Understanding Geometric Algebra

160  Camera Imaging and Conformal Transformations

Z

O

(X, Y, Z)

(x, y)

−1

1

FIGURE 9.20 Stereographic projection of a unit sphere from the south pole.

X =

2x

1 + x

2

+ y

2

, Y =

2y

1 + x

2

+ y

2

, Z =

1 − x

2

− y

2

1 + x

2

+ y

2

.

(2) Show that this corresp ondence is an inversion with respect to a sphere of radius

√

2 surr ounding the south pole (0, 0, −1).

9.2. Consider an xyz coordinate system associated with the fisheye lens camera that satis-

fies Eq. (9.7) with the or igin O at its lens center and the z-axis along its optical axis.

Show that a point (X, Y, Z) on a unit sphere surrounding the orig in is imaged at a

point (x, y) given by

x =

2fX

1 + Z

, y =

2fY

1 + Z

,

where we assume ...