Calculator Inputs
Example Data Table
Example model: x = 2t + 1 and y = 3t - 4. The Cartesian result is y + 4 = 1.5(x - 1).
| t | x(t) | y(t) | Check |
|---|---|---|---|
| -2 | -3 | -10 | -10 + 4 = 1.5(-3 - 1) |
| 0 | 1 | -4 | -4 + 4 = 1.5(1 - 1) |
| 2 | 5 | 2 | 2 + 4 = 1.5(5 - 1) |
Formula Used
Line: x = at + b, y = ct + d. Then t = (x - b) / a, so y - d = (c / a)(x - b).
Vertical parabola: x = at + b, y = ct² + d. Then y - d = (c / a²)(x - b)².
Horizontal parabola: x = at² + b, y = ct + d. Then x - b = (a / c²)(y - d)².
Ellipse: x = h + a cos(t), y = k + b sin(t). Then (x - h)²/a² + (y - k)²/b² = 1.
Hyperbola: x = h + a cosh(t), y = k + b sinh(t). Then (x - h)²/a² - (y - k)²/b² = 1.
Reciprocal curve: x = h + a/t, y = k + bt. Then (x - h)(y - k) = ab.
The residual column substitutes every sampled point into the final Cartesian equation. Values near zero confirm a consistent elimination.
How to Use This Calculator
- Select the parametric curve family that matches your problem.
- Enter the coefficients shown in the selected model.
- Set a parameter interval. Use radians for trigonometric models.
- Choose the number of sample points for the table and graph.
- Press the submit button and read the Cartesian equation above the form.
- Use the CSV or PDF buttons to save your result.
What This Calculator Does
A parametric curve gives x and y through a third value. That value is usually called t. The curve can be easy to draw, yet harder to compare with a normal equation. This calculator removes the parameter for supported curve families. It then writes a Cartesian equation in x and y.
Why Parameter Elimination Matters
Parameter elimination is useful in algebra, calculus, physics, and engineering. A moving object may have horizontal and vertical positions. Each position depends on time. Removing time shows the actual path. The final equation can reveal a line, parabola, ellipse, hyperbola, or reciprocal curve.
Advanced Checks Included
The calculator does more than print one equation. It shows the substitution idea. It creates a numerical table. It checks residual error at many parameter values. A small residual means the derived equation matches the sampled parametric points. This is helpful when teaching or testing solutions.
Graph And Export Value
The graph gives a quick visual check. Straight lines look direct. Parabolas show bending. Ellipses close into loops. Hyperbolas split into branches when the range allows it. Export buttons help you save the work. You can download a table as a CSV file. You can also create a printable report for study notes.
Best Practice
Choose the correct curve model first. Enter nonzero scale values where needed. Keep the parameter range wide enough to show the shape. For trigonometric curves, use radians. For classroom examples, compare the generated steps with your hand solution. This builds stronger algebra skills.
Reading The Result
The Cartesian equation should be read with the domain note. Some equations describe a full curve, while the parameter range may draw only part of it. For example, an ellipse may be complete, yet a short interval shows an arc. A hyperbola model may show one branch.
Common Mistakes
Do not mix degrees with radians. Do not enter zero where division is required. Check signs in shifted forms. A term like x minus h means the curve is shifted right by h. A term like y minus k means the curve is shifted upward by k. Use the residual column to catch entry mistakes.
FAQs
What does it mean to eliminate a parameter?
It means removing t from two parametric equations. The final equation uses only x and y, so the curve is written in Cartesian form.
Can this calculator solve every parametric equation?
No. It supports common forms with reliable symbolic elimination. Unusual functions may need a computer algebra system or manual algebra.
Why do some models require nonzero coefficients?
Some formulas divide by a coefficient. If that value is zero, the elimination step becomes invalid or the curve collapses to a simpler case.
Should trigonometric inputs use degrees or radians?
Use radians. The graph and sample table calculate sine, cosine, hyperbolic sine, and hyperbolic cosine with radian-based parameter values.
What is the residual column?
The residual is the result of substituting sampled points into the Cartesian equation. A value near zero means the equation checks well.
Why does the graph show only part of a curve?
The graph follows the selected parameter range. A small interval may draw only a segment, arc, or one visible branch of the full equation.
Can I use negative scale values?
Yes, many models allow negative scales. The squared Cartesian form may hide orientation, but the parametric graph still shows direction.
What does the PDF export include?
It includes the model, Cartesian equation, elimination step, domain note, residual check, graph image, and a sample of table values.