Derivation of the 2D Rotation Matrix Transforming point P from a global to a local coordinate system (Global to Local) X Y X’ Y’ q P Y Sin q X Cos q Y Cos q X Sin q X’ = X Cos q + Y Sin q Y’ = Y Cos q - X Sin q

Derivation of the 2D Rotation Matrix Transforming point P from a global to a local coordinate system (Local to Global) X Y X’ Y’ q P X’ Sin q Y’ Cos q Y’ Sin q X’ Cos q X = X’ Cos q - Y’ Sin q Y = X’ Sin q + Y’ Cos q

Conversion of Equations to Matrix Form From Global to Local: X’ = X Cos q + Y Sin q X’ Y’ = Cos Sin -Sin Cos X Y Y’ = Y Cos q - X Sin q From Local to Global: X = X’ Cos q - Y’ Sin q X Y = Cos -Sin Sin Cos X’ Y’ Y = X’ Sin q + Y’ Cos q

Rotation Matrix Convention From Global to Local: X’ Y’ = Cos Sin -Sin Cos X Y X’ Y’ = X Y or [R’] From Local to Global: X Y = Cos -Sin Sin Cos X’ Y’ X Y = X’ Y’ or [R]

X’ and Y’ are unit vectors, which means that they each have a length of 1. The first column of [R] represents the location of the endpoint of X’ The x and y components of X’ are sometimes represented as L1 and M1, which indicates that they are components of a unit vector. The second column of [R] represents the location of the endpoint of Y’ The x and y components of X’ are sometimes represented as L1 and M1, which indicates that they are components of a unit vector. Y Y’ = Cos -Sin Sin Cos [R] q X’ = L1 [R] L2 M1 M2 q X

[R] represents the orientation of the local coordinate system with respect to the global coordinate system. For example, a local coordinate system fixed to the pelvis would tell us the orientation of the pelvis in the room’s coordinate system. It can also represent the orientation of one local coordinate system with respect to another local coordinate system. For example, it can express the orientation of the thigh with respect to the pelvis. [R] is used to transform a point in local coordinates to global coordinates. In this orientation, the columns contain the x and y components of the axes. = L1 [R] L2 M1 M2 Conversely, [R’] is equal to the transpose of [R], and is used to transform a point in global coordinates to local coordinates. In this orientation, the rows contain the x and y components of the axes. = L1 [R’] M1 L2 M2

We can extend the [R] matrix to three dimensions We can extend the [R] matrix to three dimensions. As with the 2D matrix, the columns are unit vectors. The first column contains the x, y, and z components of the X-axis, the second column contains the x, y, and z components of the Y-axis, and third column contains the x, y, and z components of the Z-axis. = L1 [R] L2 M1 M2 M3 L3 N1 N2 N3 Again, [R’] is equal to the transpose of [R], and is used to transform a point in global coordinates to local coordinates. In this orientation, the rows contain the x, y, and z components of the three axes. = L1 [R’] M1 L2 M2 N2 N1 L3 M3 N3