Ellipse general to standard form calculator
Enter the six coefficients of a general second-degree equation and get the ellipse in standard form: x² + 4y² − 4 = 0 becomes an ellipse centred at the origin with semi-axes 2 and 1. Rotated ellipses get their angle too. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 1
- B
- 0
- C
- 4
- D
- 0
- E
- 0
- F
- -4
- H
- 0
- K
- 0
- Semi major
- 2
- Semi minor
- 1
Textbooks, contest problems, and CAS dumps often hand you a messy general conic Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equals zero and ask for the ellipse in standard form. Completing the square by hand is fine when B is zero and the numbers are friendly; it is easy to slip when the center is off-origin, when axes are swapped, or when a non-zero xy term means the figure is rotated. This ellipse general to standard form calculator takes the six coefficients, checks that the discriminant B squared minus four AC is negative, solves for the unique center, diagonalizes the quadratic form, and returns the semi-major and semi-minor axes, the rotation angle, human-readable standard and general equations, and a short formula block.
From general conic coefficients to ellipse standard form
The general second-degree equation in two variables is A x squared plus B x y plus C y squared plus D x plus E y plus F equals zero. Its type is governed by the discriminant delta equals B squared minus four A C: negative for ellipse type, zero for parabola type, and positive for hyperbola type.
Ellipse type alone is not enough—the locus can still be a single point, empty, or degenerate—so this calculator also recovers the center by solving the linear system from the partial derivatives, evaluates the translated constant term g at that center, and reads the semi-axis squares from the eigenvalues of the quadratic-form matrix. After translation the equation becomes A X squared plus B X Y plus C Y squared plus g equals zero.
The eigenvalues lambda one and lambda two of the matrix with A and C on the diagonal and B over two off-diagonal satisfy lambda i times the squared coordinate along each principal axis equals negative g, so each semi-axis squared is negative g over lambda i. When B is zero the axes align with the coordinate frame and the familiar complete-the-square form appears: X squared over a squared plus Y squared over b squared equals one, with center h and k.
When B is not zero a rotation by angle theta aligns the figure with its principal axes, and the response reports that angle in degrees and radians together with the standard equation written in the rotated coordinates. A circle is the special case where the two semi-axes match; rotation is then reported as zero because every angle is equivalent. Knowing both the general coefficients and the standard geometry matters because grading rubrics, CAD notes, and shader uniforms switch freely between expanded polynomials and center-axis descriptions.
Practical tips, limits, and related searches
Then shift the same figure so the center sits at one negative two and verify that completing the square by hand matches h, k, and both semi-axes. When a problem includes an xy term, trust the reported rotation and principal semi-axes rather than trying to guess orientation from a sketch alone; the eigenvector convention normalizes the major-axis angle into a stable half-turn range so repeated calls do not flip sign without reason. Keep units consistent—metres, pixels, or abstract game units—because the calculator never converts units for you.
Always treat non-ellipses as hard failures in auto-graders: a hyperbola with positive discriminant must not be accepted as a flattened ellipse, and a point locus must not be reported as zero-radius axes. This landing targets people searching for an ellipse general form to standard form calculator, conic coefficient converter, or ellipse center and semi-axes from A B C D E F; it is not a three-point ellipse fit, not a full multi-conic classifier with plots, and not a substitute for symbolic CAS step lists.
Round-trip checks are easy after a success: rebuild the general polynomial from the recovered geometry when B is zero and confirm coefficients match within floating tolerance, and confirm that area pi times semi-major times semi-minor is positive.
What you can do with it
Homework checks for general-to-standard ellipse form
Students paste textbook coefficients and compare hand-completed squares and rotation angles against a deterministic center, semi-axes, and standard equation string.
CAD and graphics cleanup of expanded conics
Recover center, semi-major, semi-minor, and rotation from expanded shader or export polynomials so colliders and drawing tools can use standard ellipse parameters.
FAQ
When is the input rejected as not an ellipse?
When B² − 4AC is not negative (parabola or hyperbola type), when the center cannot be solved, or when both semi-axis squares are not strictly positive (point or imaginary locus).
Does a non-zero B (xy term) work?
Yes. The solver diagonalizes the quadratic form, reports the principal-axis rotation, and writes the standard form in the rotated coordinates.
Is the calculator free?
The browser tool is free and runs locally on the same pure module.
What does the discriminant tell me?
B squared minus four times A times C sorts the curve: negative means ellipse type, zero a parabola and positive a hyperbola. For x² + 4y² − 4 = 0 it is −16, so the equation can describe an ellipse.
What do I get besides the standard equation?
The centre, the semi-major and semi-minor axes with their squares, the rotation angle in degrees and radians, and the general equation written out again, so you can copy whichever form your worksheet or drawing needs.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/geom/ellipse-standard-form \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":1,"b":0,"c":4,"d":0,"e":0,"f":-4}'const res = await fetch("https://api.kit.forhosting.com/geom/ellipse-standard-form", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 1,
"b": 0,
"c": 4,
"d": 0,
"e": 0,
"f": -4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/ellipse-standard-form",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 1,
"b": 0,
"c": 4,
"d": 0,
"e": 0,
"f": -4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/ellipse-standard-form", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":1,"b":0,"c":4,"d":0,"e":0,"f":-4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":1,"b":0,"c":4,"d":0,"e":0,"f":-4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/ellipse-standard-form", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 1,
"b": 0,
"c": 4,
"d": 0,
"e": 0,
"f": -4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.ellipse_standard_form",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Hyperbolas, parabolas, point ellipses, and imaginary loci fail with invalid_input instead of inventing axes.
From general conic coefficients to ellipse standard form
Both of those quantities must be strictly positive for a real non-degenerate ellipse; otherwise the request is rejected.
How to call the API and what comes back
Numbers are rounded to twelve decimal places for stable goldens across runtimes. Send a JSON body with six required finite numbers a, b, c, d, e, and f for the coefficients A, B, C, D, E, and F in A x squared plus B x y plus C y squared plus D x plus E y plus F equals zero. Each field may arrive as a JSON number or as a numeric string; aliases A through F and coeff_a through coeff_f map to the same six slots so clients that already use uppercase math notation do not need a rename layer. Absolute values must stay within the published limit of one trillion so intermediate products remain finite in IEEE double precision on the Worker and in the browser. On success the response echoes the six coefficients, returns h and k plus a nested center object, semi_major and semi_minor with their squares, rotation_deg and rotation_rad, is_circle, the conic discriminant, human-readable standard_equation and general_equation strings, and a formula object documenting the closed forms. Failed validation—missing fields, non-finite values, out-of-range magnitudes, a non-negative discriminant, a singular quadratic matrix, or coefficients that do not yield two positive semi-axis squares—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Prefer this endpoint when your input is already expanded coefficients; if you already hold center and semi-axes and only need foci or perimeter, use the dedicated ellipse foci or Ramanujan perimeter tools instead of forcing a detour through the general form.
Practical tips, limits, and related searches
Start with an axis-aligned textbook ellipse such as x squared over four plus y squared equals one, which expands to x squared plus four y squared minus four equals zero, and confirm the API returns center at the origin, semi-major two, semi-minor one, and rotation zero. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners type impossible coefficients at $0.002 only on success.
Integration use cases
Exam generators and LMS auto-graders
Oracle that accepts only real non-degenerate ellipses and rejects hyperbolas, parabolas, and point loci with invalid_input for reproducible marking scripts.
Technical questions
What inputs does the ellipse general-to-standard form calculator need?
Six finite coefficients a, b, c, d, e, f for Ax² + Bxy + Cy² + Dx + Ey + F = 0. Aliases A–F and coeff_a–coeff_f are accepted.
What is returned on success?
Center (h, k), semi_major, semi_minor, rotation in degrees and radians, is_circle, discriminant, standard_equation, general_equation, echoed coefficients, and a short formula block.