Coplanarity checker
Check whether four points in space lie on one flat plane: (0, 0, 0), (1, 0, 0), (0, 1, 0) and (1, 1, 0) do, while lifting the last one to (1, 1, 1) makes a tetrahedron of volume 1/6. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X1
- 0
- Y1
- 0
- Z1
- 0
- X2
- 1
- Y2
- 0
- Z2
- 0
- Scalar triple
- 0
- Tetrahedron volume
- 0
Four points in three-dimensional space are coplanar when a single flat plane contains every one of them. CAD validators, game-engine face builders, surveying pipelines, and linear-algebra courses constantly need a reliable yes-or-no answer for that property without dragging in a full computer-algebra system. This coplanarity checker accepts the twelve Cartesian coordinates of points A, B, C, and D, builds the three edge vectors from A, evaluates the scalar triple product AB · (AC × AD), and reports whether that product is numerically zero. When the product vanishes, the tetrahedron they span has zero volume and the points share one plane; when it does not, they form a solid tetrahedron of positive volume.
What coplanarity means and how the scalar triple product decides it
Two distinct points always define a line, and three non-collinear points always define a unique plane in Euclidean three-space. A fourth point either sits on that plane or does not. Algebraically, take vectors AB, AC, and AD from the first point to the other three.
The cross product AC × AD is a normal to the plane of A, C, and D (or zero if those three are collinear). The dot product of AB with that normal is the scalar triple product AB · (AC × AD). Geometrically it equals six times the signed volume of the tetrahedron with vertices A, B, C, and D.
If the absolute value of the triple product is zero, the volume is zero and the four points are coplanar; if it is nonzero, they form a tetrahedron with positive volume and cannot lie in one plane. Collinear triples and repeated points still yield a zero triple product, which correctly classifies them as coplanar in the weak sense that some plane (in fact infinitely many) contains them. Units cancel in the comparison: metres, millimetres, or pure abstract numbers all work as long as every coordinate uses the same scale.
Where a four-point flatness check helps
Mesh and CAD pipelines use a coplanarity checker before merging four vertices into a single polygonal face, catching non-planar quads that would distort normals or break flat-face constraints. Survey and photogrammetry scripts verify that control points intended to share a floor or wall plane actually do before computing areas or elevations. Linear-algebra and solid-geometry courses need a deterministic oracle that marks coplanar versus tetrahedral configurations without students hand-computing nine multiplications under exam pressure.
Game engines and level editors often store four corner handles for a panel or portal and must reject lifts that break planarity when the designer nudges one vertex. Failed validation does not consume balance. Keep requests small, pin the rounded triple product and volume in tests, and you get reproducible coplanarity decisions for grading, CI geometry gates, and edge workers without standing up a full computational-geometry stack of your own.
What you can do with it
CAD and mesh planarity gates
Reject non-planar quads before face merge or export by testing the four corner vertices of each candidate polygon face.
Classroom and exam auto-grading
Mark solid-geometry items that ask whether four given points are coplanar with a deterministic scalar-triple oracle.
Survey and control-point checks
Verify that four stakes or photogrammetry markers intended for one floor or wall plane actually share that plane.
FAQ
What formula does the coplanarity checker use?
It uses the scalar triple product of edge vectors from the first point: AB · (AC × AD). Points are coplanar iff that product is zero; tetrahedron volume is its absolute value divided by six.
What if a coordinate is not a number?
Non-numeric strings, Infinity, NaN, empty values, and missing coordinates are rejected and are not billed.
Are collinear or repeated points considered coplanar?
Yes. If three or four points collapse so the triple product is zero, they still lie in some plane (often infinitely many), so coplanar is true with zero tetrahedron volume.
What does the tetrahedron volume tell me?
Four points always span a tetrahedron, and its volume is the triple product divided by six. A volume of zero means the shape is flat and the points share a plane; any positive volume means they do not.
How small a gap still counts as flat?
A tiny tolerance of one billionth absorbs the rounding noise of ordinary coordinates, so the answer does not flip by accident. A real lift of a point off the plane is still reported as not coplanar.
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/coplanar-points \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x1":0,"y1":0,"z1":0,"x2":1,"y2":0,"z2":0,"x3":0,"y3":1,"z3":0,"x4":1,"y4":1,"z4":0}'const res = await fetch("https://api.kit.forhosting.com/geom/coplanar-points", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x1": 0,
"y1": 0,
"z1": 0,
"x2": 1,
"y2": 0,
"z2": 0,
"x3": 0,
"y3": 1,
"z3": 0,
"x4": 1,
"y4": 1,
"z4": 0
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/coplanar-points",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x1": 0,
"y1": 0,
"z1": 0,
"x2": 1,
"y2": 0,
"z2": 0,
"x3": 0,
"y3": 1,
"z3": 0,
"x4": 1,
"y4": 1,
"z4": 0
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/coplanar-points", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x1":0,"y1":0,"z1":0,"x2":1,"y2":0,"z2":0,"x3":0,"y3":1,"z3":0,"x4":1,"y4":1,"z4":0}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x1":0,"y1":0,"z1":0,"x2":1,"y2":0,"z2":0,"x3":0,"y3":1,"z3":0,"x4":1,"y4":1,"z4":0}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/coplanar-points", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x1": 0,
"y1": 0,
"z1": 0,
"x2": 1,
"y2": 0,
"z2": 0,
"x3": 0,
"y3": 1,
"z3": 0,
"x4": 1,
"y4": 1,
"z4": 0
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.coplanar_points",
"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
Non-numeric or missing coordinates fail with invalid_input so automation never mistakes garbage for geometry.
What coplanarity means and how the scalar triple product decides it
This endpoint uses a small absolute epsilon of one times ten to the minus nine so floating-point noise from ordinary coordinate scales does not flip the boolean, while a clear geometric lift of a point off the plane still reports not coplanar. The response echoes the four points, the rounded triple product, the absolute tetrahedron volume, a coplanar boolean, and a short formula string you can show next to the answer in teaching UIs.
Input contract, response shape, and error behaviour
Example: A at the origin, B at (1,0,0), C at (0,1,0), and D at (1,1,0) are coplanar with triple product zero and volume zero because all four have z equal to zero. Lift D to (1,1,1) and the product becomes one, volume becomes one sixth, and coplanar is false. POST a JSON body with twelve required finite numbers: x1, y1, z1 for point A; x2, y2, z2 for B; x3, y3, z3 for C; and x4, y4, z4 for D. Each field may arrive as a JSON number or as a numeric string such as "1.5" or "-2e-3"; Infinity, NaN, empty strings, missing fields, and non-numeric text fail with invalid_input naming the offending field, and those failures are not billed. Common aliases such as ax, ay, az through dx, dy, dz and p1x through p4z are declared on the form layer so CAD-style and spreadsheet-style payloads map cleanly. On success the body includes coplanar as a boolean; scalar_triple as the signed product AB · (AC × AD) rounded to twelve decimal places for stable goldens; tetrahedron_volume as the absolute value of that product divided by six; points as an object with arrays a, b, c, and d; and formula restating the decision rule. Pair the response with your own plane-equation or normal-vector display if you need a, b, c, d coefficients of ax+by+cz+d=0; this capability answers the coplanarity question only and does not emit plane fits or plot data.
Technical questions
How do I check if four 3D points are coplanar?
Send finite coordinates x1,y1,z1 through x4,y4,z4. The API computes AB·(AC×AD) and returns coplanar true when the absolute value is near zero, plus the triple product and tetrahedron volume.
What does the coplanarity checker cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.