Transform rectangular curves to polar form with guided algebra. Inspect roots, angles, and graph points. Export clean reports for lessons, homework, and technical review.
Calculator Input
Enter coefficients for Ax² + Bxy + Cy² + Dx + Ey + F = 0.
Missing terms should be entered as zero.
Formula Used
Start with the rectangular equation:
Ax² + Bxy + Cy² + Dx + Ey + F = 0
Use the polar substitutions:
x = r cosθ, y = r sinθ, x² + y² = r²
After substitution, the grouped polar equation becomes:
Move every term of the rectangular equation to the left side.
Match the equation with Ax² + Bxy + Cy² + Dx + Ey + F = 0.
Enter zero for every missing term.
Select the angle range and step size.
Choose whether to show all roots or one branch.
Press the convert button.
Review the polar equation, graph, and sample table.
Use CSV or PDF export for records and assignments.
Example Data Table
These examples show how common rectangular equations fit the coefficient form.
Curve
Rectangular Equation
A
B
C
D
E
F
Polar Structure
Circle
x² + y² - 4x + 2y - 11 = 0
1
0
1
-4
2
-11
r² + r(-4cosθ + 2sinθ) - 11 = 0
Line
2x + 3y - 6 = 0
0
0
0
2
3
-6
r(2cosθ + 3sinθ) - 6 = 0
Ellipse
4x² + 9y² - 36 = 0
4
0
9
0
0
-36
r²(4cos²θ + 9sin²θ) - 36 = 0
Parabola
y² - 4x = 0
0
0
1
-4
0
0
r²sin²θ - 4r cosθ = 0
Rectangular to Polar Equation Guide
Why Polar Conversion Matters
Rectangular and polar systems describe the same plane in different ways. A rectangular equation uses horizontal and vertical distances.
A polar equation uses distance from the origin and the angle from the positive axis. Converting between them helps you see symmetry,
rotation, and radial behavior more clearly.
How the Calculator Works
This calculator focuses on second degree rectangular equations. It supports lines, circles, parabolas, ellipses, hyperbolas, and many
mixed conics. Enter the coefficients for Ax² + Bxy + Cy² + Dx + Ey + F = 0. The tool replaces x with r cos θ and y with r sin θ.
It then groups the equation by powers of r.
Understanding Roots
The grouped polar form is usually a quadratic in r. That means each angle can have two possible radii. Some angles can have one radius,
while others have none. The calculator checks the discriminant for each angle. It lists real roots only. This makes the table useful
for graphing and checking domain behavior.
Reading the Graph
The graph gives a fast visual check. A circle may look simple in rectangular form, but its polar version can show shifting from the
origin. A line often becomes a rational radial equation. A rotated conic may include the mixed sine cosine term from Bxy. These details
are easy to miss by hand.
Better Sampling Tips
Use small angle steps for smoother plots. Use larger steps for quick checking. Results may change when the curve crosses the pole.
Negative r values are valid in polar coordinates. They point in the opposite direction from the entered angle. Keep them when studying
full curves.
Export and Verification
The export tools help with reports. CSV is useful for spreadsheets. PDF is useful for notes and homework. Always compare the displayed
polar equation with the original rectangular equation. Then inspect several angle rows. This gives you both algebraic and numeric
confidence.
Accuracy Advice
For best accuracy, write every equation in standard form first. Move all terms to the left side. Combine like terms. Enter missing
terms as zero. Avoid rounded coefficients when exact values are known. After conversion, test one or two rectangular points. Their
polar coordinates should satisfy the new equation. This simple check prevents sign mistakes and misplaced constants. It also improves
trust in every plotted result.
FAQs
1. What does this calculator convert?
It converts a rectangular equation into polar form by replacing x with r cosθ and y with r sinθ.
2. What equation format should I use?
Use Ax² + Bxy + Cy² + Dx + Ey + F = 0. Enter zero for any missing coefficient.
3. Can it handle lines?
Yes. Set A, B, and C to zero. The calculator then solves the linear polar equation for r.
4. Why are there two r values?
Many converted equations become quadratic in r. A quadratic can produce two real radius values for one angle.
5. Are negative r values wrong?
No. A negative r is valid in polar coordinates. It points opposite the direction of the entered angle.
6. What does no real root mean?
It means the discriminant is negative for that angle. The curve has no real radius there.
7. How do I get a smoother graph?
Use a smaller angle step, such as one or two degrees. Smaller steps create more sampled points.
8. What should I export?
Use CSV for spreadsheet analysis. Use PDF when you need a clean report for notes or homework.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.