Parametric Equation of a Circle

Parametric Equation of a Circle:

1. Definition:

• Parametric equations describe the x and y coordinates of a point on a curve using a third parameter variable.
• For a circle, parametric equations define x and y in terms of an angle or parameter $t$.

2. Parametric Equations for a Circle:

• General Form:
• $x=h+r\mathrm{cos}\left(t\right)$
• $y=k+r\mathrm{sin}\left(t\right)$
• $h,k$ represent the center of the circle, $r$ is the radius, and $t$ is the parameter or angle.

3. Explanation:

• $x$ Coordinate: $h+r\mathrm{cos}\left(t\right)$ represents the x-coordinate of a point on the circle with center $\left(h,k\right)$ and radius $r$ at angle $t$.
• $y$ Coordinate: $k+r\mathrm{sin}\left(t\right)$ represents the y-coordinate of the same point on the circle.

4. Parameter $t$ and Angle:

• $t$ usually varies from $0$ to $2\pi$ for one complete revolution around the circle.
• $t$ is akin to the angle measured from the positive x-axis in the counterclockwise direction.

5. Example:

• Circle with Center $\left(3,-2\right)$ and Radius $4$:
• $x=3+4\mathrm{cos}\left(t\right)$
• $y=-2+4\mathrm{sin}\left(t\right)$
• $0\le t\le 2\pi$ for a full revolution.

6. Interpretation:

• For each $t$ value in the range, the parametric equations generate corresponding x and y coordinates that trace the circumference of the circle with center $\left(3,-2\right)$ and radius $4$.
• As $t$ varies from $0$ to $2\pi$, the point on the circle moves counterclockwise around its circumference, completing one revolution.