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In geometry, a plane is the ideal body has only two dimensions, and contains infinitely many points and lines, is a fundamental geometric entities together with point and line.

It can only be defined or described in relation to other similar geometric elements. He often described drawing on the principles characteristic to determine the relationships between fundamental geometric entities. .- When it comes to a plane, you are referring to the geometric surface that has no volume (ie, which is only two-dimensional) and has an infinite number of lines and dots that cross from one side to another. However, when the term is used in the plural, they are talking about material that is developed as a graphical representation of different types of surfaces. Plans are especially used in engineering, architecture and design as it used to lay out flat on a surface other surfaces that are regularly in three dimensions.

A plane is defined by the following geometric elements:

Three points are not aligned. A line and a point outside it. Two parallel lines. Two intersecting lines.

The plans are named with a letter of the Greek alphabet.

Usually represented graphically for easy viewing, as a figure bounded by jagged edges (indicating that the drawing is a part of an infinite surface).

A plane is defined by the following geometric elements: a point and two vectors

Point P = (x1, y1, z1) Vector u = (a1, b1, c1) Vector v = (a2, b2, c2) (x, y, z) = (x_1, y_1, z_1) + m (a_1, b_1, c_1) + n (a_2, b_2, c_2) \, \!

This is the vector of the plane, but the most used is reduced, the result of equating to zero the determinant formed by the two vectors and the generic point X = (x, y, z) with the given point. Thus the plane equation is:

\ Begin {vmatrix} (\ mathbf {X} - \ mathbf {P}) \ \ \ mathbf {u} \ \ \ mathbf {v} \ end {vmatrix} = 0 => \ begin {} x-P_x vmatrix & y, & z-P_z P_y \ \ u_x & u_y & u_z \ \ v_x & v_y & v_z \ end {vmatrix} = 0 => A x + B y + C z + D = 0

Where (A, B, C) is a vector perpendicular to the plane coincides with the vector product of vectors u and v. the formula to find the equation when it is at the origin is:

a (x-h) + b (y-k) + c (z, j) = 0 \,

[Edit] Relative position between two planes

If we have a level 1 with a point A and a normal vector 1 and 2 also have a plane with a point B and a normal vector 2.

Their relative positions can be:

Plans consistent: the same direction as the normal vectors and the point A belongs to the plane 2. Parallel planes: if they have the same direction and the normal vectors point A does not belong to plane 2. Secant planes, if the normal vectors do not have the same direction.

All constructions with ruler and compass are five successive applications of basic constructions, using in each the points, lines and circles that were created in previous phases. These five unique possible constructions are basic constructions

Create the line segment joining two existing points (actually, the line: remember that the rule is of infinite length). Create the circle centered at a given point and whose circumference touches another point given Creating the point where two lines intersect non-parallel. Create the point, or the pair of points, which intersect (do) a line and a circle. Create the point, or the pair of points, which intersect (if they do) two circles.

For example, from two given points, you can create a line or two circles can be created (each point makes the center of a circle and end of another). If we opt for the two circles, the intersection will result in two new points. If we draw line segments between the original points and one of the new items, we will have built an equilateral triangle. So the problem: "build an equilateral triangle given one side (or the endpoints of one side) is trivially solvable with ruler and compass.

There are many ways to show that something is impossible. The strategy to be followed in this paper to present an informal outline of the proofs of the impossibility of the classical problems is to first determine the limits of the rule and compass, what can be done and what can not be done with them-and then show that to solve the problems should be overcome such limits.

Using a ruler and compass can be defined coordinates in the plane. It starts from two points to be considered "given", and draw the line through them. Result is called "X axis", and defines the length between the two points as unit length.

Therefore, having two points as input data is equivalent to having a coordinate axis and a unit of length.

Now a simpler constructions with ruler and compass is to draw a line perpendicular to a given, so it does just that, so we get a "Y".

So, having two points as data is equivalent to having a Cartesian coordinate system with axes X and Y, and drive away.

On the other hand, a point (x, y) in the Euclidean plane can be identified with the complex number x + yi. In the construction with ruler and compass, you start with a line segment of unit length. If you are able to build a given point, any point in the complex plane, then we can say that this point is a constructible complex number.

For example, if two data points as complex numbers 1, - 1, 1 + i, 1 - i, and so on. are easily constructible.

In fact, known constructions of Euclidean geometry can be built as complex numbers x + yi where x and y are rational numbers. More generally, using the same constructs, one can, given two complex numbers a and b, construct a + b, a - b, a × b, and a / b.

This shows that the constructible numbers form a field, it is therefore a subfield of complex numbers. It can be shown something else: given a constructible length is possible to construct its conjugate and square root.

As seen, the only ways to build new items is as the intersection of two lines or one line and a circle or two circles. Using the equations of lines and circles, can be shown that the points that intersect lie in a quadratic extension of the smaller body, F, containing two points on the line, the center of the circle and the radius of circle. That is, the intersection points are of the form x + y \ sqrt {k}, where x, yyk are in F.

Since the body of constructible points is closed under square roots, contains all points that can be obtained by a finite sequence of quadratic extensions of the body with rational coefficients of complex numbers. It said in the previous paragraph, one can show that all constructible point can be obtained by such a sequence of extensions. As a corollary, we find that the degree of the minimal polynomial for a constructible number (and therefore for any constructible length) is a power of 2. In particular, any constructible point or length is an algebraic number, but not any algebraic number can be built.







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