Math Coordinate Converter: Step-by-Step Examples & FormulasConverting between coordinate systems is a fundamental skill in mathematics, physics, engineering, and computer graphics. Different problems are more naturally described in Cartesian, polar, cylindrical, or spherical coordinates; being able to translate points and vectors between these systems simplifies calculations and provides geometric insight. This article explains the four most common coordinate systems, derives conversion formulas, and works through step-by-step examples for each pair of conversions. It also covers common pitfalls, how to convert vector components (not just point positions), and quick reference formulas.
Overview of coordinate systems
- Cartesian (rectangular): A point in 2D is (x, y); in 3D it’s (x, y, z). Axes are mutually perpendicular.
- Polar (2D): A point is (r, θ), where r ≥ 0 is the radial distance from origin and θ is the angle measured from the positive x-axis (usually in radians).
- Cylindrical (3D): Extends polar to 3D: (r, θ, z) with z same as Cartesian z.
- Spherical (3D): A point is (ρ, θ, φ) or (r, θ, φ) depending on notation; here we use ρ for radial distance from origin, θ for the azimuthal angle in the x–y plane (same as polar), and φ for the polar (inclination) angle measured from the positive z-axis.
2D conversions: Cartesian ⇄ Polar
Formulas:
- From Cartesian (x, y) to polar (r, θ):
- r = √(x^2 + y^2)
- θ = atan2(y, x) (returns angle in (-π, π], handles quadrants)
- From polar (r, θ) to Cartesian (x, y):
- x = r cos θ
- y = r sin θ
Notes:
- Use atan2 to avoid ambiguity: θ = atan2(y, x).
- For r = 0, θ is undefined; choose a convention (e.g., θ = 0).
- Be careful with angle units (degrees vs radians).
Example 1 — Cartesian to Polar:
- Given point (x, y) = (−1, √3):
- r = √[(-1)^2 + (√3)^2] = √(1 + 3) = 2
- θ = atan2(√3, −1) = 120° = 2π/3
- Polar: (2, 2π/3)
Example 2 — Polar to Cartesian:
- Given (r, θ) = (5, −π/6):
- x = 5 cos(−π/6) = 5 * (√3/2) = (5√3)/2
- y = 5 sin(−π/6) = 5 * (−1/2) = −5/2
- Cartesian: ((5√3)/2, −5/2)
3D conversions: Cartesian ⇄ Cylindrical ⇄ Spherical
Cartesian ⇄ Cylindrical
- Cartesian (x, y, z) → Cylindrical (r, θ, z):
- r = √(x^2 + y^2)
- θ = atan2(y, x)
- z = z
- Cylindrical → Cartesian:
- x = r cos θ
- y = r sin θ
- z = z
Example:
- Convert (x, y, z) = (3, −3, 4) to cylindrical:
- r = √(9 + 9) = √18 = 3√2
- θ = atan2(−3, 3) = −π/4
- Cylindrical: (3√2, −π/4, 4)
Cartesian ⇄ Spherical (ρ, θ, φ) with φ measured from positive z-axis
Formulas:
- Cartesian → Spherical:
- ρ = √(x^2 + y^2 + z^2)
- θ = atan2(y, x)
- φ = arccos(z / ρ) (for ρ ≠ 0)
- Spherical → Cartesian:
- x = ρ sin φ cos θ
- y = ρ sin φ sin θ
- z = ρ cos φ
Notes:
- Some texts swap θ and φ conventions (θ as polar, φ as azimuthal). Confirm convention before using formulas.
- For ρ = 0, angles are undefined—pick a convention if needed.
Example:
- Convert Cartesian (x, y, z) = (1, 1, √2) to spherical:
- ρ = √(1 + 1 + 2) = √4 = 2
- θ = atan2(1, 1) = π/4
- φ = arccos(z/ρ) = arccos(√2 / 2) = π/4
- Spherical: (2, π/4, π/4)
Cylindrical ⇄ Spherical
- Cylindrical (r, θ, z) → Spherical (ρ, θ, φ):
- ρ = √(r^2 + z^2)
- φ = atan2(r, z) (because tan φ = r / z, φ measured from z-axis)
- θ = θ
- Spherical → Cylindrical:
- r = ρ sin φ
- z = ρ cos φ
- θ = θ
Example:
- Given cylindrical (r, θ, z) = (2, π/6, 2√3):
- ρ = √(4 + 12) = √16 = 4
- φ = atan2(2, 2√3) = atan2(1, √3) = π/6
- Spherical: (4, π/6, π/6)
Converting vectors vs positions
- Positions: use the point formulas above.
- Vectors: when converting components, be careful—component transformation depends on basis vectors which differ between coordinate systems.
- Example: A vector V = Vx i + Vy j in Cartesian. In polar, unit vectors vary with θ: e_r = cos θ i + sin θ j, e_θ = −sin θ i + cos θ j.
- To get polar components (Vr, Vθ):
- Vr = Vx cos θ + Vy sin θ
- Vθ = −Vx sin θ + Vy cos θ
- In 3D, cylindrical and spherical vector components require similar projections onto local unit vectors; use rotation matrices or Jacobians for systematic conversion.
Jacobian determinants and differential elements
- 2D Cartesian to polar: dx dy = r dr dθ. Jacobian = r.
- 3D Cartesian to spherical: dx dy dz = ρ^2 sin φ dρ dφ dθ. Jacobian = ρ^2 sin φ.
- 3D Cartesian to cylindrical: dx dy dz = r dr dθ dz. Jacobian = r.
Use Jacobians when changing variables in integrals.
Common pitfalls and tips
- Angle ambiguity: always prefer atan2(y, x).
- Units: keep angles consistent (radians for calculus).
- Zero radius: r = 0 or ρ = 0 leaves angle(s) undefined.
- Conventions: confirm which angle is polar vs azimuthal in spherical formulas.
- Numerical stability: when computing φ = arccos(z/ρ) for small ρ, use atan2 or handle zero to avoid division by zero.
Quick reference formulas
Cartesian (x, y, z) → Polar/Cylindrical/Spherical:
- r = √(x^2 + y^2)
- θ = atan2(y, x)
- ρ = √(x^2 + y^2 + z^2)
- φ = arccos(z / ρ)
Polar/Cylindrical/Spherical → Cartesian:
- x = r cos θ = ρ sin φ cos θ
- y = r sin θ = ρ sin φ sin θ
- z = ρ cos φ
Worked comprehensive example
Convert point P with Cartesian coordinates (−2, 2, −2√2) to cylindrical and spherical.
-
Cylindrical:
- r = √(4 + 4) = √8 = 2√2
- θ = atan2(2, −2) = 3π/4
- z = −2√2
- Cylindrical: (2√2, 3π/4, −2√2)
-
Spherical:
- ρ = √(4 + 4 + 8) = √16 = 4
- θ = 3π/4 (same azimuth)
- φ = arccos(z / ρ) = arccos(−2√2 / 4) = arccos(−√2/2) = 3π/4
- Spherical: (4, 3π/4, 3π/4)
Implementation tips for a converter tool
- Use atan2 for angle calculations.
- Normalize angles to a chosen range when presenting (e.g., [0, 2π) or (−π, π]).
- Provide both radians and degrees outputs.
- Handle edge cases: r = 0, ρ = 0, near-zero floating errors.
- For vector conversions, compute unit basis vectors and project components.
- Include an option to choose spherical convention (which angle is polar vs azimuthal).
Summary
Converting coordinates is largely a matter of applying trigonometric relationships and being careful with angle conventions and special cases. Keep the conversion formulas and Jacobians handy, prefer atan2 for robust angle computation, and remember that vector component conversions require projection onto the local basis.
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