3D Geometric Transformation - Welcome to the ：三维几何变换-欢迎来到.ppt

3D Geometric Transformation Point in 3D space Position (x, y, z) Color (r, g, b) Normal (Nx, Ny, Nz) Homogenous Coordinates Position (x,y,z,w) Usually (x,y,z,1) Transformations Translation Scaling Rotation Projection 3D Translation 3D Scaling 3D Sheering 3D Rotation 3D Rotation 3D Rotation Composition of 3D Transformations Composition of 3D Transformations * (x,y,z) N (x,y,z) (x’,y’,z’) T(dx,dy,dz)= 1 0 0 dx 0 1 0 dy 0 0 1 dz 0 0 0 1 To translate the point (x,y,z) by the offset (dx, dy, dx) T(dx,dy,dz) . (x,y,z,1)T =(x+dx, y+dy, z+dz,1) (x,y,z) (x’,y’,z’) S(sx,sy,sz)= sx 0 0 0 0 sy 0 0 0 0 sz 0 0 0 0 1 To scale the vector (x,y,z) by the factors sx,sy,and sz S(sx,sy,sz) . (x,y,z,1)T =(x*sx, y*sy, z*sz,1) (x,y,z) (x’,y’,z’) Sz(hx,hy)= 1 0 hx 0 0 1 hy 0 0 0 1 0 0 0 0 1 To sheer the point (x,y,z) along the x and y axes H(hx,hy) . (x,y,z,1)T = (x+hx*z, y+hy*z, z, 1) Sheer along one or more axes (x,y,z) (x’,y’,z’) Rx(j) = 1 0 0 0 0 cos(j) -sin(j) 0 0 sin(j) cos(j) 0 0 0 0 1 A 90o Rotation of (0,1,0,1) will produce (0,0,1,1) To rotate a point (x,y,z) around the x axis by angle j x (x,y,z) (x’,y’,z’) Ry(j) = cos(j) 0 sin(j) 0 0 1 0 0 -sin(j) 0 cos(j) 0 0 0 0 1 A 90o Rotation of (1,0,0,1) will produce (0,0,1,1) To rotate a point (x,y,z) around the y axis by angle j (x,y,z) (x’,y’,z’) Rz(j) = cos(j) -sin(j) 0 0 sin(j) cos(j) 0 0 0 0 1 0 0 0 0 1 A 90o Rotation of (1,0,0,1) will produce (0,1,01) To rotate a point (x,y,z) around the z axis by angle j x T = t00 t01 t02 t03 t10 t11 t12 t13 t20 t21 t22 t23 t30 t31 t32 t33 We should split the transformation into its generic composing transformations. Then carry these transformation one by one. General 3D transformation looks as the following *