Difference between revisions of "Equation of a Circle"
(Created page with "* [https://www.khanacademy.org/math/geometry/xff63fac4:hs-geo-conic-sections/hs-geo-circle-expanded-equation/a/circle-equation-review Circle Equation Review], Khan Academy * [...") |
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| + | ==Cartesian coordinates== | ||
| + | [[Image:Circle center a b radius r.svg|thumb|right|Circle of radius ''r'' = 1, centre (''a'', ''b'') = (1.2, −0.5)]] | ||
| + | |||
| + | ;Equation of a circle | ||
| + | In an ''x''–''y'' [[Cartesian coordinate system]], the circle with centre [[Coordinate system|coordinates]] (''a'', ''b'') and radius ''r'' is the set of all points (''x'', ''y'') such that | ||
| + | : <math>(x - a)^2 + (y - b)^2 = r^2.</math> | ||
| + | |||
| + | This [[equation]], known as the ''equation of the circle'', follows from the [[Pythagorean theorem]] applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |''x'' − ''a''| and |''y'' − ''b''|. If the circle is centred at the origin (0, 0), then the equation simplifies to | ||
| + | : <math>x^2 + y^2 = r^2.</math> | ||
| + | |||
| + | ==Parametric form== | ||
| + | The equation can be written in [[parametric equation|parametric form]] using the [[trigonometric function]]s sine and cosine as | ||
| + | : <math>x = a + r\,\cos t,</math> | ||
| + | : <math>y = b + r\,\sin t,</math> | ||
| + | where ''t'' is a [[parametric variable]] in the range 0 to 2{{pi}}, interpreted geometrically as the [[angle]] that the ray from (''a'', ''b'') to (''x'', ''y'') makes with the positive ''x'' axis. | ||
| + | |||
| + | An alternative parametrisation of the circle is | ||
| + | : <math>x = a + r \frac{1 - t^2}{1 + t^2},</math> | ||
| + | : <math>y = b + r \frac{2t}{1 + t^2}.</math> | ||
| + | |||
| + | In this parameterisation, the ratio of ''t'' to ''r'' can be interpreted geometrically as the [[stereographic projection]] of the line passing through the centre parallel to the ''x'' axis (see [[Tangent half-angle substitution]]). However, this parameterisation works only if ''t'' is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted. | ||
| + | |||
| + | ==Resourcs== | ||
* [https://www.khanacademy.org/math/geometry/xff63fac4:hs-geo-conic-sections/hs-geo-circle-expanded-equation/a/circle-equation-review Circle Equation Review], Khan Academy | * [https://www.khanacademy.org/math/geometry/xff63fac4:hs-geo-conic-sections/hs-geo-circle-expanded-equation/a/circle-equation-review Circle Equation Review], Khan Academy | ||
* [https://www.khanacademy.org/math/geometry/xff63fac4:hs-geo-conic-sections/hs-geo-circle-standard-equation/v/radius-and-center-for-a-circle-equation-in-standard-form Standard Form of Circle Equation], Khan Academy | * [https://www.khanacademy.org/math/geometry/xff63fac4:hs-geo-conic-sections/hs-geo-circle-standard-equation/v/radius-and-center-for-a-circle-equation-in-standard-form Standard Form of Circle Equation], Khan Academy | ||
Revision as of 14:29, 18 October 2021
Cartesian coordinates
- Equation of a circle
In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that
This equation, known as the equation of the circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|. If the circle is centred at the origin (0, 0), then the equation simplifies to
Parametric form
The equation can be written in parametric form using the trigonometric functions sine and cosine as
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a + r\,\cos t,}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = b + r\,\sin t,}
where t is a parametric variable in the range 0 to 2Template:Pi, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x axis.
An alternative parametrisation of the circle is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a + r \frac{1 - t^2}{1 + t^2},}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = b + r \frac{2t}{1 + t^2}.}
In this parameterisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x axis (see Tangent half-angle substitution). However, this parameterisation works only if t is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted.
Resourcs
- Circle Equation Review, Khan Academy
- Standard Form of Circle Equation, Khan Academy
