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A Prototype Optical Tracking System Investigation and Development - page 35 / 170

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3.1 Supporting theory

21

Lens aperture

The lens aperture determines how much light enters the lens. This is normally specified with a quantity named ‘f-number’ and sometimes written f/#. This number is a ratio of focal length to the clear aperture φ0 (the effective lens diameter ) as shown by

f

number =

f φ

.

(3.5)

large f-number corresponds to a small aperture and therefore lets less light enter the

lens.

small f-number gives a large aperture and lets more light into the lens.

detailed

discussion on optics is given in Born and Wolf [39].

3.1.2

Quaternions

Quaternions are used in this research to describe the orientation of coordinate frames. Quaternions are the outcome of research by W.R. Hamilton in the 18th century [40]. They are hyper-complex numbers and form a 4 dimensional vector space over the real numbers.

The four components of a quaternion q can be written

q = w + xi + yj + zk.

(3.6)

Here w represents a scalar and x, y, z are often combined into a vector v. The terms i, j, and k are similar to the imaginary term i in a complex number and are orthogonal with respect to each other.

Quaternions can be used to elegantly describe rotations. Other representations such as Eu- ler ngles, rotation matrices, and the angle and axis representation exist [41–43]. However, quaternions do not suffer from a degenerative problem known as gimbal lock [41–43]. This

can affect the composition of rotations using the Euler

ngle and rotation matrix represen-

tation of rotations resulting in a loss of one degree of freedom at some orientations.

quaternion of unit length represents a rotation. The length or norm of a quaternion is similar to the standard vector definition, i.e.,

|q| = pw2 + x2 + y2 + z2.

(3.7)

Quaternion multiplication, unlike multiplication of the real numbers, does not commute, i . e . , f o r t w o q u a t e r n i o n s q 1 , q 2 , q 1 q 2 6 = q 2 q 1 . T o d e t e r m i n e t h e p r o d u c t o f t h e t w o q u a nions the components of each quaternion are multiplied term-by-term. The multiplication t e r -

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