Ready to convert
Convert a supported parametric model
Choose the matching pair. The labels adapt to the selected relationship.
Supported conversion templates
Linear pair
x = at + b and y = ct + d become a line or vertical relation.
Quadratic and cubic
A linear x rule lets the calculator replace t with an x expression.
Circle and ellipse
Cosine and sine combine through the Pythagorean identity.
Hyperbola
Secant and tangent use sec²(t) − tan²(t) = 1.
Exponential
A linear parameter in x turns the exponential into y as a function of x.
Sine curve
The calculator preserves amplitude, phase, vertical shift, and horizontal angular rate.
Example data table
These examples show the template output before applying any parameter interval.
| Model | Parametric form | Rectangular result | Key note |
|---|---|---|---|
| Line | x = 2t + 1, y = 3t − 2 | y = 1.5x − 3.5 | Solve t from x first. |
| Parabola | x = t − 2, y = t² + 1 | y = (x + 2)² + 1 | Shifted form stays readable. |
| Cubic | x = t + 1, y = t³ | y = (x − 1)³ | Replace every occurrence of t. |
| Circle | x = 1 + 3cos(t), y = 2 + 3sin(t) | (x − 1)² + (y − 2)² = 9 | Equal radii form a circle. |
| Ellipse | x = 4cos(t), y = 2sin(t) | x²/16 + y²/4 = 1 | Axes need not match. |
| Hyperbola | x = 2sec(t), y = 3tan(t) | x²/4 − y²/9 = 1 | Keep both branches. |
| Exponential | x = t, y = 5e^(2t) − 1 | y = 5e^(2x) − 1 | The shift remains outside. |
| Sine | x = 2t, y = 4sin(3t) + 1 | y = 4sin(1.5x) + 1 | Angular rate becomes d/a. |
Formula used
The core method is parameter elimination. Rearrange one coordinate for t. Then substitute that expression into the other coordinate.
For polynomial, exponential, and sine templates, substitute t = (x − b) / a into the y equation. Keep exclusions, branches, ranges, and stated parameter limits with the final relation.
How to use this calculator
- Select the template matching your x(t) and y(t) equations.
- Enter every coefficient exactly, including negative signs and shifts.
- Set the decimal places needed for your result.
- Add an optional parameter interval when only part of a curve applies.
- Choose a sample t value to verify one coordinate pair.
- Press the conversion button and inspect the substitution steps.
- Copy or export the completed conversion after checking restrictions.
Turning parameters into rectangular forms
Parametric equations describe both coordinates through one shared parameter. They are useful for motion, tracing, and curve design. A rectangular equation removes that parameter. It links x and y directly. This makes the curve easier to classify. It also helps compare graphs, intercepts, and restrictions. The process begins by isolating the parameter. Then substitute that expression into the other coordinate rule. The final result should contain x and y only.
Eliminate the parameter
For a linear pair, solve x = at + b for t. The solution is t = (x − b) / a. Put this into y = ct + d. The answer becomes a straight line. Its slope is c divided by a. Its intercept follows after simplifying. When a equals zero, x stays fixed. The rectangular result is a vertical line instead.
Parabolas often use x as a linear parameter rule. Substitute t = (x − b) / a into y. A squared parameter becomes a squared x expression. This produces vertex form or expanded quadratic form. Keep the shifted expression first. It is easier to read. Then expand only when standard form is needed. The same idea works for cubic parametric curves.
Use identities for conic curves
Circles and ellipses use sine and cosine relationships. Start with x minus the horizontal center. Divide by the horizontal radius. Do the same for y and the vertical radius. Square both expressions. Add them together. The identity cosine squared plus sine squared equals one. This removes the parameter immediately. Equal radii give a circle. Unequal radii give an ellipse.
Preserve the curve's limits
Some conversions require extra care with restrictions. A hyperbola made from secant and tangent has branch limits. The rectangular equation alone may hide the allowed points. Exponential and sine models also have useful range information. Parameter intervals can limit a curve segment. Always keep stated t bounds. Then test the resulting point at a sample parameter value. It should satisfy the converted equation.
Graph checks reveal mistakes before they become answers. Plot parameter values. Mark pairs. Compare them with the rectangular graph. Notice whether the path crosses itself. A rectangular equation usually describes the full locus. It may not record direction or starting position. Parameter bounds restore that missing detail. This matters for motion problems. It also matters for arcs, loops, and partial curves. Use domain and range statements whenever they sharpen the result. Rounded values are useful for checking. Exact coefficients are better for the final equation. Avoid cancelling factors without checking excluded values. A cancelled factor can remove a point or hide a restriction.
Check the final equation
This calculator uses guided templates instead of guessing arbitrary symbolic syntax. Select the template matching your two equations. Enter each coefficient with its sign. Use a nonzero linear x coefficient when elimination requires division. Choose a sample t value for a quick check. The result panel shows the substitution route, rectangular equation, and conditions. Export the calculation when you need a record. Careful substitution keeps every converted relation reliable and clear.
Frequently asked questions
1. What is a rectangular equation?
It is an equation containing x and y without a parameter. It describes the location of the curve on a coordinate plane.
2. Does the rectangular equation keep curve direction?
Usually no. A rectangular relation shows the locus of points. It may not reveal where the motion starts, ends, or which direction t follows.
3. Why must a be nonzero in many templates?
The calculator solves x = at + b for t. Division by a is required. When a is zero, x remains fixed instead.
4. Why are radians used for sine, cosine, secant, and tangent?
Standard parametric identities are written with radian measures. Enter t in radians for the trigonometric templates.
5. Can a parameter interval change the result?
Yes. It can restrict the rectangular curve to one arc, segment, branch, or motion path. Keep the interval with your final answer.
6. What identity converts a circle or ellipse?
Use cos²(t) + sin²(t) = 1. After normalizing x and y, the parameter cancels through this identity.
7. Why does the hyperbola template show branches?
Secant cannot have magnitude below one. That condition creates separated left and right branches in the rectangular hyperbola.
8. Can this tool accept any typed symbolic equation?
No. It uses guided coefficient templates. This avoids unsafe expression parsing and gives reliable, traceable substitutions for common function families.
9. Should I round during substitution?
Keep full values while working whenever possible. Round the displayed result at the end, using the selected decimal precision.
10. What does the sample point check do?
It evaluates x(t) and y(t) at your selected t value. The resulting coordinate should satisfy the rectangular relation.
11. What should I inspect before exporting?
Check signs, shifts, parameter bounds, branches, and the sample point. Careful substitution keeps every converted relation reliable and clear.