图形学第五章.pptVIP

5. Two-Dimensional Geometric Transformations 5-1 BASIC TRANSFORMATIONS Translation p(x,y), p(x, y) x=x+ tx, y=y+ ty (5-1) P= x P’= x y y T= tx ty P’=P+T Rotation p(x,y), p(x, y) x’=rcos(?+θ) =rcos ? cosθ-rsin ? sinθ y’=rsin(?+θ) =rcos ? sinθ+rsin ? cosθ x=rcos ? , y=rsin ? x’= xcosθ – ysinθ y’= xsinθ + ycosθ P = R * P Rotation of a point about an arbitrary pivot position (xr , yr) x’=xr+ (x-xr) cosθ - (y-yr) sinθ y’=yr+ (x-xr) sinθ + (y-yr) cosθ (5-9) Scaling x’=x*sx , y’=y*sy (5-10) P’=S*P (5-12) * Coordinates for the fixed point (Xf, Yf) x’=xf + (x-xf) sx, y = yf + (y - yf ) sy (5-13) 5-2 MATRIX RFPRESENTATIONS AND HOMOGENEOUS COORDINATES P = M1 * P + M2 (5-15) we represent each Cartesian coordinate position (x, y) with the homogeneous coordinate triple (xh, yh, h), where ? x=xh/h y=yh/h , (5-16) (x, y) (xh, yh, 1) For translation P = T(tx, ty) * P (5-18) Rotation (the coordinate origin) P’=R(?) * P (5-20) Scaling (the coordinate origin) P = S(sx, sy) *P (5-22) 5-3 COMPOSITE TRANSFORMATIONS Translations If two successive translation vectors (tx1, tyl) and (tx2, ty2) are applied to a coordinate position P, the final transformed location p is calculated as ? P = T(tx2, ty2) * {T(txl, tyl) * P} = {T(tx2, ty2) * T(tx1, ty1)} * P (5-24) T(tx2, ty2) * T(tx1, tyl) = T(tx1 + ty2, ty1+ ty2) (5-25) Rotation Two successive rotations applied to point P produce the transformed position ? P = R(θ2) *{R(θ1) *P} = {R(θ2) *R(θ2)}*P (5-26) R(θ2) *R(θ1) = R(θ1 +θ2) (5-27) Scalings Concatenating transformation matrices for two successive scaling operations produces the following composite scaling m