Analytical Geometry

The analytical geometry is an approach of the Géométrie in which one represents the objects by equations or inequations. The plan or space is necessarily provided with a Repère.

The analytical geometry allows contrary to represent mathematical functions in the shape of curves, graphs. It is thus fundamental for the Physique and the Infographie.

See also: Location in the plan and space, Chart

History

Analysis in geometry

The analytical term of geometry , in opposition to the synthetic Geometry , referred with the methods of Analysis and synthesis practiced by the Greek geometricians. It gradually came from there to merge with its privileged method, the method of the coordinates.

In Greek mathematics, the analyzes consists starting from the sought object, by supposing its existence, so as to establish its properties. It is necessary to proceed in this way until producing enough properties to characterize the object. One can then reverse the situation, by not making more the assumption of existence and by introducing it indeed the object by the means of the index properties: it is the phase of synthesis, which must lead to the proof of existence.

The difficulty practices which limited progress of the geometricians is the lack of a formalism adapted to the description of the relations between geometrical magnitudes. François Viète, at the end of the 16th century unifies calculation on the numbers and calculation on the geometrical magnitudes through an invaluable tool, the literal Calcul. The principle of the reduction to the calculus is posed, it still misses a systematic method to exploit it.

Method of the coordinates

Rene Descartes proposes to solve the problems of geometry by the automatic appeal with the calculus. In its Geometry of 1637, it gives the principle of it. It is a question of representing known and unknown sizes by letters, and of finding as many relations between known sizes and unknown factors that there are unknown factors with the problem. One recognizes an analytical step well there, leading to systems of equations which it is a question of reducing to only one equation. Descartes gives interpretations of the cases on or under-given. Its handling, however, is limited to the algebraic equations, which it classifies per degree, and cannot be applied to the curves which it describes as mechanics (today known as transcendent S).

Pierre de Fermat is the first to make, at the same time, a systematic use of the Coordonnée S themselves to solve the problems of loci. It utilizes in particular the first equations of right-hand sides, parabolas or hyperboles. It presents these ideas in AD locus planos and solidos isagoge , in 1636, text published after its death.

In the notations of Descartes, contrary to Fermat, the constants are continuously noted has , B , C , D ,… and the variables X , there , Z . He is opposed in that to the tradition of the time and a reader of today is some less diverted.

Plane analytical geometry

The plan refines is provided with a reference mark $(O,\; \backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\})$; X indicates the X-coordinate of a point, and there the ordinate of this point.

Right-hand side

A right refines (i.e. a line with the usual direction, a whole of points) is represented by a simple equation with two unknown factors:

ax + by + C = 0 (1)

If C is null, then the line passes by the origin O . If two lines are parallel, then their coefficients has and B is proportional. If B is not null, this equation can be rewritten:

there = has ′·X + B ′

has ′   =  -   a/b is called the directing coefficient or the slope of the right-hand side, and B ′   =  -   c/b is called ordered in the beginning ( offset or intercept in English); two parallel straight lines have the same directing coefficient. With this form there, one sees easily that the line passes by the item (0, B ′), which also is called ordered in the beginning (the term thus indicates at the same time the point and the ordinate of this point). If has is null, one has a horizontal line

there = B ′

passing by the point (0, B ′). If B is null, one has a vertical line

X = - c/a

passing by the point (-   C / has , 0).

To plot a straight line starting from its equation, it is enough to know two points. Simplest is to take the intersection with the axes, i.e. to consider X   successively; =  0 and there   =  0 (except if the line is parallel to an axis, in which case to trace it is commonplace). One can also take the ordinate in the beginning and a “distant” point (i.e. at the edge of the figure traced on paper, for example to consider X   =  10 if one goes up to 10), or two distant points (on each board figure); indeed, more the points are moved away, more the layout of the right-hand side is precise.

A vectorial line (i.e. a whole of Vector S colinéaires, proportional between them) is represented simply by an equation of right-hand side with C no one:

with the ₁ + drunk ₂ = 0

where U ₁ and U ₂ is the components of the vectors. One from of deduced that for a line closely connected or vectorial, the vector of components

$\backslash \; vec\; \{U\}\; =\; \backslash \; begin\; \{pmatrix\}\; -\; B\; \backslash \; \backslash \; has\; \backslash \; end\; \{pmatrix\}$

is a directing vector of the right-hand side. If the reference mark is orthonormé, according to a property of the produces scalar, the vector

$\backslash \; vec\; \{NR\}\; =\; \backslash \; begin\; \{pmatrix\}\; has\; \backslash \; \backslash \; B\; \backslash \; end\; \{pmatrix\}$

is a normal vector with the right-hand side.

Whatever the reference mark, if has ( x_A , y_A ) is a point of the right-hand side and $\backslash \; vec\; \{U\}$ a directing vector, then for any point M ( x_M , y_M ) of the right-hand side, one has

$\backslash \; overrightarrow\; \{AM\}\; =\; K\; \backslash \; cdot\; \backslash \; vec\; \{U\},\; \backslash \; K\; \backslash \; in\; \backslash \; mathbb\; \{R\}$

since $\backslash \; overrightarrow\; \{AM\}$ is colinéaire with $\backslash \; vec\; \{U\}$. This gives us a parametric equation of the right-hand side:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; (x\_M\; -\; x\_A)\; =\; K\; \backslash \; cdot\; u\_1\; \backslash \; \backslash \; (y\_M\; -\; y\_A)\; =\; K\; \backslash \; cdot\; u\_2\; \backslash \; end\; \{matrix\}\; \backslash \; right.$

who can be written

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; x\_M\; =\; u\_1\; \backslash \; cdot\; K\; +\; x\_A\; \backslash \; \backslash \; y\_M\; =\; u\_2\; \backslash \; cdot\; K\; +\; y\_A\; \backslash \; end\; \{matrix\}\; \backslash \; right.$ (2)

by eliminating the parameter K , one finds an equation of the form (1).

Not

A point is represented by a system of two simple equations with two unknown factors:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; X\; =\; has\; \backslash \; \backslash \; there\; =\; B\; \backslash \; end\; \{matrix\}\; \backslash \; right.$

what is logical since, a point being the intersection of two lines not-parallels, its coordinates must check the equations of the two lines: the reduction of this system of equations gives the form above. This is obviously the representation of the point ( has , B ).

Half-plane

A half-plane is represented by an inequation of the first degree with two unknown factors:

ax + by + C > 0

if the sign > is replaced; by a sign =, one obtains the equation of the right-hand side which delimits the half-plane; if the sign > is replaced; by the sign < (or if the sign of the coefficients is reversed), the complementary half-plane is obtained.

Intersection of right-hand sides

The plan is reported to a reference mark. A right (not vertical) can be defined by an equation: $y\; =\; ax\; +\; b$

If one considers 2 lines defined by the equations $y\; =\; ax\; +\; b$ and $y\; =a\text{'}\; X\; +b\text{'}$ one can know if there is an intersection or not thanks to one of the 3 following cases:

• If $a=a\text{'}$ and $b\; \backslash \; not=b\text{'}$ then the lines are parallel and there is no intersection.
• If $a=a\text{'}$ and $b=b\text{'}$ then the 2 lines are confused and there is thus an infinity of points of intersection.
• If $a\; \backslash \; not=a\text{'}$, whatever $b$ and $b\text{'}$, there is inevitably a point of intersection. One obtains like coordinates of the point of intersection:

$x\; =\; \backslash \; frac\; \{(B\; -\; b\text{'})\}\{(a\text{'}\; -\; a)\}$ and $y\; =\; \backslash \; frac\; \{(a\text{'}\; B\; -\; ab\text{'})\}\{(a\text{'}\; -\; a)\}$ The demonstration is done thanks to the resolution of a system of two equations to two unknown factors: there = ax + B and there = a' X + b'.

Half-line

A half-line is characterized by an equation and an inequation

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; ax\; +\; by\; +\; C\; =\; 0\; \backslash \; \backslash \; a\text{'}\; X\; +\; b\text{'}\; there\; +\; C\; >\; 0\; \backslash \; end\; \{matrix\}\; \backslash \; right.$

with at least   has; ≠ has ′ or B   ≠ B ′ . A half-line is indeed the intersection of a line and a half-plane delimited by a line nonparallel with the first. The resolution of the system obtained by replacing the sign “> ” by a sign “=” gives the coordinates of the point end of the half-line, i.e. the coordinates of the point has of a half-line If has ′ is not-no one, one can bring back oneself to a system of the type

(two systems representing of the complementary half-lines), if not with a system of the type

With a parametric equation, that returns to the equation (2) by adding the condition K   >   0 or K   <   0.

The circle and the disc

The Cercle of center has and of ray R is the whole of the points located at a distance R of has . Its equation is thus:

$(x-x\_A)\; ^2\; +\; (there\; there\; \_A)\; ^2\; =\; r^2$

that one can write:

$y\; =\; y\_A\; +\; \backslash \; sqrt\; \{r^2\; -\; (x-x\_A)\; ^2\},\; \backslash \; X\; \backslash \; in$

This form bears the name “of Cartesian equation of the circle”. Its parametric equation is

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; X\; =\; x\_A\; +\; R\; \backslash \; cdot\; \backslash \; cos\; \backslash \; theta\; \backslash \; \backslash$

there = y_A + R \ cdot \ sin \ theta \ end {matrix} \ right. where θ is a reality, which can be taken on an interval of width 2π; one takes in general] - π, π] or The equation of the disc is obtained by replacing the “equal” sign by a “lower or equal” sign.

Analytical geometry in space

The space refines is provided with a reference mark $(O,\; \backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\},\; \backslash \; vec\; \{K\})$; X indicates the X-coordinate of a point, there the ordinate and Z the dimension.

Plan

A plan refines (i.e. a plan with the usual direction in geometry, composed of points) is represented by a simple equation with three unknown factors:

ax + by + cz + D = 0 (3)

If two plans are parallel between them, then their coefficients has , B and C is proportional. If D is null, then the plan passes by O . If C is nonnull, the equation can be written

Z = has ′·X + B ′·there + C ′

with ′   has; =  -   a/c , B ′   =  -   b/c and C ′   =  -   d/c . If C is null, then there is a vertical plan.

A vectorial plan (i.e. a Coplanar whole of vectors S) is represented by an equation

with the ₁ + drunk ₂ + Cu ₃ = 0

where U ₁, U ₂ and U ₃ is the components of a vector. The following vectors are vectors of the vectorial plan, and so at least two coefficients of the equation of the plan are nonnull, two of these vectors form a base of the plan:

$\backslash \; vec\; \{U\}\; =\; \backslash \; begin\; \{pmatrix\}\; -\; B\; \backslash \; \backslash \; has\; \backslash \; \backslash \; 0\; \backslash \; end\; \{pmatrix\}$

$\backslash \; vec\; \{v\}\; =\; \backslash \; begin\; \{pmatrix\}\; 0\; \backslash \; \backslash \; -\; C\; \backslash \; \backslash \; B\; \backslash \; end\; \{pmatrix\}$

$\backslash \; vec\; \{W\}\; =\; \backslash \; begin\; \{pmatrix\}\; -\; C\; \backslash \; \backslash \; 0\; \backslash \; \backslash \; has\; \backslash \; end\; \{pmatrix\}$

(the base obtained is not a priori orthonormée). These vectors form also vectors of a plan refines whose equation has the same coefficients has , B and C that the equation of the vectorial plan.

If two of the coefficients are null, then the equation is reduced to the one of the three following forms:

U ₁ = 0, which represents the vectorial plan $(\backslash \; vec\; \{J\},\; \backslash \; vec\; \{K\})$;

U ₂ = 0, which represents the vectorial plan $(\backslash \; vec\; \{I\},\; \backslash \; vec\; \{K\})$;

U ₃ = 0, which represents the vectorial plan $(\backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\})$.

In the same way,

ax + D = 0 represents a plan refines parallel with $(\backslash \; vec\; \{J\},\; \backslash \; vec\; \{K\})$, whose equation can be written X = - d/a

by + D = 0 represents a plan refines parallel with $(\backslash \; vec\; \{I\},\; \backslash \; vec\; \{K\})$, whose equation can be written there = - d/b

cz + D = 0 represents a plan refines parallel with $(\backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\})$, of which the equation Z = - d/c can be written.

In all the cases, if the reference mark of space is orthonormal, the vector

$\backslash \; vec\; \{NR\}\; =\; \backslash \; begin\; \{pmatrix\}\; has\; \backslash \; \backslash \; B\; \backslash \; \backslash \; C\; \backslash \; end\; \{pmatrix\}$

is a normal vector in the plan

Whatever the reference mark, if the plan passes by a point has ( x_A , y_A , z_A ) and is provided with an unspecified base $(\backslash \; vec\; \{U\},\; \backslash \; vec\; \{v\})$, then for any point M ( x_M , y_M , z_M ), one has

$\backslash \; overrightarrow\; \{AM\}\; =\; K\; \backslash \; cdot\; \backslash \; vec\; \{U\}\; +\; T\; \backslash \; cdot\; \backslash \; vec\; \{v\},\; \backslash \; (K,\; T)\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; ^2$

since $\backslash \; overrightarrow\; \{AM\}$, $\backslash \; vec\; \{U\}$ and $\backslash \; vec\; \{v\}$ is coplanar. This led to the parametric equation

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; (x\_M\; -\; x\_A)\; =\; K\; \backslash \; cdot\; u\_1\; +\; T\; \backslash \; cdot\; v\_1\; \backslash \; \backslash \; (y\_M\; -\; y\_A)\; =\; K\; \backslash \; cdot\; u\_2\; +\; T\; \backslash \; cdot\; v\_2\; \backslash \; \backslash \; (z\_M\; -\; z\_A)\; =\; K\; \backslash \; cdot\; u\_3\; +\; T\; \backslash \; cdot\; v\_3\; \backslash \; end\; \{matrix\}\; \backslash \; right.\; \backslash \; Leftrightarrow\; \backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; x\_M\; =\; u\_1\; \backslash \; cdot\; K\; +\; v\_1\; \backslash \; cdot\; T\; +\; x\_A\; \backslash \; \backslash \; y\_M\; =\; u\_2\; \backslash \; cdot\; K\; +\; v\_2\; \backslash \; cdot\; T\; +\; y\_A\; \backslash \; \backslash \; z\_M\; =\; u\_3\; \backslash \; cdot\; K\; +\; v\_3\; \backslash \; cdot\; T\; +\; z\_A\; \backslash \; end\; \{matrix\}\; \backslash \; right.$ (4)

Right-hand side

A line being the intersection of two not-parallels plans, it is described by a system of two simple equations with three unknown factors

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; ax\; +\; by\; +\; cz\; +\; D\; =\; 0\; \backslash \; \backslash \; a\text{'}\; X\; +\; b\text{'}\; there\; +\; it\; Z\; +\; of\; =\; 0\; \backslash \; end\; \{matrix\}\; \backslash \; right.$ (5)

The line is contained in the two plans, it is thus orthogonal with the normal vectors $\backslash \; vec\; \{N\_1\}$ and $\backslash \; vec\; \{N\_2\}$ of the two plans. The vector product $\backslash \; vec\; \{U\}\; =\; \backslash \; vec\; \{N\_1\}\; \backslash \; wedge\; \backslash \; vec\; \{N\_2\}$ of the normal vectors thus provides a directing vector of the right-hand side. If the reference mark is orthonormé direct, the vector $\backslash \; vec\; \{U\}$ has as components:

$\backslash \; vec\; \{U\}\; =\; \backslash \; begin\; \{pmatrix\}\; bc\text{'}-b\text{'}\; C\; \backslash \; \backslash \; ca\text{'}-it\; has\; \backslash \; \backslash \; ab\text{'}-a\text{'}\; B\; \backslash \; end\; \{pmatrix\}$

So in addition one knows a point has ( x_A , y_A , z_A ) and a directing vector $\backslash \; vec\; \{U\}$ of the right-hand side, then if M ( x_M , y_M , z_M ) is a point of the right-hand side, it checks:

$\backslash \; overrightarrow\; \{AM\}\; =\; K\; \backslash \; cdot\; \backslash \; vec\; \{U\},\; \backslash \; K\; \backslash \; in\; \backslash \; mathbb\; \{R\}$

since $\backslash \; overrightarrow\; \{AM\}$ and $\backslash \; vec\; \{U\}$ is colinéaires. One thus obtains the parametric equation

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; x\_M\; =\; u\_1\; \backslash \; cdot\; K\; +\; x\_A\; \backslash \; \backslash \; y\_M\; =\; u\_2\; \backslash \; cdot\; K\; +\; y\_A\; \backslash \; \backslash \; z\_M\; =\; u\_3\; \backslash \; cdot\; K\; +\; z\_A\; \backslash \; end\; \{matrix\}\; \backslash \; right.$ (6)

Not

A point is described by a system of three simple equations with three unknown factors:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; X\; =\; has\; \backslash \; \backslash \; there\; =\; B\; \backslash \; \backslash \; Z\; =\; C\; \backslash \; end\; \{matrix\}\; \backslash \; right.$

The point being the intersection of three convergent plans, its coordinates must check the three equations; the reduction of this system gives the form above. This system of equations represents the point of course ( has , B , C ).

See too

• Euclidean Geometry

• Solid geometry
• vectorial Geometry
• Location in the plan and space
• Volumes of revolution (cone, Cylinder, Sphere, Ellipsoidal, Paraboloid)

References

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