Library / Geometry, coordinates, surveying and trigonometry / GEO-25
Geometry, coordinates, surveying and trigonometry
Reflection in a vertical line
The x-coordinate of the mirror image of the point (x, y) in the vertical line x = h.
- Ref
- GEO-25
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- x′ [cm]
Reflecting a point in the vertical line x = h keeps its y-coordinate and moves its x-coordinate to the other side by the same distance: x′ = 2h − x. The line is the perpendicular bisector of the segment joining the point and its image. A point on the line stays where it is. The y-coordinate is unchanged, so it is in the derived rows as y′ = y.
The x-coordinate of the mirror image of the point (x, y) in the vertical line x = h. The image is as far from the line as the point, on the other side, and keeps the same y.
Calculator
- Image y-coordinate (unchanged)−2 cm
- Distance from the point to the mirror line5 cm
- Distance between the point and its image10 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.
- Reflecting (8, −2) in the line x = 3 gives (−2, −2), 10 cm away from the original.
- A point already on the mirror line (x = h) maps to itself.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Mirror line x = h | −10 to 10 cm | −28 | 12 | 40 | +0.4 cm | −40 % |
| x Point x | −10 to 20 cm | −14 | 16 | 30 | +0.6 cm | +30 % |
| y Point y | −10 to 10 cm | −2 | −2 | 0 | +0.4 cm | 0 % |
Sweep of Point x
| Point x (cm) | Image x-coordinate (cm) |
|---|---|
| −10 | 16 |
| −6.25 | 12.25 |
| −2.5 | 8.5 |
| 1.25 | 4.75 |
| 5 | 1 |
| 8.75 | −2.75 |
| 12.5 | −6.5 |
| 16.25 | −10.25 |
| 20 | −14 |
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 the same amount in the opposite direction. Raising it by 0.5 (to 8.5) moves the result by +25 %, from −2 to −2.5 cm.
Point y
The y-coordinate of the original point, in cm. It does not change x′; the image keeps the same y. Does not change x′. The image y-coordinate equals this value.
Mirror line x = h
The x-position of the vertical mirror line x = h, in cm. Moves the image by twice the change. Raising it by 0.5 (to 3.5) moves the result by −50 %, from −2 to −1 cm.
Worked example with the values above
- The point is 5 cm from the line x = 3.
- Double the line position: 2h = 6.
- x′ = 6 − 8 = −2 cm, with y′ = −2 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