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The necessity to measure lands determined the first steps of Geometry. The Greek philosopher Eudemo de Rodes, of century IV B.C., one of the first historians of sciences, counts that the Egyptians measured its lands to follow the regimen of annual floodings of the river Nile. In fact, the term provém of the words Greeks geo (land) and metron (measured).
In the modern direction, Geometry is disciplines it mathematics that has for objective the study of the space and the forms in it contained.
Historical aspects. Of the old cultures of Egypt and the Mesopotâmia, Geometry consisted simply of a set of empirical rules. The Greeks, among which Euclides was distinguished, in century III B.C., had systemize all the existing knowledge on the subject and had established its beddings in a set of axioms of which, according to deductive principles, if they got excessively the results. The quarrel of the principles of Euclidean Geometry led to the construction, in century XIX, of new geometric systems, called not-Euclidean geometrias, and discharged in the generalization of its methods and its application the more abstract spaces each time.
Elements and geometric figures
Geometry, in any of its boardings, presents a series of common primary elements. The concepts, point are distinguished immediate in this level, line (straight line, curve etc.), surface, segment and others that, agreed, form all the geometric figures. The geometrias descriptive and projetiva classic if occupy of the representation and the properties of the figures and its projections. Some basic geometric figures are distinguished in them.
Polygons. A polygon of n sides (being bigger or equal n the three) is defined by n commanded points of a plan (A1, A2,… An) calls vertices, between which it cannot have three colineares consecutive. The n segments (A1A2, A2A3,… AnA1) is called sides, and its intersection forms the vertices.
Polygon is a closed line, that is, divides the plan in two regions, one interior and another exterior to the polygon. The difference between them is that any half-straight line whose origin is a point in the interior region cuts at least a side of the polygon, what does not happen necessarily if the point will be in the exterior region. In function of the number of sides (or angles), the polygons are classified in triangles, quadrilaterals, pentagons, hexagons, heptágonos, octógonos, eneágonos, decagons etc.
Triangles. The polygons of three sides receive the name from triangles. They can be equilateral (when the three sides are equal, that is, have length the same), isosceles (two equal sides) or scalene (three different sides). In accordance with the measure of its angles, the triangles are divided in acutângulos (if all the angles are minors that 90o), rectangles (if one of the angles is straight, that is, equal 90°) and obtusângulos (if one of its angles is greater that 90°). The three angles of a triangle always add 180°.
Two triangles are similar when the lengths of the sides of one of them are respectively proportional to the sides of the other. For this, the necessary and enough condition is that the two triangles have the three respectively equal angles. In the truth, as the addition of the angles is always 180°, it is enough that one of the triangles has two respectively equal angles the two angles of the other triangle to be similar.
The property of the similarity allows to demonstrate some referring laws to the rectangular triangles. The ABC triangle is considered. Triangles ABC, ABP and the ACP -- where AP is the height of hipotenusa (opposing side to the straight angle) - they are similar for having the equal angles. Consequently, its sides are proportional.
From there two important theorems are inferred: a leg (side that is not hipotenusa) is the proportional average between hipotenusa and the projection of it on this (theorem of the leg, of Euclides); e the height of hipotenusa is the proportional average between the two parts where it divides this last one (theorem of the height, of Euclides). When applying itself repetidamente the theorem of the leg, deduces the basic theorem of Pitágoras: the addition of the squares of the legs is equal to the square of hipotenusa.
According to another demonstration of the theorem, the area of the four rectangular triangles is equal to the area of the square of side to less the area of the side square c~- b:
The perpendicular to each one of the sides of a triangle that passes for the opposing vertex calls height. The three heights of a triangle pass for one same point, called ortocentro. Bissetriz is the segment, contained in the triangle, that divides the internal angle in two equal angles. The point of intersection of the bissetrizes of the three angles calls incentro, for being the center of the enrolled circumference in the triangle.
The medium ones, or segments that join each vertex with the medium point of the opposing side, are cut in the barycentre, or center of gravity of the triangle. Finally, the mediatrizes of the sides (perpendicular that they pass for the medium point of each side) are cut in circuncentro, or center of the circumscribed circumference to the triangle. These points if represent graphically as it shows the figure.
To calculate the area of a triangle, half of one of its sides for the corresponding height to this side is multiplied it. If the lengths of the three sides, b are known and c, can be calculated the area for the formula of Heron:
where p is the semiperimeter of the triangle, that is, the half of the addition of the lengths of the three sides.
Regular quadrilaterals and polygons. All polygon is called quadrilateral four sides. The quadrilaterals are classified in keystone parallelograms, trapezes and, as they have two, one respectively or no pair of parallel sides. The parallelograms can be squared, rectangular, losangos (or gaps) and rhomboids. If the angles between the sides will be straight, the parallelograms will be called squared and rectangular. If they will not be, they will be called losangos and rhomboids. The squares and losangos have the four equal sides. In the rectangles and rhomboids the sides are equal two the two.
To determine the area, or surface (s), of a parallelogram, is multiplied it base (any side) for the height (distance between parallel sides). The area of losango also can be determined as the half of the product of its diagonal lines (diagonal it is the straight line segment that joins two not consecutive vertices).
The trapezes can be isosceles, when the two not parallel sides are equal, scalene, when they are different, and rectangles, when they have two straight angles. The area of a trapeze is the half of the addition of its bases (parallel sides) multiplied by the height (distance between the bases).
A polygon is to regulate if all its angles, as well as its sides, will be equal. The regular polygons if characterize for the fact to be able to be enrolled or circumscribed to a circumference. The perpendicular to any of its sides that pass for the center of the polygon (and that it coincides with the ray of the enrolled circumference) calls apothem. When multiplying the apothem for the half of the perimeter (it adds of all the sides), gets it area of the polygon.
Circumference. Circumference is called all plain and closed curve whose points are equidistant of an interior point called center. The portion of the plan that is in the interior of a circumference calls circle. The segments that join the center with any point of the circumference call rays, and the ones that join colon any of the circumference, ropes. Ropes of big length, that they are the ones that passes for the center, are called diâmetros, each one of them resultant also of the union of two rays straight-line.
The linear length of a circumference is equal the two times its ray, multiplied for a called irrational number, that valley 3,14159… In the calculations, costuma to leave indicated it without substituting for its approach value. The area of the circle is the product of the number for the square of the ray.
The calculation of -- that any is the quotient enters the linear length of a circumference and its diameter -- if it makes from the succession of perimeters of regular polygons (of three, four, five, six etc. sides), enrolled and circumscribed in circumferences of equal ray the 1. The two successions of perimeters (of the enrolled polygons and of the circumscribed ones) have as have limited the number, that would be the perimeter of a polygon with a so great number of sides that would coincide with a circumference. Archimedes demonstrated that the value of was understood enters.
By means of computers and series of powers, already he was possible to calculate one hundred a thousand decimal houses of. For the fifteen first houses we have equal the 3,141592653589793… The number is irrational, for not having dízimas periodic, and transcendente, for not being solution of no algebraic equation with entire coefficients.
The portion of circle understood between an arc and two rays calls circular sector, and limited for an arc and a rope, circular segment. The areas of these surfaces by means of the formulas are calculated that if follow to the figure.
Polyhedrons. The solids limited for plain polygons call polyhedrons, that are called regular when its faces are equal regular polygons. It has five types of regular polyhedrons: with triangular faces (the tetrahedron, four faces; octaedro, with eight; e the icosahedron, with twenty); with square shaped faces (the cube, or hexaedro, with six faces); e with pentagonal faces (dodecaedro, with 12 faces).
Polyhedral surface is called that one formed by a finite number of polygons or faces, and that it satisfies two conditions: (1) each side of a face also belongs to one another face, and alone to one, contiguous one; (2) two contiguous faces do not belong to one same plan.
A surface with these characteristics is closed, a time that is demarcated by the polygons and allows to distinguish between interior and exterior points it. Polyhedron is the set of the interior points to a polyhedral surface. The common points the three or more faces (and therefore the three or more sides of the polygons that composes the faces) call vertices. Each side of polygon, common to the polygon of a contiguous face, calls edge.
Prism is called all polyhedron that has two equal faces and parallel bars of n sides (bases) and n lateral faces in parallelogram form. If the lateral faces will be perpendicular to the bases, the prisms are called straight; in case that contrary, they are called oblique. If the bases of a prism are parallelograms, this prism are a parallelopiped.
If the polyhedron is formed by a polygon of n sides (base) and n triangular faces with a common vertex, calls pyramid. The common point is the vertex of the pyramid, and its distance until the base, the height. If the pyramid will be cut by a parallel plan to the base, gets two polyhedrons: one another pyramid, minor, and a trunk of pyramid. The two faces parallel bars of the pyramid trunk are similar polygons of n sides, and the n lateral faces is trapezial.
Sphere, cylinder and cone. All closed surface formed by equidistant points of an interior point called center is a spherical surface. This geometric figure also can be defined as the surface generated for a circumference that turns having one of its diâmetros as axle. Sphere is the set of the points of a spherical surface and of the interior points it. The intersection of a sphere with a plain one forms a circle, that will be maximum if the plan to pass for the center of the sphere, and lesser the more in such a way distant to pass of the center.
Cylinder is a body generated for a rectangle that turns, having one of its sides as axle. The cylinder is demarcated by two circular bases and a lateral surface.
If a funny rectangular triangle having as axle one of its legs, it generates a cone, that is demarcated by two surfaces only: the circular base and the lateral surface.
The objective of analytical Geometry is to study the geometric problems by means of resources of the mathematical analysis. The method if bases on the principle according to which all point of a plan can be defined by a commanded pair of real numbers that represent in the distance of this point to the origin. In the system of cartesian coordinates (thus called in homage its creator, René Descartes), the origin if points out in the intersection enters two perpendicular axles called axle of the abscissas (or axle of the x) and axle the commanded ones (or axle of the y). Quadrants are called the four regions of the plan delimited by the two axles.
Abscissa of the point is in the distance between it and the axle of commanded, and has positive or negative signal in function of the semiplan where if it finds (to the right of the axle of the commanded ones, positive; to the left, negative). Analogamente, is defined commanded it of the point as in the distance between it and the axle of the abscissas, that also can have positive or negative signal (above of the axle of the abscissas, positive; below, negative). The point of intersection of the two axles calls origin the reference system, and its coordinates are (0, 0).
When representing the point, that is a geometric being, by means of a pair of cartesian coordinates, who is an algebraic being, plain analytical Geometry becomes possible to represent lines straight lines and curves by means of equations.
Applications of analytical Geometry. The concept of co-ordinated e, in particular, of cartesian coordinates, invaded all the domínios of the mathematics and sciences applied by means of the notion of graph of a function. Currently, such graphs are modified, corrected or extended in the screens of the modern computers, becoming automatic the analysis of a type any of function that admits graphical representation.
Equation of a straight line. In a system of cartesian coordinates, a line straight line can be represented by an equation that estabeleça a true relation for any pair of co-ordinated that it defines a point of this straight line. For example, all the co-ordinated points of (0, y) are on the axle of the commanded ones (axle of the y) and have equal abscissa the zero. Thus, the axle of the commanded ones is a straight line defined for the equation x = 0. In the same way, the axle of the abscissas (axle of the x) is a straight line defined for the equation y = 0.
Any another straight line that pass for the origin of the cartesian system has the reason between constant x and y and can be defined by the equation y = MX, where m is a constant. The reason between x and y also can be express for the equation ax + by = 0.
Any straight line can be represented by a gotten equation in the way that if it describes to follow. A point (x1, y1) of the straight line is overcome and, from it, a new pair of perpendicular, parallel axles is traced to the axles of the cartesian system, with the origin in the coordinates (x1, y1). If the coordinates that say respect to this axle are (x', y'), the equation of the straight line will have the seen form already, but now with the new 0 variable substituting the presented ones, that is, ax' + by' = 0. As x' = x - x1 and y' = y - y1, the equation of the straight line, in terms of the original coordinates, is a general linear expression equaled the zero that includes a term constant (c).
It is had, then, the equation:
(x - x1) + b (y - y1) = 0
ax + by + c = 0
To verify if data point P of co-ordinated (x, y) it is contained in one determined straight line, it is enough to substitute 0 variable x and y of the equation for the values of the coordinates of P. If the equality will be satisfied, the point belongs to the straight line. In case that contrary, it does not belong. In other words, the express equation co-ordinated the necessary and enough condition that a point of (x, y) must satisfy to belong to the straight line that it defines.
Analytical Geometry also allows to verify the conditions of parallelism, perpendicularismo and intersection of two straight lines, calculation of in the distance between colon, among others applications. All line straight line has an equation with the given form and all equation with this form represents a line straight line. The equations of this type, where the powers of x and y are unitárias (equal to one), are called linear equations or first degree.
Conical. A conical surface is generated by a straight line when turning in lathe to a axle that cuts it. The intersection of the generators of this cone of revolution with plans that do not pass for its vertex generates known curves as conical sections. In accordance with the type of intersection, the conical surfaces can be: a circle, if the plan will be parallel to the base of the cone; an ellipse, when the plan cuts to all the generators of the cone; a parabola, when the plan is parallel to an only generator; e one hipérbole, when he is parallel to the axle and it cuts two generators.
As well as the linear equations they represent a straight line, the equations of as degree of Ax2 type + Bxy~+ Cy2 + Dx~+ Ey~+ F~= 0 represents the conical ones. , B, C, D, and and are constants that define conical a specific one and depend on the eccentricity, of the position of the focus and the line of direction. The general equation sample that a conical one completely is determined when knows five of its points. Of it equations of ellipses can be deduced, parabolas or hipérboles, depending on the lesser, equal or bigger eccentricity to be that the unit.
These curves also can be defined as geometric places -- set of points that satisfy a property definitive -- e its equations are gotten from these definitions. To arrive at the equations of the conical ones, the basic processes of analytical Geometry for the calculation of in the distance are used between colon, especially the theorem of Pitágoras on the relations between the sides of a rectangular triangle.
Circle is the geometric place of the points that are equidistant of a given point, called center (c). Being C (, b) the center, and r the ray, the equation of the circle will be (x~-) 2 + (y~- b) 2 = r2
Ellipse is the set of points such that the addition of the distances of any of these points colon fixed called focos it is constant. Its equation is:
where and the b is the semiaxles biggest and lesser of the ellipse. The eccentricity of the ellipse is defined by and is always lesser that 1.
The geometric place of the points of the plan whose difference of distances colon fixed (focos) it is constant, calls hipérbole, whose equation is
Hipérbole has two asymptotes -- curves that if approach indefinitely to a straight line -- whose equations are.
The eccentricity of hipérbole is defined of analogous form to the one of the ellipse, and in this in case that it is always bigger that 1.
The parabola is formed by all the equidistant points of a point, or focus, and of a called straight line line of direction. In the distance p between the focus and the line of direction calls parameter the parabola. In simpler its in case that, the parabola has as equation y2 = 2px.
The equations of the ellipse, hipérbole and the parabola had been gotten for the particular case where the axles of the coordinates are the axles of symmetry of the conical one that it originated them (the parabola has one alone axle of symmetry, that is taken as the axle of the x; in this in case that, one assumes that the curve passes for the origin). If the conical one does not have these axles as reference, its equation has the designated form previously already
Ax2 + Bxy~+ Cy2 + Dx~+ Ey~+ F~= 0
By means of a translation and a rotation of the axles of coordinates, this equation can be transformed into another one, of reduced form.
Differently of Discardings, that analytical Geometry based on the numerical correspondence of the localization of the points of the geometric figures, Gaspard Monge used a geometric treatment purely to establish the correspondence enters the points of the three-dimensional space and the points of two perpendicular plans between itself, that they form a reference dihedron. Thus, each point in the space is projected ortogonalmente on each one of the two plans of the dihedron, originating the projections horizontal and vertical.
By the method of Monge, a figure of the three-dimensional space is studied by means of its projections in the plans of the dihedron. Being these plans struck one on the other for the rotation of one of them around its intersection (called line land), the projections appear drawn in one alone plan, call épura.
This conception reduced to a small number of abstract principles and invariable all the geometric operations that appear immediate in the usual representations of intersection with plan, spherical, conical cylindrical surfaces with etc., in the perspective, the drawings technician, the study of shades and too much graphical representations. In its Géometrie descriptive, Monge gave some examples of the job of the projections in the demonstration of the properties of the figures of three dimensions, beyond launching the seed for the modern studies of the transformations of the geometric figures. Between the disciples and continuators of Monge they are J.D. Gergonne, Brianchon, Carnot and Poncelet. Gino Loria imagined one third plan, of profile in relation to the two plans of Monge, approaching with this the method of descriptive Geometry to the one of the three-dimensional analytical Geometry of Clairaut.
Classic Geometry has enters its basic principles the fifth postulate of Euclides, or postulate of the parallel bars, that deserved speculations of geometricians of all the times. In century XVIII, Girolamo Saccheri and Johann Lambert they had formulated some hypotheses that they looked to substitute or to explain that postulate. But the work of bigger repercussion was of Legendre, that made a complete revision of the Elements of Euclides, in a version that widely was divulged in the Europe and served of base for all the courses of elementary Geometry of the Brazilian intermediate schools. The postulate of the parallel bars received from Legendre the following statement, equivalent to the one of Euclides: “For a given point a parallel to a given straight line can only be traced.”
In 1829 the works of Lobatchevski had been published in Russian, in which a new Geometry with the substitution of the fifth postulate of Euclides for another one was structuralized, not equivalent, what it originated a different Geometry of the Euclidean one. It was called by the author of imaginary Geometry and later it passed to be known as hyperbolic not-Euclidean Geometry.
The statement of the fifth postulate of the Geometry of Lobatchevski is the following one: “The straight lines of a plan can be classified in two groups in relation to one given straight line of the same plain: the group of that they intercept and the group of that they do not intercept this given straight line. The straight lines limits of these groups are called parallel to the given straight line.” Exemplificando: either straight line r and a point P, exterior the r. Passing for P we imagine straight lines as s, t, u, v that intercept r and other straight lines x, y, z…, that they do not intercept r. When separating the two groups, Lobachevski admitted straight lines m and n as parallel bars the r.
Therefore, while for an exterior point to a straight line in Euclidean Geometry if it only admits a straight line that does not intercept the given straight line, in lobachevskiana Geometry has a infinity of them. In other words, while in elementary Geometry the addition of internal angles of a triangle is equal 180o, in the Geometry of Lobachevski this addition is minor who 180o.
Without knowing the works of the Russian geometrician, the Hungarian János Bolyai had fond the analogous conclusions on the possibility to structuralize a new Geometry without logical contradiction. At the same time, Gauss wrote letters where she tells to have fond the similar conclusions. Thus, the honor of the creation of not-Euclidean Geometry is divided between these three mathematicians.
In 1854, Riemann presented to the world the one second not-Euclidean Geometry, known as elliptical Geometry, in which if it admits that “for an exterior point to a straight line if it cannot trace no parallel” or that “the addition of the internal angles of a triangle is greater that 180o.”
The not-Euclidean geometrias had been created without concern with the real world, in abstract way, therefore, without aiming at to a practical purpose. However, with the conceptualization of bending of surface, given for Gauss, it can be said that in the surfaces of null bending (plain and surfaces you designed) it is possible to establish a Geometry in the molds of the Euclidean one; in the surfaces of constant and positive bending (spherical surface) it is verified Geometry of Riemann, where the Euclidean straight line is substituted by the maximum circle and the plan for the spherical surface; in the surfaces of constant and negative bending (pseudo-sphere) a Geometry of the type of Lobachevski is applied.
source: ©Encyclopaedia Britannica of Brazil Ltda Publications.