# Analytical geometry

The analytic geometry (also vector geometry ) is a branch of geometry , the algebraic tools (especially from linear algebra ) provides for solving geometric problems. In many cases it makes it possible to solve geometrical problems purely arithmetically without using visualization as an aid.

On the other hand, geometry that establishes its propositions on an axiomatic basis without reference to a number system is called synthetic geometry .

The methods of analytical geometry are used in all natural sciences , but above all in physics , for example in the description of planetary orbits . Originally, analytical geometry only dealt with issues of plane and spatial ( Euclidean ) geometry. In the general sense, however, analytic geometry describes affine spaces of arbitrary dimensions over arbitrary bodies .

## The coordinate system

Points and their coordinates in a Cartesian coordinate system of the plane

A coordinate system is a crucial aid in analytical geometry. In practice, a Cartesian coordinate system is usually used . For some simple questions, such as the determination of straight line intersections, the examination of straight lines for parallelism or the calculation of partial ratios , an oblique coordinate system would be sufficient. A Cartesian coordinate system is essential if distances or angles are to be calculated.

## Vectors

Many calculations of analytical geometry are standardized and simplified by the methods of vector calculation. Although all analytical geometry was invented without vectors and of course can still be practiced without vectors and, conversely, vector space can be defined as an abstract-algebraic construct with no geometric reference, the use of vectors in Cartesian coordinate systems seems so natural that “Linear Algebra and Analytical Geometry ”in the upper secondary level and in the mathematical-physical-technical basic studies in general as one course.

## Coordinate and parameter equations

More complicated geometric structures such as straight lines , planes , circles , spheres are understood as point sets and described by equations . These can be coordinate equations or parametric equations.

Implicit coordinate equation
A calculation expression that depends on the coordinates is set equal to 0.${\ displaystyle (x, y, \ dotsc)}$
Example (straight line of the drawing plane)
${\ displaystyle 2x-3y + 7 = 0}$
Explicit coordinate equation
One of the coordinates is expressed by the other.
Example (level in space)
${\ displaystyle z = 2x-5y + 3}$
Explicit coordinate equations have the disadvantage that case distinctions often have to be made; so it is for example
in the plane impossible to show a parallel to the -axis in the form .${\ displaystyle y}$${\ displaystyle y = mx + t}$
Parametric equation
The position vector of any point of the structure is given by a vector arithmetic expression that contains one or more parameters.${\ displaystyle {\ vec {x}} = {\ overrightarrow {OX}}}$${\ displaystyle X}$
Example (straight in the room): ${\ displaystyle {\ vec {x}} = {\ begin {pmatrix} 2 \\ - 4 \\ - 3 \ end {pmatrix}} + \ lambda {\ begin {pmatrix} -2 \\ 0 \\ 5 \ end {pmatrix}}}$

## Analytical geometry of the plane

### Points in the plane

Each point on the plane is described by two coordinates , e.g. B. . The coordinates are usually called (in this order) the -coordinate (also: abscissa ) and the -coordinate (also: ordinate ). The terms and are also used . ${\ displaystyle P}$${\ displaystyle P (2 \ mid -1 {,} 5)}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle x_ {1}}$${\ displaystyle x_ {2}}$

The combined coordinates of points form ordered pairs in the flat case .

### Straight lines in the plane

Coordinate equation (implicit)
${\ displaystyle n_ {x} x + n_ {y} y + n_ {0} = \, \! 0}$
One also speaks of the normal (en) form of the straight line equation, since the vector is perpendicular (normal) to the straight line.${\ displaystyle {\ begin {pmatrix} n_ {x} \\ n_ {y} \ end {pmatrix}}}$
Parametric equation
${\ displaystyle {\ overrightarrow {OX}} = {\ overrightarrow {OA}} + \ lambda {\ vec {u}}}$
Here is the position vector of any arbitrary but fixed point on the straight line (support point); is a so-called direction vector, i.e. a vector whose direction is parallel to the straight line.${\ displaystyle {\ overrightarrow {OA}} = {\ vec {A}}}$${\ displaystyle {\ vec {u}}}$

### Second order curves in the plane

By an (implicit coordinate) equation of the second degree

${\ displaystyle Ax ^ {2} + By ^ {2} + Cxy + Dx + Ey + F = 0}$

a conic section is generally given. Depending on the values ​​of the coefficients, this can be an ellipse (special case: circle ), a parabola or a hyperbola .

## Analytical geometry of Euclidean space

### Points in space

Each point in space is determined by three coordinates , e.g. B. . Each point is assigned its position vector , which is the connection vector between the origin of the coordinate system and the given point. Its coordinates correspond to those of the point , but are written as a column vector: ${\ displaystyle P}$${\ displaystyle P = P (4 \ mid -0 {,} 5 \ mid -3)}$${\ displaystyle P}$ ${\ displaystyle {\ overrightarrow {OP}}}$${\ displaystyle P}$

${\ displaystyle {\ overrightarrow {OP}} = {\ begin {pmatrix} 4 \\ - 0 {,} 5 \\ - 3 \ end {pmatrix}}}$

The coordinates are referred to as -, - and -coordinates or -, - and -coordinates (in that order) . ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$${\ displaystyle x_ {1}}$${\ displaystyle x_ {2}}$${\ displaystyle x_ {3}}$

The combined coordinates of points form 3-tuples in the spatial case .

### Straight lines in space

Coordinate equations
Lines in space cannot be described by a single coordinate equation. A straight line can always be understood as the intersection (intersection) of two planes and coordinate equations of these two planes (see below) can be used to clearly define the straight line.
Parametric equation
${\ displaystyle {\ overrightarrow {OX}} = {\ overrightarrow {OA}} + \ lambda {\ vec {u}}}$

So the equation has the same form as in the two-dimensional case.

### Levels in space

Coordinate equation (implicit)
${\ displaystyle n_ {x} x + n_ {y} y + n_ {z} z + n_ {0} = \, \! 0}$
This type of plane equation is called normal form , since the vector is perpendicular (normal) to the plane.${\ displaystyle {\ vec {n}} = {\ begin {pmatrix} n_ {x} \\ n_ {y} \\ n_ {z} \ end {pmatrix}}}$
Parametric equation
${\ displaystyle {\ overrightarrow {OX}} = {\ overrightarrow {OA}} + \ lambda {\ vec {u}} + \ mu {\ vec {v}}}$
${\ displaystyle {\ overrightarrow {OA}} = {\ vec {A}}}$is the position vector of any arbitrary but fixed point on the plane (support point); and are linearly independent direction vectors (or span vectors ), that is, vectors parallel to the plane that "span" the plane.${\ displaystyle {\ vec {u}}}$${\ displaystyle {\ vec {v}}}$

### Second order surfaces in space

The general coordinate equation of the second degree

${\ displaystyle Ax ^ {2} + By ^ {2} + Cz ^ {2} + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0}$

describes a surface of the second order . The most important special cases are:

## Generalization: Analytical geometry of any affine space

The concepts of analytical geometry can be generalized by allowing coordinates from any body and any dimensions .

If there is a vector space over a body and an associated affine space , then a -dimensional subspace of can be described by the parameter equation ${\ displaystyle V}$${\ displaystyle K}$${\ displaystyle R}$${\ displaystyle V}$${\ displaystyle k}$${\ displaystyle R}$

${\ displaystyle {\ overrightarrow {OX}} = {\ overrightarrow {OA}} + \ lambda _ {1} {\ vec {v_ {1}}} + \ dotsb + \ lambda _ {k} {\ vec {v_ {k}}}}$.

It is the position vector of an arbitrary, but fixed chosen point of the subspace (base); the vectors are linearly independent vectors, that is, a basis of the subspace of , which belongs to the considered subspace of . ${\ displaystyle {\ overrightarrow {OA}} = {\ vec {A}}}$${\ displaystyle {\ vec {v_ {1}}}, \ dotsc, {\ vec {v_ {k}}}}$${\ displaystyle V}$${\ displaystyle R}$

For is the equation of a straight line, for the equation of a plane. If 1 is smaller than the dimension of or , one speaks of a hyperplane . ${\ displaystyle k = 1}$${\ displaystyle k = 2}$${\ displaystyle k}$${\ displaystyle R}$${\ displaystyle V}$

In analogy to the second-order curves (conic sections) of plane geometry and the second-order surfaces of spatial geometry, so-called quadrics are also considered in -dimensional affine space, i.e. , second-order hypersurfaces (with the dimension ), which are defined by second-degree coordinate equations are defined: ${\ displaystyle n}$${\ displaystyle n-1}$

${\ displaystyle \ sum _ {i = 1} ^ {n} \ sum _ {j = 1} ^ {n} A_ {ij} x_ {i} x_ {j} + \ sum _ {i = 1} ^ { n} B_ {i} x_ {i} + C = 0}$

## Typical tasks in analytical geometry

### Incidence Review

The aim here is to determine whether a given point belongs to a given set of points (e.g. to a straight line).

#### In two-dimensional space

The straight line with the explicit coordinate equation

${\ displaystyle y = 2 \ cdot x-3}$

to be viewed as.

The point lies on this straight line, as can be seen by inserting the coordinates and (point sample): ${\ displaystyle P (2 | 1)}$${\ displaystyle x = 2}$${\ displaystyle y = 1}$

${\ displaystyle 1 = 2 \ cdot 2-3}$

The point, however, is not on the straight line. For and is namely ${\ displaystyle S (4 | 2)}$${\ displaystyle x = 4}$${\ displaystyle y = 2}$

${\ displaystyle 2 \ neq 2 \ cdot 4-3}$.

#### In three-dimensional space

It should be checked whether the point lies on the straight line with the following parameter form : ${\ displaystyle Q (-1 \ mid 9 \ mid -2)}$

${\ displaystyle {\ vec {x}} = {\ begin {pmatrix} -7 \\ 7 \\ - 6 \ end {pmatrix}} + t {\ begin {pmatrix} 3 \\ 1 \\ 2 \ end { pmatrix}}}$ .

If the position vector of is used, this leads to the following 3 equations: ${\ displaystyle {\ vec {x}}}$${\ displaystyle {\ overrightarrow {OQ}}}$${\ displaystyle Q}$

${\ displaystyle {\ begin {matrix} -1 & = & - 7 & + & t \ cdot 3 & \ Rightarrow & t = 2 \\ 9 & = & 7 & + & t \ cdot 1 & \ Rightarrow & t = 2 \\ - 2 & = & - 6 & + & t \ cdot 2 & \ Rightarrow & t = 2 \\\ end {matrix}}}$

Since has the same value in all three cases (here ), lies on the straight line. ${\ displaystyle t}$${\ displaystyle 2}$${\ displaystyle Q}$

### Determination of the intersection of two sets of points

Determining the intersection of two sets of points (e.g. the intersection of two straight lines) amounts to solving a system of equations . Depending on the form in which the two sets of points are described, the procedure varies a little:

case 1
Both sets of points are given by coordinate equations.
In this case, the intersection is described by the totality of the coordinate equations.
Case 2
Both sets of points are given by parameter equations.
The intersection is obtained by equating the right-hand sides of these equations.
Case 3
One of the point sets is given by a coordinate equation, the other by a parametric equation.
In this case, the individual coordinates of the vector parameter equation are inserted into the vector equation.

#### In two-dimensional space

It should be checked whether and where the graphs of the functions and intersect. And corresponds to : ${\ displaystyle g (x)}$${\ displaystyle f (x)}$${\ displaystyle g (x) = 2x ^ {2} -2}$${\ displaystyle f (x) = 2x + 2}$

In order to calculate the points of intersection, the function terms of the equations of the two functions are now set equal. In this way one finds the coordinate (s) for which the two functions have the same coordinate: ${\ displaystyle x}$${\ displaystyle y}$

${\ displaystyle {\ begin {matrix} 2x ^ {2} & - & 2 & = & 2 \ cdot x & + & 2 \\ 2x ^ {2} & - & 2 \ cdot x & - & 4 & = & 0 \\ x ^ {2} & - & x & - & 2 & = & 0 \\\ end {matrix}}}$

Solving this quadratic function leads to the solutions: and . ${\ displaystyle x_ {1} = - 1}$${\ displaystyle x_ {2} = 2}$

Substituting it into one of the two initial equations gives the intersection points at: and . ${\ displaystyle S_ {1} (- 1 | 0)}$${\ displaystyle S_ {2} (2 | 6)}$

#### In three-dimensional space

It should be checked whether and at what point the two straight lines and intersect. The two straight lines are defined as follows: ${\ displaystyle g_ {1}}$${\ displaystyle g_ {2}}$

${\ displaystyle g_ {1}: \; {\ vec {x}} = {\ begin {pmatrix} 20 \\ - 30 \\ 1200 \\\ end {pmatrix}} + s \ cdot {\ begin {pmatrix} -100 \\ 225 \\ - 3 \\\ end {pmatrix}} \ qquad \ left (s \ in \ mathbb {R} \ right)}$
${\ displaystyle g_ {2}: \; {\ vec {x}} = {\ begin {pmatrix} -2480 \\ 105 \\ 1167 \\\ end {pmatrix}} + t \ cdot {\ begin {pmatrix} 150 \\ 120 \\ 1 \\\ end {pmatrix}} \ qquad \ left (t \ in \ mathbb {R} \ right)}$

As in two-dimensional space, the two equations are also equated here:

${\ displaystyle {\ begin {pmatrix} 20 \\ - 30 \\ 1200 \\\ end {pmatrix}} + s \ cdot {\ begin {pmatrix} -100 \\ 225 \\ - 3 \\\ end {pmatrix }} = {\ begin {pmatrix} -2480 \\ 105 \\ 1167 \\\ end {pmatrix}} + t \ cdot {\ begin {pmatrix} 150 \\ 120 \\ 1 \\\ end {pmatrix}} }$
${\ displaystyle s \ cdot {\ begin {pmatrix} -100 \\ 225 \\ - 3 \\\ end {pmatrix}} - t \ cdot {\ begin {pmatrix} -150 \\ - 120 \\ - 1 \ \\ end {pmatrix}} = {\ begin {pmatrix} -2500 \\ 135 \\ - 33 \\\ end {pmatrix}}}$

The vector equation can be broken down into the following 3 equations:

${\ displaystyle {\ begin {array} {ccccl | l} -100 \ cdot s & - & 150 \ cdot t & = & - 2500 & \ cdot 3/50 \\ 225 \ cdot s & - & 120 \ cdot t & = & 135 &: 5 \\ -3 \ cdot s & - & 1 \ cdot t & = & - 33 & \ cdot (-2) \\\ end {array}} \ Leftrightarrow {\ begin {array} {ccccl} -6 \ cdot s & - & 9 \ cdot t & = & -150 \\ 45 \ cdot s & - & 24 \ cdot t & = & 27 \\ 6 \ cdot s & + & 2 \ cdot t & = & 66 \\\ end {array}}}$

Adding the first and last equation yields or . From the first equation thus results by employing so . This solution also satisfies the second equation because . ${\ displaystyle -7 \ cdot t = -84}$${\ displaystyle t = 12}$${\ displaystyle -6 \ cdot s = -42}$${\ displaystyle s = 7}$${\ displaystyle 45 \ times 7-24 \ times 12 = 27}$

The position vector of the intersection of the straight lines is obtained by inserting one of the two calculated parameters ( ) into the corresponding straight line ( ): ${\ displaystyle s}$${\ displaystyle g_ {1}}$

${\ displaystyle {\ begin {pmatrix} 20 \\ - 30 \\ 1200 \\\ end {pmatrix}} + 7 \ cdot {\ begin {pmatrix} -100 \\ 225 \\ - 3 \\\ end {pmatrix }} = {\ begin {pmatrix} -680 \\ 1545 \\ 1179 \\\ end {pmatrix}} \ Rightarrow S \ left (-680 | 1545 | 1179 \ right)}$

## history

Analytical geometry was founded by the French mathematician and philosopher René Descartes . Major enhancements are thanks to Leonhard Euler , who particularly dealt with the curves and surfaces of the second order. The development of vector calculation (by Hermann Graßmann among others ) made the vector notation common today possible.

David Hilbert has proven that three-dimensional analytical geometry is completely equivalent to (synthetic) Euclidean geometry in the form he specified . In practical terms it is far superior to this. In the first half of the 20th century, therefore, the view was held that geometry as it has been taught since Euclid was only of historical interest.

Nicolas Bourbaki even went one step further: he completely dispensed with geometrical concepts such as point, straight line, etc. and considered everything necessary to have been said with the treatment of linear algebra . Of course - as always with Bourbaki - the needs of applied mathematics are completely ignored.