Nonlinear Projections - Transformations and Projections in Computer Graphics - page 148

with parametric methods should either notice that cos 2 u +sin 2 u = 1 or should refer to
[Salomon 05]. A complete sphere of radius k is created when this half-circle is rotated
360 ◦ about the x axis. The parametric equation of the sphere is therefore the product
of the half-circle with the matrix that rotates about the x axis,

⎛ ⎞

10 0

⎝ ⎠ = k (cos u, sin u sin w, sin u cos w ) ,

k (cos u, 0 , sin u ) 0 os w − sin w (4.3)

0 in w cos w

180 ◦ and 0 360 ◦ .

for 0 ≤ u ≤ ≤ w ≤

y

k

x

r

z

u

w

z

k

u

x

(a) (b)

Figure 4.8: Analysis of the Angular Fisheye Projection.

The word barycentric is derived from barycenter , meaning “center of gravity,” be-
cause such weights are used to calculate the center of gravity of an object. Barycentric
weights have many uses in geometry in general and in curve and surface design in
particular.
Figure 4.8b shows the half-circle in the xz plane and how it is rotated. It is clear
that the angle w of a point P on the sphere is one of the parameters of the projected
point P ∗ . This angle determines the distance r of P ∗ from the center of the radius- k
circle. In the figure, r equals k sin w , but the point is that for w = 0 we want r =0,
while for w =90 ◦ we want r = k/ 2 and not r = k . Thisisbecause r values from k/ 2to
k correspond to w values in the “right” hemisphere (i.e., from 90 ◦ to 270 ◦ ). Thus, for
w values in the interval [0 , 90], we write r = 2 sin w , and Table 4.9 lists the expressions
of r for the remaining three intervals of w .
Once we have r , we still need to decide where in the radius- k circle to place P ∗ ,
andthisisdeterminedby u . This angle varies in the interval [0 , 180 ◦ ], and P ∗ has

180 ◦ ) or the “bottom” half (if to be placed either in the “top” half (if 0 ≤ w ≤

180 ◦ ≤ 360 ◦ ) of the circle, as indicated by Table 4.9.

w ≤