By Raynaud M. (Ed), Shioda T. (Ed)

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GEOMETRY. --- DEFINITIONS. 1. If a block of wooden or stone be minimize within the sbape represented iu Fjg. 1, it is going to have six fiat faces. each one face of the block is termed a floor; and if those faces are made soft by means of sharpening, in order that, whilst a straight-edge is utilized to anybody of them, the immediately side in every thing will contact the skin, the faces are referred to as airplane surfaces, or planes.

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The position of a locator can be retrieved in object and world space as follows. xform -query -objectSpace - t r a n s l a t i o n ; xform -query -worldSpace - t r a n s l a t i o n ; This Page Intentionally Left Blank 3 Vectors A vector has both a direction and a magnitude. The magnitude of a vector is simply its length. Vectors are often used to define the difference between points, which is the displacement from the first point to the second. A series of vectors (combined with an origin) can also define a coordinate flame.

Thus, to rotate the vector by +45 degrees (~r/4 radians), the additional angle is added to the existing angle. v ' = (r, 0 + av/4) 46 C H A P T E R 4 Rotations These polar coordinates can then be converted back to Cartesian coordinates. Fortunately, there is a faster way to perform a rotation if the complete series of operations is taken into account. Given the additional angle of rotation &, the final angle is 0 + &. The conversion of the final angle back to Cartesian coordinates can be defined as x ' = r cos(O + (b) y'= r sin(O + (b) By applying trigonometric identities, this can be expanded to x' = r cos(O) cos(d)) -- r sin(O) s i n ( ~ ) y' = r cos(O) s i n ( d ) ) + r sin(O) cos(d)) Referring to the equations for converting from polar to Cartesian coordinates, it is known that x = r cos(O) y = r sin(O) and thus all instances of these in the equation can be replaced by x and y as follows.

D i s t a n c e T o ( pl ). 8 A vector that has a length of 1 is called a unit vector. To turn any vector into a unit vector, it needs to be normalized. This is done as follows. u Ilull Each component of the vector is divided by the vector's current length. The vector's length will then be 1. Unit vectors are represented with the "hat" character (^) above them. For a 3D vector the calculation is as follows. Ux Uy Uz ) Vectors with a length of 1 are also sometimes referred to as normal vectors. This can cause some confusion, in that the term normal has another meaning with regard to vectors.