Circle Equation in Polar Form

The polar form of the circle equation is expressed as: $$ r^2 + r_c^2 - 2 \cdot r \cdot r_c \cos (\alpha - \alpha_c) - R^2 = 0 $$ where \( R \) is the radius of the circle, and \( r_c \) and \( r \) represent the distances from the origin (pole) to the center \( C \) and to any point \( P \) on the circle, respectively.
the circle in polar form

By varying the polar coordinates (α, r), all the points on the circle can be determined.

Here, the angle α and the segment length r serve as the variables.

a practical example

    Proof

    Consider any circle on the plane.

    The circle has a radius of \( R = 2 \).

    an example of a circle with radius R=2

    The center of the circle \( C \) is located at the polar coordinates (αc, rc).

    $$ C: (\alpha_c, r_c) = (36.87°, 5) $$

    Here, the angle αc = 36.87° and the segment \( r_c \) = 5.

    the polar coordinates of the circle's center

    Now, take any point \( P \) on the circle.

    $$ P: (\alpha, r) = (60.27°, 4.8) $$

    Point \( P \) has polar coordinates α = 60.27° and \( r \) = 4.8.

    point P on the circle

    The radius \( R \), along with segments \( OP \) and \( OC \), forms triangle \( OPC \).

    Applying the cosine rule to triangle \( OPC \), we get:

    $$ \overline{PC}^2 = \overline{OP}^2 + \overline{OC}^2 - 2 \cdot \overline{OP} \cdot \overline{OC} \cdot \cos(\alpha - \alpha_c) $$

    Given that the sides of the triangle are \( PC = R \), \( OC = r_c \), and \( OP = r \), we have:

    $$ R^2 = r^2 + r_c^2 - 2 \cdot r \cdot r_c \cdot \cos(\alpha - \alpha_c) $$

    Rearranging the terms, we arrive at:

    $$ r^2 + r_c^2 - 2 \cdot r \cdot r_c \cdot \cos(\alpha - \alpha_c) - R^2 = 0 $$

    This is the final form of the circle's equation in polar coordinates, which completes our proof.

    Verification: By substituting the polar coordinate values α = 60.27°, αc = 36.87°, \( r = 4.8 \), \( r_c = 5 \), and \( R = 2 \) into the equation, we confirm that: $$ R^2 = r^2 + r_c^2 - 2 \cdot r \cdot r_c \cdot \cos(\alpha - \alpha_c) $$ $$ 5^2 = 4.8^2 + 5^2 - 2 \cdot 4.8 \cdot 5 \cdot \cos(60.27° - 36.87°) $$ $$ 25 = 23.04 + 25 - 2 \cdot 24 \cdot \cos(23.4°) $$ $$ 25 = 48 - 44 $$ $$ 25 = 25 $$ This verifies the equation is correct.

    And so on.

     
     

    Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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