Polar Coordinate Systems - 3D Math Primer for Graphics and Game Development - page 215

7. Convert the following 3D Cartesian coordinates to (canonical) cylindrical
form:

(a) (1, 1, 1)
(b) (0,−5, 2)
(c) (−3, 4,−7)
(d) (0, 0,−3)

8. Convert the following spherical coordinates (r, θ, φ) to Cartesian form ac-
cording to the standard mathematical convention:

(a) (4, π/3, 3π/4)
(b) (5,−5π/6, π/3)
(c) (2,−π/6, π)
(d) (8, 9π/4, π/6)

9. Interpret the spherical coordinates (a)-(d) from the previous exercise as
(r, h, p) triples, switching to our video game conventions.

1. Convert to canonical (r, h, p) coordinates.
2. Use the canonical coordinates to convert to Cartesian form (using the
video game conventions).

10. Convert the following 3D Cartesian coordinates to (canonical) spherical
form using our modified convention:
(a) (

√ √

3,− √ 2, 2 2)

√
3, 6,−4)
(c) (−1,−1,−1)
(d) (2,−2

(b) (2

√

3, 4)

(e) (− √ 3,− √

√
2)

3, 2

(f) (3, 4, 12)

11. What do the “grid lines” look like in spherical space? Assuming the spher-
ical conventions used in this topic, describe the shape defined by the set of
all points that meet the following criteria. Do not restrict the coordinates
to the canonical set.

(a) A fixed radius r = r 0 , but any arbitrary values for h and p.
(b) A fixed heading h = h 0 , but any arbitrary values for r and p.
(c) A fixed pitch p = p 0 , but any arbitrary values for r and h.
12. During crunch time one evening, a game developer decided to get some
fresh air and go for a walk. The developer left the studio walking south
and walked for 5 km. She then turned east and walked another 5 km.
Realizing that all the fresh air was making her light-headed, she decided to
return to the studio. She turned north, walked 5 km and was back at the