1.1. LOCATING POINTS IN SPACE 11 To invert these relations, we note that, since ρ ≥ 0 and 0 ≤ φ ≤ π by convention, z and r completely determine ρ and φ : ρ = √ r 2 + z 2 θ = θ φ = arccos z ρ . (1.8) The ambiguities in spherical coordinates are the same as those for cylindrical coordinates: the origin has ρ = 0 and both θ and φ arbitrary; any other point on the z-axis ( φ = 0 or φ = π ) has arbitrary θ , and for points oF the z-axis, θ can (in principle) be augmented by arbitrary even multiples of π . Thus, the point P with cylindrical coordinates r = 4 θ = 5 π 6 z = 4 has spherical coordinates ρ = 4 √ 2 θ = 5 π 6 φ = π 4 . Combining Equations (
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Coordinate system, Polar coordinate system, φ, Coordinate systems