Arbitrary Rotations in 3 D Lecture 18 Wed

Arbitrary Rotations in 3 D Lecture 18 Wed, Oct 8, 2003

Rotations about an Arbitrary Axis Consider a rotation through an angle about a line through the origin. We must assign a positive direction to the line. We do this by using a vector rather than a line.

Rotations about an Arbitrary Axis The function call gl. Rotatef(angle, vx, vy, vz) will create a single matrix that represents a rotation about the vector v = (vx, vy, vz). There are some clever ways of obtaining this matrix. We will learn an elementary (nonclever) method.

Rotations about an Arbitrary Axis To find the matrix of an rotation through an angle about a vector v emanating from the origin n n Rotate about the y-axis so that v is in the xy-plane. Let this angle be and call the new vector v’. Rotate about the z-axis so that v’ is aligned with the positive x-axis. Let this angle be – .

Rotations about an Arbitrary Axis n n n Rotate about the x-axis through angle . Rotate about the z-axis through angle . Rotate about the y-axis through angle –.

Rotations about an Arbitrary Axis Find the matrix of a rotation of angle about unit vector v = (vx, vy, vz). v

Rotations about an Arbitrary Axis Rotate v about the y-axis through angle to get vector v’. Call this matrix Ry( ). v

Rotations about an Arbitrary Axis Rotate v’ about the z-axis through angle – to get vector v’’. Call this matrix Rz(– ). v’ v’’

Rotations about an Arbitrary Axis Rotate about the x-axis through angle . Call this matrix Rx( ). v’’

Rotations about an Arbitrary Axis Then apply Rz(- )– 1 followed by Ry( )– 1. The matrix of the rotation is the product Ry( )– 1 Rz(- )– 1 Rx( )Rz(- )Ry( ) This is the same as Ry(- )Rz( )Rx( )Rz(- )Ry( )

Example: Rotation Find the matrix of the rotation about v = (1/3, 2/3) through 90. v projects to (1/3, 0, 2/3) in the xzplane. = tan– 1(2). cos( ) = 1/ 5, sin( ) = 2/ 5.

Example: Rotation The matrix of this rotation is 1/ 5 Ry( ) = 0 – 2/ 5 0 0 1 0 0 2/ 5 0 1/ 5 0 0 1

Example: Rotation v rotates into the vector v’ = ( 5/3, 2/3, 0). = tan– 1(2/ 5). cos(- ) = 5/3, sin(- ) = -2/3.

Example: Rotation The matrix of this rotation is 5/3 Rz(- ) = -2/3 0 0 2/3 5/3 0 0 0 0 1

Example: Rotation Now apply the original rotation of 90 to the x-axis. The matrix is Rz( ) = 1 0 0 0 1 0 0 -1 0 0 0 1

Example: Rotation Reverse the rotation through angle . 5/3 Rz( ) = 2/3 0 0 -2/3 5/3 0 0 0 0 1

Example: Rotation Reverse the rotation through angle . 1/ 5 Ry(- ) = 0 2/ 5 0 0 1 0 0 -2/ 5 0 1/ 5 0 0 1

Example: Rotation The product of these five matrices is the matrix of the original rotation. R( ) = 1/9 8/9 -4/9 0 -4/9 7/9 0 8/9 1/9 4/9 0 0 1

Example: Rotation How can we verify that this is correct? If P is a point on the axis of rotation, then the transformed P should be the same as P. If v is orthogonal to the axis, then the transformed v should be orthogonal to v. If we apply the transformation 4 times, we should get the identity.

Example: Rotation A point on the axis is of the form (t, 2 t, 1). The matrix maps this point to (t, 2 t, 1).

Example: Rotation Compute R( )2 = R( )4 = -7/9 4/9 0 4/9 -1/9 8/9 0 4/9 8/9 -1/9 0 0 1 1 0 0 0 0 1

Special Properties of Rotation Matrices Every rotation matrix has the following properties. n n Each row or column dotted with itself is 1. Each row (column) dotted with a different row (column) is 0. A matrix with this property is called orthonormal. Its inverse equals its transpose.

Special Properties of Rotation Matrices Verify that R( ) is orthonormal. Consider each row of the matrix to be a point. Where does the matrix map each row? These are the points that map to the points (1, 0, 0), (0, 1, 0), and (0, 0, 1). What about the columns? What do they represent?