Calculator
Example Data Table
| Model | Rectangular form | Parametric form | Usual parameter range |
|---|---|---|---|
| Circle | (x - 2)² + (y - 3)² = 25 | x = 2 + 5cos(t), y = 3 + 5sin(t) | 0 ≤ t ≤ 2π |
| Ellipse | (x² / 36) + (y² / 9) = 1 | x = 6cos(t), y = 3sin(t) | 0 ≤ t ≤ 2π |
| Line | 2x + 3y - 6 = 0 | x = t, y = (-2t + 6) / 3 | All real t |
| Parabola | (x - 1)² = 8(y + 2) | x = 1 + t, y = -2 + t² / 8 | All real t |
Formula Used
Circle: (x - h)² + (y - k)² = r² becomes x(t) = h + rcos(t), y(t) = k + rsin(t).
Ellipse: ((x - h)² / a²) + ((y - k)² / b²) = 1 becomes x(t) = h + acos(t), y(t) = k + bsin(t).
Line: Ax + By + C = 0 becomes x(t) = t, y(t) = (-At - C) / B when B is not zero.
Vertical line: x = c becomes x(t) = c, y(t) = t.
Vertical parabola: (x - h)² = 4p(y - k) becomes x(t) = h + t, y(t) = k + t² / 4p.
Hyperbola: The calculator uses cosh²(t) - sinh²(t) = 1 for branch based conversion.
How to Use This Calculator
Select the rectangular equation model first. Then enter only the values used by that model. For a circle, use h, k, and r. For an ellipse, use h, k, a, and b. For a general line, use A, B, and C. Next, set the t start, t end, and t step values. Press Calculate to view equations and sample points. Use the CSV or PDF buttons to export the same result.
Rectangular to Parametric Equation Conversion
What the Calculator Does
This calculator changes a rectangular equation into parametric form. A rectangular equation uses x and y directly. A parametric equation uses a third variable, usually t. The new form writes x and y as separate functions of t. This is useful in calculus, analytic geometry, motion study, graphing, and coordinate conversion.
Why Parametric Form Helps
Parametric form can describe direction and movement. It can also describe curves that are hard to write as one function of x. A circle is a simple example. Its rectangular form has squared terms. Its parametric form uses sine and cosine. That form gives smooth points around the curve.
Supported Curve Types
The tool supports lines, circles, ellipses, parabolas, and hyperbolas. Each model has its own trusted conversion rule. For a line, the calculator can set x equal to t. Then it solves for y. For a vertical line, x stays constant and y becomes t. For conic sections, the calculator uses standard center, radius, axis, and vertex values.
Using Parameters Carefully
The parameter range affects the points shown in the table. For a complete circle or ellipse, a range from 0 to 2π is common. For lines and parabolas, all real t values are valid. The table only shows a selected sample range. It does not limit the real equation unless you choose that limit for your own graph.
Reading the Result
The result section shows the original rectangular form, x(t), y(t), domain guidance, and the method used. It also lists sample coordinates. These points help verify the conversion. You can copy the equations into graphing tools. You can also download reports for lessons, homework, records, or project notes.
FAQs
What is a rectangular equation?
A rectangular equation uses x and y coordinates directly. Examples include y = 2x + 1, x² + y² = 25, and (x² / 9) + (y² / 4) = 1.
What is a parametric equation?
A parametric equation writes x and y as separate functions of another variable. This variable is often t. It can represent position, time, direction, or curve progress.
Can every rectangular equation be converted?
Many equations can be converted, but not always in one simple way. This calculator supports common lines and conic sections with standard conversion formulas.
Why does a circle use sine and cosine?
Sine and cosine satisfy cos²(t) + sin²(t) = 1. This identity matches the centered circle equation after scaling by the radius.
What t range should I use?
Use 0 to 2π for one full circle or ellipse. Use any practical range for lines, parabolas, and hyperbolas when making sample points.
What happens if B equals zero in a line?
If B is zero, the general line may be vertical. The calculator then keeps x constant and lets y equal t when A is valid.
Are the sample points exact?
The formulas are exact, but displayed decimal points are rounded. Rounding keeps the table readable and easier to export.
Can I download my result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report containing equations, method, and sample points.