Affine Transformation

Affine Transformations In this lecture, we will continue with the discussion of the remaining affine transformations and composite transformation matrix

Reflection  Reflection produces a mirror image of an object  It is also a rigid body transformation  Reflection is relative the axis of the reflection. In 2-D, the axis are either x and y whereas 3-D includes z axis.  Reflection is actually a special case of scaling (with the negative scale factor)

Reflection (cont.) For 2-D reflection, the transformation matrix F has the Following form F(x) = F(y) = About x-axisabout y-axis

Reflection (cont.) For 3-D reflection, the transformation matrix F has the following form F(z) = Reflection about z-axis

Reflection (cont.) Reflection can be generalized by concatenating rotation and reflection matrices. Example: If reflection at y=x axis (45 degree), the transformations involved are: 1. Clockwise rotation of 45 degree 2. Reflection about x axis 3. Counter clockwise rotation of 45 degree

Shearing  Distort an object by moving one side relative to another  It neither rigid body nor orthogonal transformation. I.e. changing a square into parallelogram in 2-D or cube into parallelepiped in 3-D space  It is normally used to display italic text using regular ones

Shearing (cont.) For 2-D shearing transformation the transformation matrix has The following form X directiony direction

Shearing (cont.) The values can be positive or negative numbers Positive: moves points to the right Negative: moves points to the left

Shearing (cont.) For 3-D space shearing transformations the number of Transformation matrix are many. Basically, to get one transformation matrix for shearing, we can Substitute any zero term in identity matrix with a value like Example below:

Example (3-D shearing) Let us consider a unit cube whose lower left corner coincides With the origin

Example (3-D shearing) Based on the diagram, we want to shear the cube at about the z axis. In this case the face of the cube that lies on the xy-coordinate plane does not move. The face that lies on the plane z=1 is translated by a vector (shx,shy). This is called xy-shear. Under the xy-shear, the origin and x- and y-unit vectors are unchanged. The transformation matrix for this transformation is:

Composition matrix  As can be seen in the previous example, we can actually compose or concatenate many affine transformation matrices to produce a single (resultant) affine transformation matrix  For example, to get a composition matrix for rotation and translation, we perform matrix multiplication.  We can build a composite transformation for a general-case transformation at any other points (besides origin)

Example Let say we want to produce an object that goes through the following transformation operations: translate by (3, -4), M 1 then rotate through 30 degree, M 2 then scale by (2, -1), M 3 then translate by (0, 1.5), M 4 and finally, rotate through –30 degree, M 5 All of these transformation can be represented by a single matrix, M M = M 5 M 4 M 3 M 2 M 1

Example (cont.) Composition matrix for affine transformations can only be produced if we use homogenous coordinates (for translation case). Notice the multiplication is done in reverse order (FILO) Exercise: Build a transformation matrix that a) rotates through 45 degrees b) then scales in x by 1.5 and in y by –2, c) and finally translates through (3,5) Find the image under this transformation of the point (1,2)

Affine transformation in OpenGL Recall our lesson on OpenGL graphics pipeline. CT = Current Transformation

OpenGL All the transformations will be performed in CT by changing the Current transformation matrix. In other way, the CT matrix is An example of composite matrix. Typically, in OpenGL, we will have the following fragment of code in our program before we start performing transformations. glMatrixMode(GL_MODELVIEW); glLoadIdentity();

OpenGL By having this code in our program, we set our matrix Mode to be GL_MODELVIEW and initialize the CT matrix to be identity CT I Once we have set this, we can perform transformation Which means we modify the CT by post-multiplication by a matrix

OpenGL CTCT.T(translation matrix) CTCT.S(scaling matrix) CTCT.R(rotation matrix) CTCT.M(arbitrary matrix) Once all of the transformation matrix multiplications have been done, the vertices will be transformed based on the final (composite) matrix In OpenGL all matrices are in 4 x 4 matrix

OpenGL The three transformation supported in most graphics system (including OpenGL) are translation, rotation with a fixed point of the origin and scaling with a fixed point of the origin. The functions for these transformation in OpenGL are: glTranslatef(dx, dy, dz); glRotatef (angle, vx, vy, vz); glScalef(sx, sy, sz);

OpenGL If we want to perform rotation at any other point using the provided functions in OpenGL, (I.e. 45 degree rotation about the line through the origin and the point (1,2,3) with a fixed point of (4,5,6).) the following code fragment shows us how to perform this transformation. glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(4.0,5.0,6.0); glRotatef(45.0,1.0,2.0,3.0); glTranslatef(-4.0,-5.0,-6.0); NB: Take note at the reverse order implementation

OpenGL For most purposes, rotation, translation and scaling can be used to form our desired object. However, in some cases the provided transformation matrices are not enough. For example, if we want to form a transformation matrix for shearing and reflection. For these transformations, it is easier if we set up the matrix directly. We can load a 4 x 4 homogeneous-coordinate matrix as the current matrix (CT)

OpenGL To load the matrix we call this function: glLoadMatrixf(myarray); Or we can multiply our shearing matrix by calling this function glMultMatrixf(myarray); Where myarray is a one-dimensional array of 16 element arranged by colomns.

OpenGL To define a user-defined matrix (for shearing and reflection) we can follow the same way as shown below: Glfloat myarray[16]; for (i=0;i<3;i++) for(j=0;j=3;j++) myarray[4*j+i] = M[i][j];