Editing Analytic geometry (section)

==Equations and curves==
{{main|Solution set|Locus (mathematics)}}
In analytic geometry, any [[equation]] involving the coordinates specifies a [[subset]] of the plane, namely the [[solution set]] for the equation, or [[locus (mathematics)|locus]].  For example, the equation ''y''&nbsp;=&nbsp;''x'' corresponds to the set of all the points on the plane whose ''x''-coordinate and ''y''-coordinate are equal.  These points form a [[line (geometry)|line]], and ''y''&nbsp;=&nbsp;''x'' is said to be the equation for this line.  In general, linear equations involving ''x'' and ''y'' specify lines, [[quadratic equation]]s specify [[conic section]]s, and more complicated equations describe more complicated figures.<ref name=intro>Percey Franklyn Smith, Arthur Sullivan Gale (1905)''Introduction to Analytic Geometry'', Athaeneum Press
</ref>

Usually, a single equation corresponds to a [[curve]] on the plane.  This is not always the case: the trivial equation ''x''&nbsp;=&nbsp;''x'' specifies the entire plane, and the equation ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;=&nbsp;0 specifies only the single point (0,&nbsp;0).  In three dimensions, a single equation usually gives a [[Surface (mathematics)|surface]], and a curve must be specified as the [[Intersection (set theory)|intersection]] of two surfaces (see below), or as a system of [[parametric equation]]s.<ref>William H. McCrea, ''Analytic Geometry of Three Dimensions''
Courier Dover Publications, Jan 27, 2012</ref> The equation ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;=&nbsp;''r''<sup>2</sup> is the equation for any circle centered at the origin (0, 0) with a radius of r.

===Lines and planes===
{{main|Line (geometry)|Plane (geometry)}}
Lines in a [[Cartesian plane]], or more generally, in [[affine coordinates]], can be described algebraically by ''linear'' equations. In two dimensions, the equation for non-vertical lines is often given in the ''[[slope-intercept form]]'':
<math display="block"> y = mx + b </math>
where:
* ''m'' is the [[slope]] or [[slope|gradient]] of the line.
* ''b'' is the [[y-intercept]] of the line.
* ''x'' is the [[independent variable]] of the function ''y'' = ''f''(''x'').

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the [[normal vector]]) to indicate its "inclination".

Specifically, let <math>\mathbf{r}_0</math> be the position vector of some point <math>P_0 = (x_0,y_0,z_0)</math>, and let <math>\mathbf{n} = (a,b,c)</math> be a nonzero vector. The plane determined by this point and vector consists of those points <math>P</math>, with position vector <math>\mathbf{r}</math>, such that the vector drawn from <math>P_0</math> to <math>P</math> is perpendicular to <math>\mathbf{n}</math>. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points <math>\mathbf{r}</math> such that
<math display="block">\mathbf{n} \cdot (\mathbf{r}-\mathbf{r}_0) =0.</math>
(The dot here means a [[dot product]], not scalar multiplication.)
Expanded this becomes
<math display="block"> a (x-x_0)+ b(y-y_0)+ c(z-z_0)=0,</math>
{{cn span|text=which is the ''point-normal'' form of the equation of a plane. |date=April 2022}} This is just a [[linear equation]]:
<math display="block"> ax + by + cz + d = 0, \text{ where } d = -(ax_0 + by_0 + cz_0).</math>
Conversely, it is easily shown that if ''a'', ''b'', ''c'' and ''d'' are constants and ''a'', ''b'', and ''c'' are not all zero, then the graph of the equation
<math display="block"> ax + by + cz + d = 0,</math>
{{cn span|text=is a plane having the vector <math>\mathbf{n} = (a,b,c)</math> as a normal.|date=April 2022}} This familiar equation for a plane is called the ''general form'' of the equation of the plane.<ref>{{citation
 | last1 = Vujičić | first1 = Milan
 | last2 = Sanderson | first2 = Jeffrey
 | editor-first1 = Milan
 | editor-first2 = Jeffrey
 | editor-last1 = Vujičić
 | editor-last2 = Sanderson
 | doi = 10.1007/978-3-540-74639-3
 | page = 27
 | publisher = Springer
 | title = Linear Algebra Thoroughly Explained
 | year = 2008| isbn = 978-3-540-74637-9
 }}</ref>

In three dimensions, lines can ''not'' be described by a single linear equation, so they are frequently described by [[parametric equation]]s:
<math display="block"> x = x_0 + at </math>
<math display="block"> y = y_0 + bt </math>
<math display="block"> z = z_0 + ct </math>
where:
* ''x'', ''y'', and ''z'' are all functions of the independent variable ''t'' which ranges over the real numbers.
* (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) is any point on the line.
* ''a'', ''b'', and ''c'' are related to the slope of the line, such that the [[vector (geometric)|vector]] (''a'', ''b'', ''c'') is parallel to the line.

===Conic sections===
{{main|Conic section}}
[[File:Hyperbler_konjugerte.png|thumb|right|250px|A hyperbola and its [[conjugate hyperbola]] ]]
In the [[Cartesian coordinate system]], the [[Graph of a function|graph]] of a [[quadratic equation]] in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form
<math display="block">Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0\text{ with }A, B, C\text{ not all zero.}  </math>
As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional [[projective space]] <math>\mathbf{P}^5.</math>

The conic sections described by this equation can be classified using the [[discriminant]]<ref>{{citation | title=Math refresher for scientists and engineers | first1=John R. | last1=Fanchi | publisher=John Wiley and Sons | year=2006 | isbn=0-471-75715-2 | pages=44–45 | url=https://books.google.com/books?id=75mAJPcAWT8C}}, [https://books.google.com/books?id=75mAJPcAWT8C&pg=PA45 Section 3.2, page 45]</ref>

<math display="block">B^2 - 4AC .</math>
If the conic is non-degenerate, then:
* if <math>B^2 - 4AC < 0 </math>, the equation represents an [[ellipse]];
** if <math>A = C </math> and <math>B = 0 </math>, the equation represents a [[circle]], which is a special case of an ellipse;
* if <math>B^2 - 4AC = 0 </math>, the equation represents a [[parabola]];
* if <math>B^2 - 4AC > 0 </math>, the equation represents a [[hyperbola]];
** if we also have <math>A + C = 0 </math>, the equation represents a [[hyperbola|rectangular hyperbola]].

===Quadric surfaces===
{{main|Quadric surface}}
A '''quadric''', or '''quadric surface''', is a ''2''-dimensional [[Surface (mathematics)|surface]] in 3-dimensional space defined as the [[locus (mathematics)|locus]] of [[root of a function|zeros]] of a [[quadratic polynomial]]. In coordinates {{nowrap|''x''<sub>1</sub>, ''x''<sub>2</sub>,''x''<sub>3</sub>}}, the general quadric is defined by the [[algebraic equation]]<ref name="geom">Silvio Levy [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html Quadrics] {{Webarchive|url=https://web.archive.org/web/20180718165310/http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html |date=2018-07-18 }} in "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'', [[CRC Press]], from [[The Geometry Center]] at [[University of Minnesota]]</ref>

<math display="block">\sum_{i,j=1}^{3} x_i Q_{ij} x_j + \sum_{i=1}^{3} P_i  x_i + R = 0.</math>

Quadric surfaces include [[ellipsoid]]s (including the [[sphere]]), [[paraboloid]]s, [[hyperboloid]]s, [[cylinder (geometry)|cylinder]]s, [[cones]], and [[plane (geometry)|planes]].