Calculator Inputs
Formula Used
A Cartesian equation uses x and y directly. A parametric equation uses a third variable, usually t. The curve is written as ordered functions x(t) and y(t). Every value of t gives one point on the curve.
- Line: if y = mx + c, then x(t) = t and y(t) = mt + c.
- Circle: if center is (h, k), then x(t) = h + r cos(t), y(t) = k + r sin(t).
- Ellipse: x(t) = h + a cos(t), y(t) = k + b sin(t).
- Vertical parabola: x(t) = t, y(t) = a(t - h)² + k.
- Segment: x(t) = x₁ + (x₂ - x₁)t, y(t) = y₁ + (y₂ - y₁)t.
How to Use This Calculator
Select the Cartesian model that matches your equation. Enter the related constants. For a line, enter slope and intercept. For a circle, enter center and radius. For an ellipse, enter center and axis values. For a segment, enter both endpoints. Choose the starting and ending value of t. Then select the number of sample points. Press the calculate button. The result appears above the form, below the header. Review the parametric form, graph, derivative check, and generated point table. Use the CSV button to export data. Use the PDF button to save a report.
Example Data Table
This example converts the Cartesian line y = 2x + 1 into x(t) = t and y(t) = 2t + 1.
| Cartesian equation | t | x(t) | y(t) | Point |
|---|---|---|---|---|
| y = 2x + 1 | 0 | 0 | 1 | (0, 1) |
| y = 2x + 1 | 2 | 2 | 5 | (2, 5) |
| y = 2x + 1 | 4 | 4 | 9 | (4, 9) |
Cartesian to Parametric Conversion Guide
What the Conversion Means
Cartesian form describes a curve with x and y. Parametric form describes the same curve with a moving parameter. This parameter is often called t. It can represent time, angle, distance, or a chosen input. A single value of t creates one point. Many values of t trace the whole curve.
Why Parametric Form Is Useful
Parametric equations are helpful when motion matters. They also make circles and ellipses easier to graph. A circle is difficult as one y function. It needs upper and lower branches. Parametric form avoids that split. One pair of functions can trace the whole shape.
Common Conversion Method
The simplest method is substitution. If y is already written in terms of x, set x equal to t. Then replace every x with t. The result is x(t) and y(t). This works well for lines, parabolas, polynomials, and many functions. If x is written in terms of y, set y equal to t instead.
Curves With Angles
Circles and ellipses use trigonometric parameters. The angle t controls position around the center. Cosine gives the horizontal coordinate. Sine gives the vertical coordinate. For a full closed curve, use t from 0 to 2π. For an arc, use a smaller interval. The graph updates from the calculated table values.
Reading the Output
The calculator returns the selected Cartesian model. It then shows the matching parametric equations. The table lists t, x(t), and y(t). These rows are useful for plotting or checking work. The graph gives a fast visual check. The derivative line shows how coordinates change with t. Export options help save the results for reports, homework, or technical notes.
FAQs
1. What is a Cartesian equation?
A Cartesian equation describes a curve using x and y coordinates. It often appears as y = f(x), x = f(y), or an implicit relation between both variables.
2. What is a parametric equation?
A parametric equation writes coordinates as functions of another variable. Usually, x = x(t) and y = y(t). Each t value creates one point.
3. Why convert Cartesian form to parametric form?
Parametric form is useful for motion, graphing, animation, and curves that fail the vertical line test. It can describe complex paths more naturally.
4. How do I convert y = f(x)?
Set x(t) = t. Then replace x with t in the original equation. The result becomes y(t) = f(t).
5. How is a circle converted?
For a circle centered at (h, k), use x(t) = h + r cos(t) and y(t) = k + r sin(t). A full circle uses 0 to 2π.
6. Can this calculator handle ellipses?
Yes. Enter center values h and k, then enter semi-axis values a and b. The calculator builds x(t) and y(t) using cosine and sine.
7. What does the sample count mean?
The sample count controls how many t values are used in the table and graph. More samples create smoother plots but larger exported files.
8. Is the exported data exact?
The formulas are exact for the selected model. The exported table is numerical because it uses sampled t values and rounded decimal output.