Utilize our Prolate Spheroid Coordinates Calculator to effortlessly convert Cartesian (x,y,z) coordinates into their equivalent Prolate Spheroidal (λ, μ, φ) values. Simply input your point's coordinates and the semi-focal distance to get instant, precise results for advanced geometry, physics, and engineering problems.
Formula:
To convert Cartesian coordinates (x, y, z) to Prolate Spheroidal coordinates (λ, μ, φ) with semi-focal distance 'a', we first calculate the distances to the foci F1(0,0,-a) and F2(0,0,a):
- r1 = √(x2 + y2 + (z + a)2)
- r2 = √(x2 + y2 + (z - a)2)
Then, the Prolate Spheroidal coordinates are given by:
- λ (Lambda) = (r1 + r2) / (2a)
- μ (Mu) = (r1 - r2) / (2a)
- φ (Phi) = atan2(y, x)
Where:
- x, y, z: Cartesian coordinates of the point.
- a: The semi-focal distance of the prolate spheroid.
- r1: Distance from the point (x,y,z) to focus F1(0,0,-a).
- r2: Distance from the point (x,y,z) to focus F2(0,0,a).
- atan2(y, x): The two-argument arctangent function, returning the angle in radians between the positive x-axis and the point (x,y).