Calculate the Cartesian equation from a pair of parametric equations by eliminating the parameter t, or convert a Cartesian equation back into parametric form.
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Parametric To Cartesian Formula
Converting parametric equations to a Cartesian equation means eliminating the parameter t so the relationship is written only in terms of x and y. The method depends on the form of the parametric equations.
For a pair of linear parametric equations, solve one equation for t and substitute it into the other:
x = a + bt, y = c + dt => y = c + (d/b)(x - a)
where a is the x-intercept term, b is the rate of change of x, c is the y-intercept term, and d is the rate of change of y. This requires b not equal to 0.
For a circle or ellipse written with cosine and sine, use the identity cos^2 t + sin^2 t = 1:
x = h + acos t, y = k + bsin t => ((x - h)/a)^2 + ((y - k)/b)^2 = 1
where h and k are the center coordinates, a is the horizontal radius, and b is the vertical radius. When a equals b the curve is a circle of radius a.
For a hyperbola written with secant and tangent, use the identity sec^2 t – tan^2 t = 1:
x = h + asec t, y = k + btan t => ((x - h)/a)^2 - ((y - k)/b)^2 = 1
where h and k are the center coordinates and a and b are the scaling constants on x and y.
When one variable is set equal to the parameter directly, substitution is immediate:
x = t, y = f(t) => y = f(x)
where f is any expression in t. Replacing every t with x gives the Cartesian equation.
Common Parametric Forms and Their Cartesian Equations
The table below lists the standard parametric forms you can eliminate the parameter from and the Cartesian equation each produces.
| Parametric form | Curve | Cartesian equation |
|---|---|---|
| x = a + bt, y = c + dt | Line | y = c + (d/b)(x – a) |
| x = rcos t, y = rsin t | Circle | x^2 + y^2 = r^2 |
| x = acos t, y = bsin t | Ellipse | (x/a)^2 + (y/b)^2 = 1 |
| x = asec t, y = b*tan t | Hyperbola | (x/a)^2 – (y/b)^2 = 1 |
| x = t, y = t^2 | Parabola | y = x^2 |
Example Problems
Example 1. Eliminate the parameter from x = 1 + 2t and y = 3 + 4t. Solve the first equation for t to get t = (x – 1)/2. Substitute into the second equation: y = 3 + 4 * (x – 1)/2 = 3 + 2(x – 1) = 2x + 1. The Cartesian equation is y = 2x + 1, a straight line.
Example 2. Eliminate the parameter from x = 5cos t and y = 3sin t. Divide to isolate the trig terms: cos t = x/5 and sin t = y/3. Substitute into cos^2 t + sin^2 t = 1 to get (x/5)^2 + (y/3)^2 = 1, an ellipse centered at the origin.
Frequently Asked Questions
What does it mean to eliminate the parameter? Eliminating the parameter means rewriting a pair of equations that both depend on t into a single equation that relates x and y directly. The parameter t no longer appears in the final Cartesian equation.
Does the Cartesian equation always describe the full curve? Not always. If the parameter t is restricted to an interval, the parametric curve may be only a portion of the full Cartesian graph. Check the range of t before assuming the two represent the exact same set of points.
Can any parametric equation be converted to Cartesian form? Many can, but some cannot be expressed as a simple closed form in x and y. Linear, conic, and direct-substitution cases convert cleanly, while more complex parameterizations may not reduce to an elementary Cartesian equation.
