How this instrument works
A point in spherical coordinates carries one distance and two angles: r is the straight-line reach from the origin, θ sweeps around the equator the way longitude does, and φ tips down from the north pole, the way colatitude does — ninety degrees minus ordinary latitude. Together the three pin down any location on, or inside, an imaginary sphere of radius r, which is exactly why physicists reach for this system whenever the thing being measured — a planet's gravity field, an electron cloud around a nucleus, a loudspeaker's radiation pattern — has no favoured axis at all, only a favoured centre.
The three formulas fall out of two right triangles stacked one inside the other. φ, measured from the vertical axis, first splits r into a height and a flat reach: z = r·cosφ climbs straight up the axis, while r·sinφ is the radius of the circle the point traces if it spins in place — call it the point's shadow on the equatorial plane. θ then resolves that shadow exactly as ordinary two-dimensional polar coordinates would, x = (r·sinφ)·cosθ and y = (r·sinφ)·sinθ, so the whole system is nothing more than flat polar coordinates lifted onto a slice, with a second angle deciding how high that slice sits.
Two settings collapse the formula usefully. Set φ to 0° or 180° and sinφ vanishes, so x and y both go to zero regardless of θ — every azimuth describes the same pole, the point (0, 0, r) or (0, 0, −r), just as every line of longitude meets at the North Pole on a globe. Set φ to 90° instead and cosφ vanishes, dropping z to zero and leaving the ordinary flat-plane conversion. The one real trap is the letters themselves: this page follows the physics convention, θ around the equator and φ down from the pole, but many calculus textbooks swap the two names, so a formula copied from another source without checking which angle its φ actually means is the single most common way this calculation goes wrong.
- Enter the point's straight-line distance from the origin into r (radial distance) — zero or positive, any unit.
- Enter the sweep around the equator into θ (azimuthal angle), toggling between degrees, radians, and turns as needed.
- Enter the tip down from the north pole into φ (polar angle from +z axis) — 0° points straight up the +z axis, 180° straight down.
- Read x, y, and z for the point's position in ordinary Cartesian space, each in the same unit r was entered in.
- Check x² + y² + z² against r² as a quick arithmetic sanity check on any hand calculation.
Worked example — a point at r = 5, θ = 45°, φ = 60°
Place a point 5 units from the origin, swept 45° around the equator and tipped 60° down from the north pole: r = 5, θ = 45° (0.7853981633974483 rad), φ = 60° (1.0471975511965976 rad). The shadow radius is r·sinφ = 5 × 0.8660254037844386 = 4.330127018922193, and because 45° is the one angle where cosθ and sinθ coincide, both trig values equal 0.7071067811865476 — so x and y come out identical: x = 4.330127018922193 × 0.7071067811865476 = 3.0618621784789726, and y = 3.0618621784789726. The height follows from cosφ alone: z = 5 × 0.5000000000000001 = 2.5000000000000004, the digits past the decimal a trace of floating-point rounding on an exact 2.5.
A quick check confirms the numbers: x² + y² + z² = 9.375 + 9.375 + 6.25 ≈ 25.000000000000004, matching r² = 5² = 25 to every digit that counts, the last digit merely rounding noise from representing 45° and 60° in radians rather than a sign of error. That x and y landed on the same value is not a coincidence of this particular angle: 45° is the one azimuth where cosθ and sinθ are equal, so any point placed there lands with x equal to y regardless of what r and φ happen to be.
Questions
What do r, θ, and φ mean in spherical coordinates?
r is the straight-line distance from the origin to the point, always zero or positive. θ is the azimuthal angle, swept around the equator from a reference direction, exactly like longitude. φ is the polar angle, measured down from the positive z-axis rather than up from the equator, so it runs from 0° at the north pole to 180° at the south pole — the complement of ordinary latitude, not latitude itself.
Why do some textbooks swap θ and φ?
There are two competing conventions and no single global standard. Physics and most engineering follow ISO 80000-2, used on this page: θ is the azimuth around the equator and φ is the polar angle from the axis. Many calculus textbooks, particularly in the US, use the opposite pairing. Before trusting a formula pulled from elsewhere, check which angle its author measures from the pole rather than assuming the letters match.
How is this different from converting cylindrical coordinates?
Cylindrical coordinates describe a point with a flat radius, one sweep angle, and a separate height that behaves exactly like ordinary z. Spherical coordinates drop that separate height entirely: r already measures out to the point in three dimensions, so a second angle, φ, says how far the point has tilted from straight up. A pipe or a drill bit, symmetric about one line, suits cylindrical coordinates; a planet or an atom, symmetric about one point, suits spherical coordinates instead.
What happens at the poles, where φ is 0° or 180°?
Both x and y collapse to zero because sinφ is zero there, leaving only z = ±r. Every value of θ describes the identical point in that case — the pole itself — the same way every line of longitude converges at the North Pole on a globe. It is the spherical analogue of the origin swallowing the angle in ordinary 2D polar coordinates.
What happens at φ = 90°?
cosφ is zero, so z drops out and the point lands exactly on the equatorial plane. What remains, x = r·cosθ and y = r·sinθ, is the ordinary two-dimensional polar-to-Cartesian conversion — spherical coordinates reduce to flat polar coordinates precisely on that one plane.
What's the most common mistake when using this formula by hand?
Mixing up sine and cosine between the two angles, or forgetting that φ is measured from the pole rather than the equator. A point with φ close to 0° sits nearly on the z-axis with a tiny shadow radius, not out near the equator — treating φ like ordinary latitude, where small angles mean near the equator, gets the geometry backwards.