Derive and understand the conversion formulas between different representations of spherical coordinates | Step-by-Step Solution

Understanding Spherical Coordinate Conversion Formulas

What We're Solving:

We need to understand how to derive and interpret the two main versions of spherical coordinate conversion formulas - the colatitude version (using angle θ from the north pole) and the latitude version (using angle λ from the equatorial plane).

The Approach:

We're going to visualize a 3D coordinate system and understand how the same point can be described using different angle measurements. The key is understanding what each angle represents and how trigonometry helps us convert between coordinate systems.

Step-by-Step Solution:

Step 1: Set up the coordinate system

• Imagine a sphere with center at the origin (0,0,0)
• The z-axis points "up" (like the North Pole)
• The x and y axes form the equatorial plane
• Any point P on or inside the sphere can be described by distance r and two angles

Step 2: Understand the colatitude version

• θ (theta) = colatitude angle, measured FROM the positive z-axis DOWN toward the equatorial plane
• φ (phi) = azimuthal angle, measured around the z-axis (like longitude)

To derive the formulas:

• The projection onto the xy-plane has length r sin θ (this is the "shadow" when light shines from above)
• x-component: r sin θ cos φ (project the xy-shadow onto x-axis)
• y-component: r sin θ sin φ (project the xy-shadow onto y-axis)
• z-component: r cos θ (direct projection onto z-axis)

Step 3: Understand the latitude version

• λ (lambda) = latitude angle, measured FROM the equatorial plane UP toward the poles
• φ (phi) = same azimuthal angle as before

To derive these formulas:

• The projection onto the xy-plane has length r cos λ (gets smaller as you go toward poles)
• x-component: r cos λ cos φ (project the xy-shadow onto x-axis)
• y-component: r cos λ sin φ (project the xy-shadow onto y-axis)
• z-component: r sin λ (direct projection onto z-axis)

Step 4: Connect the two systems Notice that θ and λ are complementary angles: θ + λ = 90° This means: sin θ = cos λ and cos θ = sin λ

The Answer:

Both formula sets are correct and equivalent! They represent the same 3D points using different angle conventions:

• Colatitude version: Measures angles down from the north pole (common in physics)
• Latitude version: Measures angles up from the equator (common in geography)

The choice depends on your field and what feels more intuitive for your specific application.

Memory Tip:

Remember "Co-latitude = 90° - latitude"!

For colatitude (θ): Think "Sin θ goes with the Shadow" (xy-plane projection) For latitude (λ): Think "Cos λ goes with the Circle" (equatorial circle projection)

The beauty of mathematics is that the same reality can be described in multiple valid ways. Both systems are just different perspectives on the same 3D space! 🌍