Library / Geometry, coordinates, surveying and trigonometry / GEO-23
Geometry, coordinates, surveying and trigonometry
Enlargement about a chosen centre
The x-coordinate of the image of the point (x, y) after an enlargement of factor k about the centre (h, j).
- Ref
- GEO-23
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- x′ [cm]
An enlargement with scale factor k about the centre C = (h, j) moves each point along the line from the centre, to P′ = C + k(P − C). Negative k places the image on the opposite side of C. The x-coordinate is h + k(x − h); the y-coordinate j + k(y − j) is in the derived rows. A factor of 0 collapses the plane to the centre.
The x-coordinate of the image of the point (x, y) after an enlargement of factor k about the centre (h, j). A negative k puts the image on the far side of the centre. The y-coordinate, j + k(y − j), is in the derived rows.
Calculator
- Image y-coordinate, j + k(y − j)7 cm
- Distance from the centre to the original point2.828 cm
- Distance from the centre to the image5.657 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Enlarging (4, 1) by −2 about (2, 3) gives (−2, 7).
- A negative factor of −2 turns the figure over as well as scaling it, so the image lies on the other side of the centre.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Centre x | −10 to 10 cm | −38 | 22 | 60 | +0.4 cm | −60 % |
| x Point x | −10 to 10 cm | −14 | 26 | 40 | +0.4 cm | +40 % |
| k Scale factor (signed) | −3 to 3 | −4 | 8 | 12 | +0.12 | −12 % |
| y Point y | −10 to 10 cm | −2 | −2 | 0 | +0.4 cm | 0 % |
| j Centre y | −10 to 10 cm | −2 | −2 | 0 | +0.4 cm | 0 % |
Sweep of Scale factor (signed)
| Scale factor (signed) | Image x-coordinate (cm) |
|---|---|
| −3 | −4 |
| −2.25 | −2.5 |
| −1.5 | −1 |
| −0.75 | 0.5 |
| 0 | 2 |
| 0.75 | 3.5 |
| 1.5 | 5 |
| 2.25 | 6.5 |
| 3 | 8 |
Sweep one variable
Variables and how each one contributes
Image x-coordinate
What the equation gives you. Everything else feeds it.
Point x
The x-coordinate of the original point, in cm. Moves the image by k times the change. Raising it by 0.5 (to 4.5) moves the result by +50 %, from −2 to −3 cm.
Point y
The y-coordinate of the original point, in cm. It does not change x′; it sets y′, shown in the derived rows. Does not change x′. It moves y′ by k times any change.
Centre x
The x-coordinate of the centre of enlargement, in cm. Moves the image by (1 − k) times the change, so it pushes the image the other way when k is above 1. Raising it by 0.5 (to 2.5) moves the result by −75 %, from −2 to −0.5 cm.
Centre y
The y-coordinate of the centre of enlargement, in cm. It does not change x′; it sets y′. Does not change x′. It moves y′ by (1 − k) times any change.
Scale factor (signed)
The signed scale factor, a plain number. Negative values put the image on the opposite side of the centre. Moves the image along the line from the centre, in proportion to the distance of the point from the centre. Raising it by 0.1 (to −1.9) moves the result by −10 %, from −2 to −1.8 cm.
Worked example with the values above
- The point is 2 cm to the right of the centre and −2 cm above it.
- Scale the offset by k = −2: x offset = −4 cm.
- x′ = 2 + −4 = −2 cm, and y′ = 7 cm.
Assumptions and limits
- Similar shapes: every length changes by the same factor k.
- Coordinates in a right-handed system with y upwards, and angles measured anticlockwise.
Common mistakes
- Applying the length scale factor to an area or volume without squaring or cubing it.
- Forgetting that a negative scale factor puts the image on the opposite side of the centre.
- Using degrees for a rotation where radians are needed, or the wrong turn direction.
Statistics and uncertainty
Scale factors act on all lengths, so area errors double and volume errors triple the error in k. For coordinate transformations, the result is as accurate as the measured positions and angles.
Where it is used
- Scaling models, maps and drawings, and finding how material, mass and cost change with size.
- Moving, turning and mirroring points in drawings and CAD.
Maths you need
- Scale factor: ratio of corresponding lengths
- Coordinates: a point’s position as (x, y)
- Sine and cosine: the horizontal and vertical parts of a rotation