Homothety dilation calculator
Find where a point lands under a homothety: the point (4, 2) scaled by 2 about the origin goes to (8, 4). Any centre and any ratio work, including negative ones that flip the point through the centre. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X
- 4
- Y
- 2
- Cx
- 0
- Cy
- 0
- K
- 2
- Image x
- 8
- Image y
- 4
- Dx
- 4
- Dy
- 2
Homothety, also called dilation or central similarity, maps every point P in the plane to an image P prime that lies on the ray from a fixed center C through P, scaled by a real ratio k. The defining vector identity is P prime equals C plus k times the vector from C to P, so distances from the center multiply by the absolute value of k while angles at the center are preserved. Students meet this map when studying similar figures, intercept theorems, and spiral similarities; designers use positive k to enlarge or shrink a figure about an anchor; negative k produces an opposite homothety that reflects through the center as it scales.
What a plane homothety does and how the image is computed
A plane homothety with center C equals the point with coordinates cx and cy and ratio k is the unique map that sends every point P with coordinates x and y to the image P prime defined by the vector formula P prime equals C plus k times open parenthesis P minus C close parenthesis. When k equals one the map is the identity and the image coincides with the original point. When k equals zero every point collapses onto the center, so the image is always C.
When k is strictly between zero and one the figure shrinks toward C while staying on the same rays. When k is greater than one the figure enlarges away from C. When k is negative the image lies on the opposite ray through C, so the map is an opposite homothety that simultaneously scales by the absolute value of k and reflects through the center; the classical case k equals negative one is a central symmetry, also called a point reflection or half-turn about C.
Homothety preserves angles at the center and multiplies all distances measured from C by the absolute value of k, which is why similar figures constructed by dilation keep the same shape. Parallel lines remain parallel, circles map to circles, and segments map to segments whose lengths scale by absolute k.
Where a homothety calculator helps
Classrooms and tutoring platforms use a homothety dilation calculator so learners can check image coordinates for arbitrary centers and ratios without sign errors when expanding k times open parenthesis x minus cx close parenthesis by hand. Exam generators need a deterministic oracle that marks wrong a non-numeric ratio instead of silently inventing a scale factor. Failed validation does not consume balance.
Keep requests small, pin the rounded image coordinates in tests, and you get reproducible dilations for homework, LMS plugins, and edge workers without standing up a full computer-algebra or graphics stack of your own.
What you can do with it
Classroom dilation checks
Let students enter a point, center, and ratio k and verify the image coordinates against the textbook identity P′ = C + k(P − C).
Exam and LMS auto-grading
Mark homothety items with a deterministic oracle that rejects non-numeric ratios and returns stable twelve-decimal image coordinates.
CAD, games, and SVG anchors
Scale a design point about an arbitrary anchor in scripts and edge workers without re-coding the dilation formula in every client.
FAQ
How do I compute the image of a point under a homothety?
Send finite numbers x, y for the point, cx, cy for the center, and k for the ratio.
Does the center have to be the origin?
No. Any finite center (cx, cy) works. The formula always dilates relative to that center, not relative to the coordinate origin unless you set cx and cy to zero.
What does a negative ratio do in a homothety?
The image lands on the opposite side of the centre, scaled by the size of the ratio. A ratio of −1 is a point reflection, so the point (2, 0) about the origin goes to (−2, 0).
What happens with a ratio of 0 or 1?
A ratio of 1 is the identity and leaves every point exactly where it is. A ratio of 0 collapses every point onto the centre, so the image is always the centre itself.
Which properties does a homothety keep?
Angles stay the same, parallel lines stay parallel and circles map to circles. Every distance measured from the centre is multiplied by the absolute value of the ratio, which is why the figures stay similar.
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/homothety \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x":4,"y":2,"cx":0,"cy":0,"k":2}'const res = await fetch("https://api.kit.forhosting.com/geom/homothety", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x": 4,
"y": 2,
"cx": 0,
"cy": 0,
"k": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/homothety",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x": 4,
"y": 2,
"cx": 0,
"cy": 0,
"k": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/homothety", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x":4,"y":2,"cx":0,"cy":0,"k":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x":4,"y":2,"cx":0,"cy":0,"k":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/homothety", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x": 4,
"y": 2,
"cx": 0,
"cy": 0,
"k": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.homothety",
"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
This homothety dilation calculator accepts the point coordinates, the center, and the ratio, returns the exact image coordinates with the intermediate displacement vector, and fails cleanly with invalid_input when the ratio or any coordinate is missing or non-numeric.
What a plane homothety does and how the image is computed
Expanding components gives image_x equals cx plus k times open parenthesis x minus cx close parenthesis and image_y equals cy plus k times open parenthesis y minus cy close parenthesis.
Input contract, response fields, and error behaviour
Example: point four comma two, center zero comma zero, ratio two yields image eight comma four because the displacement vector is four comma two and doubling it from the origin produces eight comma four. Example with a non-origin center: point five comma one, center one comma one, ratio three yields image thirteen comma one, because the displacement is four comma zero, times three is twelve comma zero, added to the center is thirteen comma one. Negative ratio example: point two comma zero, center zero comma zero, ratio negative one yields image negative two comma zero, the central symmetry of the point. POST a JSON body with five required finite numbers: x and y for the point to dilate, cx and cy for the homothety center, and k for the ratio. Each field may arrive as a JSON number or as a numeric string; empty strings, missing keys, non-numeric text, and non-finite values such as NaN or Infinity fail with invalid_input naming the field. Absolute values larger than one trillion are rejected to keep intermediate products finite on the Worker isolate. On success the body echoes x, y, cx, cy, and k; reports image_x and image_y as the dilated coordinates; includes dx and dy as the displacement from center to the original point before scaling; and sets method to homothety_dilation with a short formula string restating the vector identity. Invalid ratio example: sending k as the string not-a-number fails with invalid_input and is not billed. Pair the response with your own drawing code if you need a sketch of rays and scaled segments; this endpoint returns coordinates only and does not emit plots or multi-step prose beyond the formula field.
Where a homothety dilation API earns its keep
CAD plugins, game editors, and SVG tooling often place a figure relative to an anchor and then scale about that anchor; calling a tiny parse API avoids re-implementing the same five-number formula in every host language.
Technical questions
What does the ratio k mean?
k multiplies vectors from the center: |k|>1 enlarges, 0<|k|<1 shrinks, k<0 is an opposite homothety (through the center), k=1 is identity, and k=0 maps every point to the center.
What happens if the ratio is non-numeric?
The task fails with invalid_input. The ratio k must be a finite number (JSON number or numeric string); text such as not-a-number is rejected and not billed.
What does the homothety dilation calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.