Lecture07-2DTransforms 计算机图形学ppt课件.ppt

Rotation Example (cont) Result: x y 0 1 2 3 4 0 1 2 3 4 5 x y 0 1 2 3 4 0 1 2 3 4 5 Triangle Vertices after Rotation: (2, 0.59), (3.41, 2), (1.29, 4.2) Initial Triangle Vertices: (1, 1), (3, 1), (3, 4) Homogeneous Coordinates Problem: Rotation, scale, and shear multiply a matrix with p: p′ = Mp Translation adds a vector to p: p′ = p + T Would like to treat all transformations the same Optimize the hardware Compose transformations Solution: homogeneous coordinates Homogeneous: “of the same or similar kind or nature” Increase point’s dimensionality ? add a third coordinate w: Two homogeneous points (p1 and p2) specify the same 2-D Cartesian point if: p1 = cp2 for some real-valued scalar number c Homogeneous Coordinates (cont.) With homogeneous coordinates, the x-y plane is a two-dimensional sub-space in 3-D Although homogeneous points have three coordinates, they correlate to positions on a 2-D plane Can use any 2-D plane that doesn’t include the origin For simplicity, choose the plane w = 1 ? [x y 1]T = [X Y 1]T x y w Homogeneous point 2-D Cartesian coordinates Homogeneous Coordinates For 2D transformations, the points are now 3-vectors And the transformation matrices are 3x3 matrices Homogeneous Coordinates: Translation What does the translation matrix look like? To translate a point p to point p’, we need with and Homogeneous Coordinates: Translation So Performing the multiply gives us: Homogeneous Coordinates: Translation From We know that Similarly, Implies And implies Homogeneous Coordinates: Translation So, This is the 2D translation matrix We apply this matrix to each vertex of a polygon to translate the polygon as a whole Does it work? Homogeneous Coordinates: Translation In this example, Polygon vertices are: x y 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Homogeneous Coordinates: Scale What does the scale matrix look like? To scale a point p to point p’, we need with and Homogeneous Coordinates: Scale So Performing the mult