Equation of a Circle - Revision history

Khanh at 18:26, 14 November 2021

2021-11-14T18:26:01Z

Khanh at 18:25, 14 November 2021

2021-11-14T18:25:48Z

Lila: /* Resourcs */

2021-10-18T20:32:10Z

Resourcs

Lila: /* Parametric form */

2021-10-18T20:31:42Z

Parametric form

Lila: /* Cartesian coordinates */

2021-10-18T20:31:13Z

Cartesian coordinates

Lila at 20:30, 18 October 2021

2021-10-18T20:30:34Z

Lila at 20:29, 18 October 2021

2021-10-18T20:29:07Z

Khanh: 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 * [..."

2021-09-17T20:51:47Z

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 * [..."

New page

* [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

@@ Line 21: / Line 21: @@
 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''&nbsp;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==
+==Resources==
 * [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

← Older revision		Revision as of 20:32, 18 October 2021
Line 24:		Line 24:
	* [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
		+	* [https://en.wikipedia.org/wiki/Circle Circle], Wikipedia

@@ Line 13: / Line 13: @@
 : <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'',&nbsp;''b'') to (''x'',&nbsp;''y'') makes with the positive ''x''&nbsp;axis.
+where ''t'' is a parametric variable in the range 0 to <math>2\pi</math>, interpreted geometrically as the angle that the ray from (''a'',&nbsp;''b'') to (''x'',&nbsp;''y'') makes with the positive ''x''&nbsp;axis.
 An alternative parametrisation of the circle is

@@ Line 3: / Line 3: @@
 [[Image:Circle center a b radius r.svg|thumb|right|Circle of radius ''r''&nbsp;=&nbsp;1, centre (''a'',&nbsp;''b'') =&nbsp;(1.2,&nbsp;−0.5)]]
 In an ''x''–''y'' Cartesian coordinate system, the circle with center 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>

@@ Line 4: / Line 4: @@
 ;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
+In an ''x''–''y'' Cartesian coordinate system, the circle with center 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,&nbsp;0), then the equation simplifies to
+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,&nbsp;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
+The equation can be written in parametric form using the trigonometric functions 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'',&nbsp;''b'') to (''x'',&nbsp;''y'') makes with the positive ''x''&nbsp;axis.
+where ''t'' is a parametric variable in the range 0 to 2{{pi}}, interpreted geometrically as the angle that the ray from (''a'',&nbsp;''b'') to (''x'',&nbsp;''y'') makes with the positive ''x''&nbsp;axis.
 An alternative parametrisation of the circle is
@@ Line 20: / Line 20: @@
 : <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''&nbsp;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.
+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''&nbsp;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-standard-equation/v/radius-and-center-for-a-circle-equation-in-standard-form Standard Form of Circle Equation], Khan Academy

← Older revision		Revision as of 20:29, 18 October 2021
Line 1:		Line 1:
		+
		+	==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