Hi Tanegashima. Welcome to PF.Tanegashima said:Homework Statement
Calculate the distance between these two points:
(3;π/2;−1) and (5;3π/2;5) (cylindrical coordinates)
(10;π/4;3π/4) and (5;π/6;7π/4) (spherical coordinates)
Do I need to put them in Cartesian coordinates and continue the calc. or can I do with integrals?
Homework Equations
->dl = dr ûr + r d∅ û∅ + dz ^k
->dl = dr ûr + r dθ ûθ + r sin (∅) û∅
The Attempt at a Solution
There are two main coordinate systems used in mathematics and science to calculate distance: spherical coordinates and cylindrical coordinates. Spherical coordinates use a radius, inclination angle, and azimuth angle to locate a point in three-dimensional space, while cylindrical coordinates use a distance from the origin, angle in the xy-plane, and height above the xy-plane.
To convert from spherical coordinates to cylindrical coordinates, you can use the following formulas:
x = ρsin(θ)cos(φ)
y = ρsin(θ)sin(φ)
z = ρcos(θ)
where ρ is the distance from the origin, θ is the inclination angle, and φ is the azimuth angle. To convert from cylindrical coordinates to spherical coordinates, you can use these formulas:
ρ = √(x²+y²)
θ = tan⁻¹(y/x)
φ = tan⁻¹(z/√(x²+y²))
To calculate the distance between two points using spherical coordinates, you can use the Haversine formula:
d = 2r arccos( sin²((φ₂-φ₁)/2) + cos(φ₁)cos(φ₂)sin²((λ₂-λ₁)/2) )
where d is the distance, r is the radius of the sphere, φ₁ and φ₂ are the latitudes of the two points, and λ₁ and λ₂ are the longitudes of the two points.
To calculate the distance between two points using cylindrical coordinates, you can use the Pythagorean theorem:
d = √((ρ₂-ρ₁)² + (z₂-z₁)²)
where d is the distance, ρ₁ and ρ₂ are the distances from the origin of the two points, and z₁ and z₂ are the heights above the xy-plane of the two points.
Yes, there are other coordinate systems that can be used to calculate distance, such as Cartesian coordinates, polar coordinates, and geographic coordinates. However, spherical and cylindrical coordinates are commonly used in scientific and mathematical calculations involving three-dimensional space.