Example: Suppose T is a Euclidean transformation T(x) = Ax + b. Find A and b if T(0,0) = G0, T(1,0) = G1, and T(0,1) = G2. In this case G0 = (2,3) and the segment G0G1 makes a 45-degree angle with the horizontal.
Some problems on Euclidean matrix transformations
What is rotation T? Given image of O and angle of rotation of x-axis.
Example: Suppose T is a Euclidean transformation T(x) = Ax + b. Find A and b if T(0,0) = G0, T(1,0) = G1, and T(0,1) = G2. In this case G0 = (2,3) and the segment G0G1 makes a 45-degree angle with the horizontal.
What is rotation T? Given center of rotation direction of image of x-axis.
Example: Suppose T is a Euclidean transformation T(x) = Ax + b. Find A and b if T(0,0) = (0,0) and the x-axis is rotated to the ray OP, where P = (5, 12)?
What is rotation T? Given center of rotation and angle of rotation
Example: Suppose T is a Euclidean transformation T(x) = Ax + b. Find A and b the center of rotation is (2,1) and the angle of rotation is 120 degrees.
What is the composition of two glide reflections?
Example 1: Suppose S is a glide reflection, S(x) = Mx + b, and suppose T is a glide reflection T(x) = Mx + c, with the same matrix M. What does this mean geometrically about the relationship between S and T? Write down a typical reflection matrix M with letters as entries. Show that M^2 = I. What kind of transformation is ST? What is its matrix formula? How about S^2?
Example 2: Suppose S is as above and U(x) is also a glide reflection, with U(x) = Nx + d. Study the matrix product MU and explain why this is a rotation matrix. Then compute SU and explain what kind of transformation this is.