Library / Geometry, coordinates, surveying and trigonometry / GEO-24
Geometry, coordinates, surveying and trigonometry
Rotation about a chosen centre
The x-coordinate of the image of a point after an anticlockwise rotation by θ about the centre (h, j).
- Ref
- GEO-24
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- x′ [cm]
Rotating the point by an anticlockwise angle θ about the centre C = (h, j) turns its offset (u, v) = (x − h, y − j) into (u cos θ − v sin θ, u sin θ + v cos θ). The x-coordinate is h + u cos θ − v sin θ. The y-coordinate, j + u sin θ + v cos θ, is in the derived rows. For a clockwise turn enter a negative angle.
The x-coordinate of the image of a point after an anticlockwise rotation by θ about the centre (h, j). The offsets u = x − h and v = y − j are measured from the centre. The y-coordinate, j + u sin θ + v cos θ, is in the derived rows.
Calculator
- Image y-coordinate, j + u sin θ + v cos θ5 cm
- Distance from the centre (unchanged by rotation)2.236 cm
- Rotation angle in degrees90 °
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.
- Turning the point 90° anticlockwise about (2, 3) puts it at (3, 5), still 2.236 cm from the centre.
- A quarter turn (π/2) maps the offset (u, v) to (−v, u), a quick check on any answer.
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 | −9 | 11 | 20 | +0.4 cm | +13.3 % |
| v Offset y − j | −10 to 10 cm | −8 | 12 | 20 | +0.4 cm | −13.3 % |
| θ Anticlockwise rotation | −6.283 to 6.283 rad | −0.2059 | 4.234 | 4.44 | +0.25132 rad | −17.6 % |
| u Offset x − h | −10 to 10 cm | 3 | 3 | 5.359e-7 | +0.4 cm | +3.57e-7 % |
| j Centre y | −10 to 10 cm | 3 | 3 | 0 | +0.4 cm | 0 % |
Sweep of Anticlockwise rotation
| Anticlockwise rotation (rad) | Image x-coordinate (cm) |
|---|---|
| −6.283 | 4 |
| −4.712 | 3 |
| −3.142 | −9.265e-5 |
| −1.571 | 1 |
| 0 | 4 |
| 1.571 | 3 |
| 3.141 | 9.266e-5 |
| 4.712 | 0.9997 |
| 6.283 | 4 |
Sweep one variable
Variables and how each one contributes
Image x-coordinate
What the equation gives you. Everything else feeds it.
Centre x
The x-coordinate of the centre of rotation, in cm. Moves the image by the same amount in x when the angle is fixed. Raising it by 0.5 (to 2.5) moves the result by +16.7 %, from 3 to 3.5 cm.
Centre y
The y-coordinate of the centre, in cm. It does not change x′; it sets y′. Does not change x′. It moves y′ by the same amount.
Offset x − h
The horizontal offset of the point from the centre, x − h, in cm. Changes x′ by cos θ times the change. Raising it by 0.5 (to 2.5) moves the result by +4.47×10−7 %, from 3 to 3 cm.
Offset y − j
The vertical offset of the point from the centre, y − j, in cm. Changes x′ by −sin θ times the change. Raising it by 0.5 (to −0.5) moves the result by −16.7 %, from 3 to 2.5 cm.
Anticlockwise rotation
The anticlockwise rotation angle, in radians. Use a negative value for a clockwise turn. Turns the offset further round the centre, which changes x′ by an amount that depends on the starting position. Raising it by 0.05 (to 1.621) moves the result by −3.37 %, from 3 to 2.899 cm.
Worked example with the values above
- cos θ = 2.6795×10−8 and sin θ = 1 for θ = 1.571 rad.
- New x offset: u cos θ − v sin θ = 5.359×10−8 − (−1) = 1.
- x′ = 2 + 1 = 3 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