Polar Distance Calculator

Polar Distance Formula:

\[ d = \sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos \Delta\theta} \]

Polar Distance (d):

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1. What is the Polar Distance Formula?

The polar distance formula calculates the straight-line distance between two points given in polar coordinates. It's derived from the law of cosines and provides the Euclidean distance between points with radii r₁ and r₂ separated by an angular difference Δθ.

2. How Does the Calculator Work?

The calculator uses the polar distance formula:

\[ d = \sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos \Delta\theta} \]

Where:

\( r_1 \) — First radius (distance from origin to first point)
\( r_2 \) — Second radius (distance from origin to second point)
\( \Delta\theta \) — Angular difference between the two points in degrees
\( d \) — Resulting polar distance between the two points

Explanation: The formula calculates the straight-line distance between two points in a polar coordinate system by applying the law of cosines to the triangle formed by the two radii and the distance between the points.

3. Applications of Polar Distance

Details: This formula is widely used in physics, engineering, navigation, and computer graphics where polar coordinates are employed. It's essential for calculating distances in radar systems, antenna positioning, and polar coordinate transformations.

4. Using the Calculator

Tips: Enter both radii in the same units (positive values), and the angular difference in degrees (0-360°). The calculator will compute the straight-line distance between the two points in polar coordinates.

5. Frequently Asked Questions (FAQ)

Q1: What if the angular difference is 0° or 180°?
A: When Δθ = 0°, the points are collinear with the origin and d = |r₁ - r₂|. When Δθ = 180°, the points are opposite and d = r₁ + r₂.

Q2: Can this formula be used for 3D polar coordinates?
A: This specific formula is for 2D polar coordinates. For 3D spherical coordinates, a more complex formula involving both angular differences is required.

Q3: What units should I use for the radii?
A: The units can be any consistent measurement (meters, feet, etc.), but both radii must use the same units.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for the given inputs, assuming perfect polar coordinate representation.

Q5: What's the relationship to Cartesian coordinates?
A: This formula is equivalent to the Euclidean distance formula when converting polar coordinates (r,θ) to Cartesian coordinates (x,y).