Polar Coordinate Systems - 3D Math Primer for Graphics and Game Development - page 211

7.3.5 Converting between Spherical and Cartesian Coordinates

Let's see if we can convert spherical coordi-
nates to 3D Cartesian coordinates. Exam-
ine Figure 7.8, which shows both spherical
and Cartesian coordinates. We first develop
the conversions using the traditional right-
handed conventions for both Cartesian and
spherical spaces, and then we show conver-
sions applicable to our left-handed conven-
tions.
Notice in Figure 7.8 that we've intro-
duced a new variable d, which is the hor-
izontal distance between the point and the
vertical axis. From the right triangle with
hypotenuse r and legs d and z, we get

z/r = cosφ,
z = r cosφ.

Figure 7.8
Spherical and Cartesian coordinates
for math people

and so we're left to compute x and y.
Consider that if φ = 90 o , we basically
have 2D polar coordinates. Let's assign x ′ and y ′ to stand for the x and y
coordinates that would result if φ = 90 o . From Section 7.1.3, we have

′ ′

x = r cosθ, y = r sinθ.

Notice that when φ = 90 o , d = r. As φ decreases, d decreases, and by
the properties of similar triangles, x/x

′ ′

= d/r. Looking at △drz
again, we observe that d/r = sinφ. Putting all this together, we have

= y/y

Converting spherical
coordinates used by
math people to 3D
Cartesian coordinates

x = r sinφ cosθ, y = r sinφ sinθ, z = r cosφ.

These equations are applicable for right-handed math people. If we
adopt our conventions for both the Cartesian (see Section 1.3.4) and spher-
ical (see Section 7.3.3) spaces, the following formulas should be used:

Spherical-to-Cartesian
conversion for the
conventions used in this
book

x = r cosp sinh, y = −r sinp, z = r cosp cosh. (7.3)

Converting from Cartesian coordinates to spherical coordinates is more
complicated, due to aliasing. We know that there are multiple sets of spher-
ical coordinates that map to any given 3D position; we want the canoni-
cal coordinates. The derivation that follows uses our preferred aviation-
inspired conventions in Equation (7.3) because those conventions are the
ones most commonly used in video games.