Science & Data

Coordinate systems

Build points and vectors, do point arithmetic, convert between coordinate systems, and compute distances, products, and angles.

point(1, 2, 3)
point(3,4) to polar
distance(point(0,0,0), point(3,4,0))
cross(point(1,0,0), point(0,1,0))

Constructors

• point(x, y, z, …) — a Cartesian point/vector of any dimension. cartesian(…) and vector(…) are aliases.
• polar(r, θ) — 2-D polar.
• cylindrical(r, θ, z) — 3-D cylindrical.
• spherical(ρ, θ, φ) — 3-D spherical (θ = inclination from +z, φ = azimuth).
• minkowski(t, x, y, z) — spacetime point, signature (−,+,+,+) with c = 1.

point(1, 2, 3)
polar(5, 90)
cylindrical(2, 90, 5)
spherical(1, 90, 0)
minkowski(1, 2, 3, 4)

Angles default to degrees (like the trig functions). An explicit angle unit is honoured:

polar(5, 1.5708 radian) to cartesian
polar(5, 100 grad) to cartesian

Arithmetic

Subtraction and addition are component-wise; multiplying or dividing by a scalar scales the point.

point(1,2,3) - point(4,5,6)
point(1,1) + point(2,3)
3 * point(1,2)
point(2,4) / 2

Conversions

Convert with to. to distance collapses a point to its scalar length from the origin; <number> to vector expands a scalar back into a point.

point(3,4) to polar
polar(5, 90) to cartesian
point(1,2,2) to spherical
point(1,2,3) - point(4,5,6) to distance
5 to vector

The function-call form also converts when given a single point:

polar(point(3,4))
cartesian(polar(5, 90))

Operations

distance(point(1,2,3), point(4,5,6))
magnitude(point(3,4))
midpoint(point(0,0), point(4,6))
dot(point(1,2,3), point(4,5,6))
cross(point(1,0,0), point(0,1,0))
angle(point(1,0), point(0,1))
normalize(point(3,4))
atan2(1, 1)
hypot(3, 4)

Minkowski spacetime

interval(p, q) is the signed interval s² (negative = timelike, zero = lightlike, positive = spacelike). to distance gives √|s²|.

interval(minkowski(5,0,0,0), minkowski(0,3,0,0))
minkowski(5,3,0,0) to distance

Common mistakes