의 기록 제거 ◊glLoadMatrix – 특정 Matrix 의 호출 ◊glTranslatef – Translate Matrix 기록 ◊glRotatef – Rotate Matrix 기록 ◊glScaled, glScalef – Scale Matrix 기록 ◊glBegin – delimit the vertices of a primitive or a group of like primitives ◊glVertex3fv