Polar Distance Calculator Graph

Polar Distance Formula:

\[ \text{Distance} = \sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_1 - \theta_2)} \]

Radius 1 (r1):

units

Radius 2 (r2):

units

Angle 1 (θ1):

degrees

Angle 2 (θ2):

degrees

Polar Distance:

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1. What is Polar Distance?

Polar distance calculates the straight-line distance between two points given in polar coordinates (radius and angle). It's derived from the law of cosines and provides the Euclidean distance between points in a polar coordinate system.

2. How Does the Calculator Work?

The calculator uses the polar distance formula:

\[ \text{Distance} = \sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_1 - \theta_2)} \]

Where:

\( r_1 \) — Radius of first point
\( r_2 \) — Radius of second point
\( \theta_1 \) — Angle of first point (in degrees)
\( \theta_2 \) — Angle of second point (in degrees)

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: Polar distance calculations are essential in various fields including physics, engineering, navigation, computer graphics, and astronomy where polar coordinate systems are commonly used to represent positions and movements.

4. Using the Calculator

Tips: Enter both radii (must be non-negative values) and both angles in degrees. The calculator will compute the straight-line distance between the two points in the polar coordinate system.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between polar and Cartesian distance?
A: Polar distance calculates straight-line distance between points given in polar coordinates (r,θ), while Cartesian distance uses rectangular coordinates (x,y). Both give the same Euclidean distance result.

Q2: Can angles be negative in polar coordinates?
A: Yes, negative angles are valid in polar coordinates and represent clockwise rotation from the positive x-axis.

Q3: What if both points have the same angle?
A: If θ1 = θ2, the distance simplifies to |r1 - r2|, as the points lie on the same radial line.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for the given inputs, though practical accuracy depends on the precision of your input values.

Q5: Can this formula be used for 3D polar coordinates?
A: No, this formula is specifically for 2D polar coordinates. For spherical coordinates (3D polar), a different formula involving azimuth and elevation angles is required.