Parametric equation of the circle

In mathematics, a parametric equation it allows to represent a curve or a surface in the plane or in the space, by means of arbitrary values or by means of a constant, called parameter, instead of using an independent variable whose values derive from those of the dependent variable. A simple example is kinematics: suppose a body that moves in a plane and, as time passes, represents a path like the one represented in Figure I. The x and y coordinates of the position of the object depend on the instant of time t . Therefore there will exist functions x and y of the variable (or parameter) t, such that x = x (t) and y = y (t); these two equations are called parametric equations of the curve and each value of t determines a point (x, y) on the plane.

Figure 2. Parametric equations of the curve
Figure 2. Parametric equations of the curve

The points (x, y) of a circle can be expressed from an individual variable θ; these individual variables are called parameter. The parametric equations of a circle with radius r ≥ 0 and center (h, k), are given by:

  • x = h + rcosθ 0 <θ <2π
  • y = k + rsinθ

The parametric equation of a circle centered at the origin and with radius r:

x2 + and2 = r2

The parametric equation with the formula of an individual variable:

  •  x = ± √ (r2 - Y2)
  • y = ± √ (r2 - x2)

Each formula expresses a partial portion of the circle:

  • y = √ (r2 - x2) (cap)
  • y = - √ (r2 - x2) (background)
  • x = √ (r2 - Y2) (right side)
  • x = - √ (r2 - Y2) (left)

 Examples

  • Determine the radius of the parametric equation of the circle for:

 x = 5sint; y = 5sint

where 0 <t <π.

The parametric equation of a circle is x2 + and2 = r2, where x = x synt; y = y cost. Thus:

(5 synth)2 + (5 cost)2 = r2

25 (without2t + cos2t) = r2

As without2t + cos2t = 1, we have

r2 = 25

r = √25 = 5

  • Determine the parametric equation of a circle with radius r = 6 and centered at the point (2,4).

 We know that (h, k) = (2,4) and r = 6, therefore, the parametric equation of the circle of the statement is:

                x = h + rcosθ → x = 2 + 6cosθ

                y = k + rsinθ → y = 4 + 6sinθ