The direction of a point needs careful attention: opposite points can produce the same ratio (y/x). A reliable derivation must preserve the signs of the Cartesian coordinates, as well as the distance from the origin.

These formulas follow the physics convention: r represents distance, θ measures the angle from the positive z-axis, and φ measures rotation around it. Other conventions may exchange the angle symbols, so their definitions matter.

Deriving Radial Distance r Formula:

For a point P with three coordinates (x, y, z), its horizontal projection is (x, y,0). Taking the xy-plane as the reference plane, let s represent the distance from the origin to that projected point:

s = √(x² + y²)

The horizontal triangle follows 2D polar coordinates. In this 2D view, the positive x-axis supplies the reference direction, and φ increases counter-clockwise when viewed from above the positive z-axis.

Now consider the right triangle with horizontal leg s, vertical leg z, and the position vector as its sloping side. The magnitude of this vector is the Euclidean distance from the origin:

r² = s² + z²

Substituting s² = x² + y² gives the radial coordinate:

r = √(x² + y² + z²)

Deriving Polar Angle θ:

The positive z-axis points in the zenith direction. Measuring θ from this zenith makes it the zenith angle, also called the inclination angle.

The vertical component therefore satisfies z = r cosθ. Dividing by the distance and applying inverse cosine gives:

θ = arccos(z / r), where r > 0.

This angle ranges from zero to π radians, or 180°, covering points above and below the xy-plane.

Deriving Azimuthal Angle φ:

The azimuth angle φ comes from the horizontal projection of the point onto the xy-plane. This creates a right triangle with horizontal component x, vertical component y, and distance s from the origin.

Using simple trigonometry, the two components are:

x = s cosφ, y = s sinφ

Dividing the second equation by the first eliminates s:

y/x = (s sinφ)/(s cosφ) = tanφ

Applying inverse tangent to both sides gives the basic expression:

φ = arctan(y / x)

However, this expression needs a quadrant check. For example, (x, y)=(4, 4) and (-4, -4) give the same ratio, although their directions differ by 180°.

Let α = arctan(y/x). To express the azimuth between zero and 360°, use:

• φ = α, if x > 0 and y ≥ 0
• φ = α + 360°, if x > 0 and y < 0
• φ = α + 180°, if x < 0

When x = 0, division is undefined: the azimuth is 90° for y>0, or 270° for y<0. If both horizontal components are zero, the azimuth arbitrary case applies because the point lies on the z-axis.

For the range -180° < φ ≤ 180°, let α = arctan(y/x), then use:

• φ = α, if x > 0
• φ = α + 180°, if x < 0 and y ≥ 0
• φ = α − 180°, if x < 0 and y < 0

When x = 0:

• φ = 90°, if y > 0
• φ = -90°, if y < 0
• φ is undefined if y = 0

Here, 180° is included and -180° is excluded because they represent the same direction.

Two special cases complete the derivation. On the z-axis, the horizontal direction is undefined, making the azimuth arbitrary; at the origin, radius zero means neither angle has a unique value.