Circles in Parametric Form
Let’s first start with showing circles in parametric form. On the right is a diagram of a circle with a center at (0,0), radius r and a point (x, y). The radius makes angle theta with the x axis.
**Note that rsin(theta) = y can also be written in parameter y(t). In this case, t is a parameter of time, represented by the angle theta.
**Note that rsin(theta) = y can also be written in parameter y(t). In this case, t is a parameter of time, represented by the angle theta.
The circle to the left has a center at (h, k), radius r and crosses a point (x, y). Everything is the same as the previous example except the circle has been translated so that the the leg opposite to the angle is no longer y, but y-h. The leg adjacent to the angle is no longer x, but x-h.
Ellipses in Parametric Form
Ellipses in parametric form are extremely similar to circles in parametric form except for the fact that ellipses do not have a radius. Therefore, we will use b to signify the radius along the y-axis and a to signify the radius along the x-axis.
**Note that this is the same for both horizontal and vertical ellipses.
**Note that this is the same for both horizontal and vertical ellipses.
Going from Parametric Form to Rectangular Form
Start out with y (t) , subtract h from both sides, and divide by b on both sides. Then square both sides.
Do the same with x (t).
Add the two equations together. Remember the Pythagorean identity cos^2(t)+sin^2(t) = 1.
**Only parameters in which the first equation includes a sin(t) and the second equation includes a cos (t) will be able to be converted into rectangular form because when you add the squares of two equations, the result must be cos^2(t)+sin^2(t) because that is the only Pythagorean identity that equals to 1 when the two values are added together.
Do the same with x (t).
Add the two equations together. Remember the Pythagorean identity cos^2(t)+sin^2(t) = 1.
**Only parameters in which the first equation includes a sin(t) and the second equation includes a cos (t) will be able to be converted into rectangular form because when you add the squares of two equations, the result must be cos^2(t)+sin^2(t) because that is the only Pythagorean identity that equals to 1 when the two values are added together.
