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rotation matrix R defined in Equation 1210 in terms of these angles are cos > cos K, sin w sin > cos K, + cos w sin K, - cos w sin > cos K, + sin w sin K, - cos > sin K, - sin w sin > sin K, + cos w cos K, cos w sin > sin K, + sin w cos K, sin > - sinw cos > cos w cos >

(1213)

When used with Type N Mortar 3,000 5,500 8,000 10,500 13,000

Although this is a common representation for rotation, determining the rotation by solving for the Euler angles leads to algorithms that are not numerically well conditioned since small changes in the Euler angles may correspond to large changes in rotation Calibration algorithms either solve for the entries of the rotation matrix or use other representations for the rotation angles such as quaternions The rotation matrix is an orthonormal matrix,

RTR=I ,

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When the materials and construction conform to these requirements, the compressive strength of the masonry assembly can be determined by referencing tables In each table, the compressive strength of the masonry assembly is based upon the mortar type and the masonry unit compressive strength Table 41 provides the compressive strength of clay masonry according to IBC 2000 Table 42 provides the compressive strength of clay masonry according to IBC 2003 and Table 43 gives the compressive strength of concrete masonry according to IBC When the criteria for using the unit strength method are not met or the specified compressive strength of masonry exceeds the values listed in the tables, the prism strength method must be

(1214)

where I is the identity matrix This means that the matrix inverse is just the transpose of the rotation matrix A calibration algorithm will produce a rigid body transformation between coordinate systems in one direction; for example, from coordinate system 1 to coordinate system 2, (1215) The inverse rigid body transform that converts coordinates in system 2 to coordinates in system 1 is (1216) where the notation Pi ,0 means the point in coordinate system i that is the origin ofthe other coordinate system Note that the inverse translation is not just -P20 but must be multiplied by the inverse rotation matrix, because the , translation P20 is in coordinate system 2 and the inverse translation must , be expressed in the same orientation as coordinate system 1

Rotation can also be specified as a counterclockwise (right-handed) rotation about the axis specified by the unit vector (w x , wy , wz ) This is a very intuitive way of viewing rotation, but it has the same problems as Euler angles in numerical algorithms The angle and axis representation can be converted into a rotation matrix for use in the formula for rigid body transformation (Equation 129), but it would be nice to have a scheme for working directly with the angle and axis representation that produced good numerical algorithms This is part of the motivation for the quaternion representation for rotation, discussed in the next section

TABLE 42 IBC 2003 Compressive Strength of Clay Masonry For Net Area Compressive Strength of Masonry (psi) 1,000 1,500 2,000 2,500 3,000 3,500 4,000

The quaternion is a representation for rotation that has been shown through experience to yield well-conditioned numerical solutions to orientation problems A quaternion is a four-element vector, (1217) To understand how quaternions encode rotation, consider the unit circle in the x-y plane with the implicit equation (1218) Positions on the unit circle correspond to rotation angles In three dimensions, the unit sphere is defined by the equation (1219) Positions on the unit sphere in three dimensions encode the rotation angles of wand <p about the x and y axes but cannot represent the twist K, about the z axis One more degree of freedom is required to represent all three rotation angles The unit sphere in four dimensions is defined by the implicit equation (1220)

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