nehe教程四.ppt

Chapter 4 2-D Transformations Cartesian coordinates Linear transformations, affine transformations Homogeneous coordinates Translation Scaling Rotation Reflection Combinations of transformations Animations Affine Transformations on vertices An affine transformation is a linear transformation followed by a translation. Its 2D general form is Homogeneous coordinates of vertices A point in homogeneous coordinates (x, y, w), w ≠ 0, corresponds to the 2-D vertex (x/w, y/w) in Cartesian coordinates. Conceive that the Cartesian coordinates axes lies on the plane of w = 1. The intersection of the plane and the line connecting the origin and (x, y, w) gives the corresponding Cartesian coordinates. Scaling relative to the origin (changing size) The new position: In general, In homogeneous coordinates Rotation about the origin In Cartesian, x’ = r cos(?+?) = r cos? cos? ? r sin? sin? = x cos? - y sin? y’ = r sin(?+?) = r sin? cos? + r cos? sin? = x sin? + y cos? Conduct a sequence of transformations Translate the right-angle vertex to the origin (Tx = -1, Ty = -1) Rotate 45o (?/4 radian) sin ?/4 = cos ?/4 = 0.7071 The computation of [ ] from [ ] [ ] is called matrix multiplication. The general form is: A sequence of transformations can be lumped in a single matrix via matrix multiplications In OpenGL, all the model transformations are accumulated in the current transformation matrix (CTM). All vertices of an object will be transformed via this matrix before the object is drawn. A system stack is provided for storing the backup copies of the CTM during execution. We usually save the CTM in the stack before the drawing of a transformed object. And restore the original CTM afterwards. To save a copy of the CTM in the stack glPushMatrix(); To overwrite the CTM with the top matrix in the stack glPopMatrix(); To specify a translation glTranslatef( double Tx, double Ty, 0.0) The system first genera