Understanding Rectangular to Polar Conversion
A rectangular equation uses x and y. A polar equation uses r and theta. Both describe the same graph. The difference is the coordinate system. Rectangular form measures horizontal and vertical distance. Polar form measures distance from the origin and angle from the positive x axis.
Why Conversion Matters
Polar form can make many curves easier to inspect. Circles centered at the origin become simple. Lines through the origin become angle rules. Spirals, limacons, and roses are also easier to compare in polar form. This calculator focuses on algebraic replacement. It changes every x into r cos theta. It changes every y into r sin theta. Then it groups the new expression by powers of r.
What the Calculator Handles
The form supports a general second degree equation. You can enter coefficients for x squared, y squared, xy, x, y, and the constant term. The tool builds a symbolic polar equation from those values. It also evaluates the equation at a chosen angle. When the expression becomes a quadratic in r, the calculator finds possible radius values. This is helpful when a curve crosses the same ray more than once.
Reading the Output
The output shows the substitution pattern first. Next, it shows the grouped polar equation. The values P, Q, and F describe the quadratic expression Pr squared plus Qr plus F equals zero. If P is near zero, the equation becomes linear. If both P and Q vanish, the chosen ray may have every point or no point, depending on F.
Practical Graphing Tips
Use several angle values to understand the curve. Check zero, thirty, forty five, sixty, and ninety degrees. Negative radius values are not always errors. In polar graphing, a negative radius points in the opposite direction. Compare the converted point with the rectangular point for confidence. Use the example table for quick tests. Export the result when you need a record for homework, reports, or notes. For best results, keep coefficients exact when possible. Fractions reduce rounding noise. Decimal entries still work well. Always review the displayed equation before exporting. Small coefficient changes can create very different polar graphs quickly.