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(1221) Each unit quaternion and its antipole -q = (-qo, -ql, -q2, -q3) represent a rotation in three dimensions The rotation matrix for rigid body transformation can be obtained from the elements of the unit quaternion:

qr -

Required Net Area Compressive Strength of Clay Masonry Units (psi) When used with Type M or S Mortar 1,700 3,350 4,950 6,600 8,250 9,900 13,200 When used with Type N Mortar 2,100 4,150 6,200 8,250 10,300

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(1222) After the unit quaternion is computed, Equation 1222 can be used to compute the rotation matrix so that the rotation can be applied to each point using matrix multiplication The unit quaternion is closely related to the angle and axis representation for rotation, described in Section 1222 A rotation can be represented as a scalar (j for the amount of rotation and a vector (w x , wy , wz ) for the axis of rotation A quaternion has a scalar part, which is related to the amount of rotation, and a vector part, which is the axis of rotation Let the axis of rotation be represented by the unit vector (w x , W y , wz ) and use i, j, and k to represent the coordinate axes so that the unit vector for the rotation axis can be represented as (1223) The unit quaternion for a counterclockwise rotation by

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about this axis is (1224) (1225)

cos 2"

(International Building Code 2003, MSJC Specification 1999, and MSJC Specification 2002)

The first term is called the scalar (real) part ofthe quaternion, and the other terms are called the vector (imaginary) part A point p = (x, y, z) in space has a quaternion representation r which is the purely imaginary quaternion with vector part equal to p, r = xi + yj

(1226)

IBC Compressive Strength of Concrete Masonry For Net Area Compressive Strength of Masonry (psi) 1,000 1,500 2,000 2,500 3,000

Let p' be point p rotated by matrix R( q),

(1227)

If r is the quaternion representation for point p, then the quaternion representation r' for the rotated point can be computed directly from the elements of quaternion q, r' = qrq*, (1228)

Required Net Area Compressive Strength of Concrete Masonry Units (psi) When used with Type M or S Mortar 1,250 1,900 2,800 3,750 4,800 When used with Type N Mortar 1,300 2,150 3,050 4,050 5,250

where q* = (qo, -qx, -qy, -qz) is the conjugate of quaternion q and quaternion multiplication is defined as

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Examining the entire operation from the crow s nest will allow you to find the problems in the system Oftentimes, the problems have become so imbedded in the process that they aren t even realized as waste If an organization is to achieve six sigma, employees must be trained to look for waste in every aspect of their jobs

(roqo - rxqx - ryqy - rzqz,

(1229)

(International Building Code 2000, IBC 2003, MSJC Specification 1999, and MSJC Specification 2002)

+ rxqo + ryqz - rzqy, roqy - rxqz + ryqo + rzqx, roqz + rxqy - ryqx + rzqo)

Rigid body transformations can be conveniently represented using a sevenelement vector, (qO, ql, q2, q3, q4, q5, q6), in which the first four elements are a unit quaternion and the last three elements are the translation If we let R( q) denote the rotation matrix corresponding to the unit quaternion in this representation, then the rigid body transformation is

used to evaluate masonry assembly compressive strength ASTM C 1314 Standard Test Method for Compressive Strength of Masonry Prisms describes the procedure for evaluating the compressive strength of masonry assemblies Test values so obtained are compared to the specified compressive strength of masonry, f m, required by the project specifications in accordance with the relevant building code The compressive strength of veneer assemblies and glass unit masonry assemblies need not be determined because their structural design does not depend upon this parameter Single wythe and multiple wythe masonry assemblies that are designed by the empirical or rule-of-thumb provisions of the applicable building code are also exempted from the requirement for determining assembly compressive strength for the same reason

(1230)

We will use quaternions in the next section to present an algorithm for solving the absolute orientation problem

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