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 + b*t, y = c + d*t => y = c + (d/b)*(x - a), b != 0
Here a and c are the coordinates at t = 0; b and d are the rates of change. If b = 0 and d is nonzero, the result is the vertical line x = a. If both rates are zero, the result is the point (a, c).
For a circle or ellipse written with cosine and sine, use the identity cos^2 t + sin^2 t = 1:
x = h + a*cos(t), y = k + b*sin(t) => ((x - h)/a)^2 + ((y - k)/b)^2 = 1
The center is (h, k), and nonzero a and b have radii |a| and |b|. If |a| = |b|, the curve is a circle of radius |a|. Negative scales change traversal, not the Cartesian curve.
For a hyperbola written with secant and tangent, use the identity sec^2 t – tan^2 t = 1:
x = h + a*sec(t), y = k + b*tan(t) => ((x - h)/a)^2 - ((y - k)/b)^2 = 1
Here h and k are the center coordinates and a and b are nonzero scaling constants. The parameter excludes t = pi/2 + n*pi for every integer n. With its full natural real domain, the parameterization covers both branches.
When one variable is set equal to the parameter directly, substitution is immediate:
x = t, y = f(t) => y = f(x)
For a supported expression f(t), replacing only variable tokens t with x gives y = f(x); function names stay intact. The calculator checks arithmetic syntax and supports common trig functions, sqrt, abs, exp, ln (natural log), log (base 10), and constants pi and e. Use * for multiplication, ^ for powers, and parentheses for functions. Trig arguments use radians. Preserve the expressionโs real domain and any extra parameter interval restrictions. For Cartesian-to-parametric conversion, select y = f(x) or x = g(y) and enter only its right-hand side.
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.
