Parametric Equation of a Circle

A circle is a set of all points in a plane that are at a given distance (radius) from a fixed point (center). The standard equation of a circle with center at the origin (0, 0) and radius r is given by:

$$ x^2 + y^2 = r^2 $$

However, this is not the only way to represent a circle. A parametric equation provides an alternative representation where each coordinate of the circle is expressed as a function of a parameter, usually denoted as t. This parameter t often represents the angle that a radius vector makes with the positive x-axis, measured in radians.

Parametric Equations

The parametric equations of a circle with center at the origin and radius r are:

$$ x(t) = r \cos(t) $$ $$ y(t) = r \sin(t) $$

where t is the parameter, typically representing the angle in radians.

If the circle is not centered at the origin but at a point (h, k), the parametric equations become:

$$ x(t) = h + r \cos(t) $$ $$ y(t) = k + r \sin(t) $$

Important Points

• The parameter t usually ranges from 0 to 2π for a complete traversal of the circle.
• The parametric form is particularly useful in computer graphics and animations where the position of a point needs to be updated based on time or another variable.
• Parametric equations allow for easy differentiation and integration, which can be useful in various applications such as calculating arc length or area.

Table of Differences and Important Points

Aspect	Standard Equation	Parametric Equations
Form	Implicit	Explicit
Center at Origin	$x^2 + y^2 = r^2$	$x(t) = r \cos(t)$ $y(t) = r \sin(t)$
Center at (h, k)	$(x-h)^2 + (y-k)^2 = r^2$	$x(t) = h + r \cos(t)$ $y(t) = k + r \sin(t)$
Parameter	Not applicable	Angle t, typically in radians
Range of Parameter	Not applicable	Usually 0 ≤ t < 2π
Applications	Geometry problems	Calculus, animations, and graphics

Examples

Example 1: Circle at the Origin

Consider a circle with a radius of 3 units centered at the origin. The parametric equations are:

$$ x(t) = 3 \cos(t) $$ $$ y(t) = 3 \sin(t) $$

For t = π/2, the point on the circle is:

$$ x(π/2) = 3 \cos(π/2) = 0 $$ $$ y(π/2) = 3 \sin(π/2) = 3 $$

So, the point is (0, 3) on the circle.

Example 2: Circle with Center at (2, -1)

Now consider a circle with a radius of 4 units and center at (2, -1). The parametric equations are:

$$ x(t) = 2 + 4 \cos(t) $$ $$ y(t) = -1 + 4 \sin(t) $$

For t = π, the point on the circle is:

$$ x(π) = 2 + 4 \cos(π) = 2 - 4 = -2 $$ $$ y(π) = -1 + 4 \sin(π) = -1 $$

So, the point is (-2, -1) on the circle.

Conclusion

The parametric equation of a circle is a powerful tool in mathematics that provides a way to describe the position of a point on a circle using a single parameter. This form is particularly useful in applications that involve motion or require the use of calculus. By understanding the parametric equations, one can easily work with circles in various mathematical contexts.