Equation converter
Choose a parametric family and enter its values
The calculator removes t using the matching identity or substitution method.
Example data
Common conversion patterns
| Parametric equations | Rectangular equation | Method |
|---|---|---|
| x = 1 + 2t, y = −3 + 4t | y + 3 = 2(x − 1) | Substitute for t |
| x = 2 + 3t², y = 1 + 2t | x − 2 = 0.75(y − 1)² | Square substitution |
| x = 4cos(t), y = 4sin(t) | x² + y² = 16 | Trig identity |
| x = 1 + 5cos(t), y = −2 + 3sin(t) | (x − 1)²/25 + (y + 2)²/9 = 1 | Ellipse identity |
Formula used
Eliminate the parameter
For linear equations, solve one equation for t. Substitute that expression into the other equation. For x = h + at and y = k + bt, the result is y − k = (b/a)(x − h).
For trigonometric curves, isolate sin(t), cos(t), sec(t), or tan(t). Then apply an identity. Circles and ellipses use sin²(t) + cos²(t) = 1. Hyperbolas use sec²(t) − tan²(t) = 1.
For parabolas, isolate t from the linear equation. Square it when the other equation contains t². The substitution preserves the algebraic curve, but a limited parameter range can trace only part of that curve.
How to use this calculator
Convert your equations in four steps
- Select the equation family that matches your x(t) and y(t) equations.
- Enter a, b, h, and k exactly as they appear in the model.
- Add a parameter interval when the curve uses a restricted domain.
- Choose precision, submit the form, then review the equation and elimination steps.
Concept guide
From moving points to familiar curves
Why elimination matters
Parametric equations describe x and y through a shared parameter. The parameter often represents time, an angle, or another changing quantity. A rectangular equation removes that extra variable. It shows a direct relationship between x and y. This form is useful when you need a standard graph, an intercept, a symmetry test, or an equation for later algebra.
Start with the equation family
Look for the structure before performing algebra. Linear expressions in t usually form a line. One squared term and one linear term often form a parabola. Sine and cosine terms point toward a circle or ellipse. Secant and tangent commonly produce a hyperbola. Recognizing the pattern helps you choose the safest identity or substitution path.
Use substitution for algebraic forms
When one equation is linear in t, solve it first. Replace t in the second equation. For example, y = k + bt gives t = (y − k)/b. Insert this into x = h + at². The result becomes x − h = a(y − k)²/b². Keep grouped shifts visible. They make the center, vertex, and orientation easier to read.
Use identities for trigonometric forms
Trigonometric models require a different route. Divide each equation by its amplitude. This isolates cosine and sine, or secant and tangent. Square the two relations. Then combine them with the relevant identity. The method converts circular motion into a circle equation. It also converts unequal amplitudes into an ellipse equation. Signs and shifts still matter, so carry them through every step.
Keep domain limits attached
A rectangular equation can include points that the original parameter never reaches. This happens when t has limits. For example, an ellipse with t from zero to π traces only its upper half. A parabola can also be traced in a single direction. Record the original interval beside the result. It tells you whether the rectangular equation represents a full curve or only a path segment.
Check the conversion
Choose a simple parameter value, such as zero or one. Compute x and y from the original equations. Substitute those coordinates into the new rectangular equation. Both sides should agree, aside from rounding. Test another value when possible. This small check catches missing squares, reversed signs, and misplaced shifts. It also confirms that your calculated curve matches the original motion.
Use the result with confidence
Rectangular form supports many next steps. You can graph the relation, locate its center or vertex, calculate intersections, or compare it with another equation. Compare the converted equation with a plotted table of points. Visual confirmation makes hidden domain limits, openings, and asymptotes easier to notice before drawing conclusions. Parametric form still remains valuable for direction and timing. Use both forms together. One describes how the point travels. The other describes the geometric set of points. That combination makes curve analysis more complete and more reliable.
Frequently asked questions
Practical checks before you graph
1. What is a rectangular equation?
It is an equation relating x and y directly. The parameter does not appear. A rectangular form describes the geometric curve without stating how a point moves along it.
2. Why remove the parameter?
Removing the parameter makes standard graphing and algebra easier. It can reveal a line, circle, ellipse, parabola, or hyperbola. The original parametric form remains useful for direction and timing.
3. Which parametric families does this calculator support?
It supports lines, two parabola orientations, circles, ellipses, and secant-tangent hyperbolas. Each family uses a reliable substitution or identity-based conversion method.
4. How does a line conversion work?
Solve x = h + at for t. Substitute that expression into y = k + bt. The result is y − k = (b/a)(x − h), provided a is not zero.
5. Why does a circle use squared terms?
The conversion uses sin²(t) + cos²(t) = 1. After isolating sine and cosine, squaring produces the shifted circle equation with radius squared on the right side.
6. What changes for an ellipse?
The cosine and sine amplitudes can differ. Their squares become separate denominators. This produces the standard ellipse form, with horizontal and vertical semi-axis lengths determined by the amplitudes.
7. Why should I enter a t interval?
An interval records which portion of a curve is traced. The rectangular equation may show the complete curve, while the original equations describe only an arc, branch, or directed segment.
8. Can two t values produce the same point?
Yes. Periodic trigonometric equations repeat points. Squared parameter terms can also merge positive and negative parameter values. The rectangular equation keeps the curve, not every travel detail.
9. What happens if a required coefficient is zero?
The selected formula may become degenerate or require a different family. The calculator flags coefficients that would create division by zero in the conversion steps.
10. How can I verify the result?
Choose one or two parameter values. Compute the coordinates from the original pair. Substitute those coordinates into the rectangular equation. Matching values confirm the algebra, apart from chosen rounding.
11. Does every parametric pair have one simple rectangular form?
No. Some pairs need implicit equations, piecewise restrictions, or numerical methods. Careful elimination produces reliable equations for confident graphing decisions.