Substitute the extreme values of the parameter (usually 0 and 1) into the parametric equations for x and y.

1. Identify the center (a,b) and radius r from the equations x(t) = a + rcos(t) and y(t) = b + rsin(t). 2. Write the equation in the form (x-a)^2 + (y-b)^2 = r^2.

Use the trigonometric identity $cos^2(t) + sin^2(t) = 1$. Solve for cos(t) and sin(t) in terms of x and y, then substitute into the identity.

Subtract the coordinates of the initial point from the coordinates of the terminal point: $(x_2 - x_1, y_2 - y_1)$.

$x(t) = cos(t)$, $y(t) = sin(t)$

0 \le t \le 2\pi

$x(t) = a + rcos(t)$, $y(t) = b + rsin(t)$

$x(t) = cos(t + c)$, $y(t) = sin(t + c)$

$x = x_1 + k(x_2 - x_1)$, $y = y_1 + k(y_2 - y_1)$, where 0 \le k \le 1

As 't' increases, the (x, y) point moves along the defined path. The rate of change of x(t) and y(t) determines the speed and direction.

Shifting the center involves adding constants to x(t) and y(t). Changing the radius involves multiplying cos(t) and sin(t) by the radius. Rotation involves adding a constant to 't'.

The direction vector (x2 - x1, y2 - y1) determines the slope and orientation of the line segment. It's scaled by the parameter 'k' to move from (x1, y1) to (x2, y2).

In the equations $x(t) = a + rcos(t)$ and $y(t) = b + rsin(t)$, (a, b) is the center and 'r' is the radius.