Library / Geometry, coordinates, surveying and trigonometry / GEO-40
Geometry, coordinates, surveying and trigonometry
Parametric point on a circle
The x-coordinate of the point at angle t, measured anticlockwise from the positive x-axis, on a circle of radius r centred at (h, k).
- Ref
- GEO-40
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- x [cm]
A point on a circle with centre (h, k) and radius r at an angle t measured anticlockwise from the positive x-axis is (h + r cos t, k + r sin t). As t runs from 0 to 2π the point traces the circle once. The x-coordinate is shown here, and the y-coordinate, k + r sin t, is in the derived rows.
The x-coordinate of the point at angle t, measured anticlockwise from the positive x-axis, on a circle of radius r centred at (h, k). The y-coordinate, k + r sin t, is in the derived rows.
Calculator
- y-coordinate, k + r sin t1.464 cm
- Angle in degrees120 °
- Distance from the origin1.773 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.
- At 120° the point is at (−1, 1.464), always 4 cm from the centre (1, −2).
- At 120° the point is above and to the left of the centre by 0.5 r and 0.866 r, which is how a bolt circle of equally spaced holes is laid out.
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 | −12 | 8 | 20 | +0.4 cm | −40 % |
| r Radius | 0.5 to 20 cm | −9 | 0.75 | 9.75 | +10% | +20 % |
| t Angle from the positive x-axis | −6.283 to 6.283 rad | −2.913 | 5 | 7.913 | +0.25132 rad | +79.9 % |
| k Centre y | −10 to 10 cm | −1 | −1 | 0 | +0.4 cm | 0 % |
Sweep of Angle from the positive x-axis
| Angle from the positive x-axis (rad) | x-coordinate of the point (cm) |
|---|---|
| −6.283 | 5 |
| −4.712 | 0.9994 |
| −3.142 | −3 |
| −1.571 | 1 |
| 0 | 5 |
| 1.571 | 1 |
| 3.141 | −3 |
| 4.712 | 0.9994 |
| 6.283 | 5 |
Sweep one variable
Variables and how each one contributes
x-coordinate of the point
What the equation gives you. Everything else feeds it.
Centre x
The x-coordinate of the centre, in cm. Moves the point by the same amount in x. Raising it by 0.5 (to 1.5) moves the result by −50 %, from −1 to −0.5 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.
Radius
The radius of the circle, in cm. Moves the point further from the centre along its angle, in proportion to cos t. Raising it by 10 % (to 4.4) moves the result by +20 %, from −1 to −1.2 cm.
Angle from the positive x-axis
The angle in radians, measured anticlockwise from the positive x-axis. Moves the point round the circle, so x changes at a rate −r sin t. Raising it by 0.05 (to 2.144) moves the result by +17.1 %, from −1 to −1.171 cm.
Worked example with the values above
- cos t = cos(2.094) = −0.5 and sin t = 0.86603.
- Offsets from the centre: r cos t = −2 cm and r sin t = 3.4641 cm.
- x = 1 + −2 = −1 cm, y = 1.4641 cm.
Assumptions and limits
- Plane Cartesian coordinates, and a circle of positive radius unless stated.
- Angles in radians measured anticlockwise from the positive x-axis.
Common mistakes
- Entering the angle in degrees where radians are needed.
- Measuring the angle from the y-axis, or clockwise, instead of anticlockwise from the x-axis.
- Forgetting to add the centre coordinates to the offsets.
Statistics and uncertainty
These are exact results from the circle and line equations, but they need a root of a difference. Where the radicand is close to zero, as near tangency, small input errors can change the answer a lot or change whether a solution exists at all.
Where it is used
- Circle fitting, wheel and cam geometry, hole patterns and clearance checks.
- Finding where a path, ray or cut meets a circular boundary.
Maths you need
- Equation of a circle: (x − h)² + (y − k)² = r²
- Quadratic discriminant: b² − 4ac decides the number of roots
- Gradient of a normal: perpendicular to the tangent