Analytic geometry

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following:[16]

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation y = 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, and y = x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.[17]

Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x² + y² = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations.[18] The equation x² + y² = r² is the equation for any circle centered at the origin (0, 0) with a radius of r.

The distance formula on the plane follows from the Pythagorean theorem.

In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x₁, y₁) and (x₂, y₂) is defined by the formula

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}},\!}$

which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula

${\displaystyle \theta =\arctan(m),}$

where m is the slope of the line.

In three dimensions, distance is given by the generalization of the Pythagorean theorem:

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},\!}$,

while the angle between two vectors is given by the dot product. The dot product of two Euclidean vectors A and B is defined by[24]

${\displaystyle \mathbf {A} \cdot \mathbf {B} {\stackrel {\mathrm {def} }{=}}\|\mathbf {A} \|\,\|\mathbf {B} \|\cos \theta ,}$

where θ is the angle between A and B.

a) y = f(x) = |x|       b) y = f(x+3)       c) y = f(x)-3       d) y = 1/2 f(x)

Transformations are applied to a parent function to turn it into a new function with similar characteristics.

The graph of ${\displaystyle R(x,y)}$ is changed by standard transformations as follows:

• Changing ${\displaystyle x}$ to ${\displaystyle x-h}$ moves the graph to the right ${\displaystyle h}$ units.
• Changing ${\displaystyle y}$ to ${\displaystyle y-k}$ moves the graph up ${\displaystyle k}$ units.
• Changing ${\displaystyle x}$ to ${\displaystyle x/b}$ stretches the graph horizontally by a factor of ${\displaystyle b}$. (think of the ${\displaystyle x}$ as being dilated)
• Changing ${\displaystyle y}$ to ${\displaystyle y/a}$ stretches the graph vertically.
• Changing ${\displaystyle x}$ to ${\displaystyle x\cos A+y\sin A}$ and changing ${\displaystyle y}$ to ${\displaystyle -x\sin A+y\cos A}$ rotates the graph by an angle ${\displaystyle A}$.

There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations.

For example, the parent function ${\displaystyle y=1/x}$ has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if ${\displaystyle y=f(x)}$, then it can be transformed into ${\displaystyle y=af(b(x-k))+h}$. In the new transformed function, ${\displaystyle a}$ is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative ${\displaystyle a}$ values, the function is reflected in the ${\displaystyle x}$-axis. The ${\displaystyle b}$ value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like ${\displaystyle a}$, reflects the function in the ${\displaystyle y}$-axis when it is negative. The ${\displaystyle k}$ and ${\displaystyle h}$ values introduce translations, ${\displaystyle h}$, vertical, and ${\displaystyle k}$ horizontal. Positive ${\displaystyle h}$ and ${\displaystyle k}$ values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.

Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.

Suppose that ${\displaystyle R(x,y)}$ is a relation in the ${\displaystyle xy}$ plane. For example,

${\displaystyle x^{2}+y^{2}-1=0}$

is the relation that describes the unit circle.

For two geometric objects P and Q represented by the relations ${\displaystyle P(x,y)}$ and ${\displaystyle Q(x,y)}$ the intersection is the collection of all points ${\displaystyle (x,y)}$ which are in both relations.[25]

For example, ${\displaystyle P}$ might be the circle with radius 1 and center ${\displaystyle (0,0)}$: ${\displaystyle P=\{(x,y)|x^{2}+y^{2}=1\}}$ and ${\displaystyle Q}$ might be the circle with radius 1 and center ${\displaystyle (1,0):Q=\{(x,y)|(x-1)^{2}+y^{2}=1\}}$. The intersection of these two circles is the collection of points which make both equations true. Does the point ${\displaystyle (0,0)}$ make both equations true? Using ${\displaystyle (0,0)}$ for ${\displaystyle (x,y)}$, the equation for ${\displaystyle Q}$ becomes ${\displaystyle (0-1)^{2}+0^{2}=1}$ or ${\displaystyle (-1)^{2}=1}$ which is true, so ${\displaystyle (0,0)}$ is in the relation ${\displaystyle Q}$. On the other hand, still using ${\displaystyle (0,0)}$ for ${\displaystyle (x,y)}$ the equation for ${\displaystyle P}$ becomes ${\displaystyle 0^{2}+0^{2}=1}$ or ${\displaystyle 0=1}$ which is false. ${\displaystyle (0,0)}$ is not in ${\displaystyle P}$ so it is not in the intersection.

The intersection of ${\displaystyle P}$ and ${\displaystyle Q}$ can be found by solving the simultaneous equations:

${\displaystyle x^{2}+y^{2}=1}$

${\displaystyle (x-1)^{2}+y^{2}=1.}$

Traditional methods for finding intersections include substitution and elimination.

Substitution: Solve the first equation for ${\displaystyle y}$ in terms of ${\displaystyle x}$ and then substitute the expression for ${\displaystyle y}$ into the second equation:

${\displaystyle x^{2}+y^{2}=1}$

${\displaystyle y^{2}=1-x^{2}}$.

We then substitute this value for ${\displaystyle y^{2}}$ into the other equation and proceed to solve for ${\displaystyle x}$:

${\displaystyle (x-1)^{2}+(1-x^{2})=1}$

${\displaystyle x^{2}-2x+1+1-x^{2}=1}$

${\displaystyle -2x=-1}$

${\displaystyle x=1/2.}$

Next, we place this value of ${\displaystyle x}$ in either of the original equations and solve for ${\displaystyle y}$:

${\displaystyle (1/2)^{2}+y^{2}=1}$

${\displaystyle y^{2}=3/4}$

${\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.}$

So our intersection has two points:

${\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;\mathrm {and} \;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).}$

Elimination: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get ${\displaystyle (x-1)^{2}-x^{2}=0}$. The ${\displaystyle y^{2}}$ in the first equation is subtracted from the ${\displaystyle y^{2}}$ in the second equation leaving no ${\displaystyle y}$ term. The variable ${\displaystyle y}$ has been eliminated. We then solve the remaining equation for ${\displaystyle x}$, in the same way as in the substitution method:

${\displaystyle x^{2}-2x+1+1-x^{2}=1}$

${\displaystyle -2x=-1}$

${\displaystyle x=1/2.}$

We then place this value of ${\displaystyle x}$ in either of the original equations and solve for ${\displaystyle y}$:

${\displaystyle (1/2)^{2}+y^{2}=1}$

${\displaystyle y^{2}=3/4}$

${\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.}$

So our intersection has two points:

${\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;\mathrm {and} \;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).}$

For conic sections, as many as 4 points might be in the intersection.

• Cross product
• Rotation of axes
• Translation of axes
• Vector space

• Coordinate Geometry topics with interactive animations