Understanding Cartesian to Polar Conversion
A Cartesian equation uses x and y coordinates. A polar equation uses r and theta. This calculator changes one form into the other. It replaces x with r cos theta. It replaces y with r sin theta. Then it organizes the expression.
The tool is useful for circles, lines, parabolas, and general conics. It can read expanded quadratic equations. These include x squared, xy, y squared, x, y, and constant terms. When that structure is found, the calculator builds a clean polar equation. It also evaluates r for a selected angle.
Why Polar Form Helps
Many curves look simpler in polar form. A circle centered at the origin becomes r equals a constant. A vertical line becomes r cos theta equals a constant. A horizontal line becomes r sin theta equals a constant. Some conics also become easier to inspect.
The result area shows the original equation first. Next, it shows the direct substitution. Then it shows the standard polar structure. For a quadratic equation, the form is q(theta) r squared plus l(theta) r plus f equals zero. This makes the calculation easier to check.
Numerical Angle Evaluation
The angle evaluation is optional. Enter theta in degrees or radians. The calculator computes possible real radius values. It can show the plus root, minus root, or both roots. If the discriminant is negative, no real radius exists for that angle.
Validation helps reduce algebra mistakes. After finding r, the tool computes x and y from the selected angle. It can also show a residual for supported equations. A residual near zero means the converted point fits the original curve.
Exports and Input Tips
The CSV export is helpful for worksheets. The PDF export is useful for reports. Both downloads include the equation, substitution, polar form, angle, and computed values.
Use expanded equations for the strongest results. Avoid hidden multiplication when possible. Write 3*x instead of 3x if you prefer clarity. The parser also understands common compact forms like 3x and x^2. For equations with functions or parentheses, use the direct substitution result. Then simplify the algebra manually if needed. This method keeps the conversion transparent. It also shows every important replacement step.
It supports quick study and classroom checks. It also helps engineering notes and future review tasks.