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
- 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)^2 + (y - b)^2 = r^2.}
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
- 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^2 + y^2 = r^2.}
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
