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The Geometry of Perspective Drawing on the Computer. Andrejs Treibergs University of Utah Department of Mathematics 155 South 1400 East, Rm. 233 Salt Lake City, UTAH 84112-0090 E-mail address:treiberg@math.utah.edu. Abstract. We present some mathematical ideas that occur in art and computer graphics.

We touch upon the geometry of similar triangles, rigid motions in three space, perspective transformations, and projective geometry. We discuss computations behind rendering objects in perspective. We then describe vanishing points, answer how to measure distance in stages, a receding direction in Similarities and Differences of the Iraqi Constitution Essay, a perspective drawing and why a circle in three space becomes an ericksons of development ellipse when drawn in perspective. We use MAPLE generated computations and graphics to illustrate the ideas. ``I freely confess that I never had a taste for study or research either in physics or geometry except in so far as they could serve as a means of arriving at some sort of knowledge of the creativity and imagination proximate causes. for the good and convenience of life, in maintaining health, in the practice of some art. having observed that a good part of the ericksons stages of development arts is based on geometry, among others the cutting of stone in architecture, that of sundials, and that of creativity perspective in particular.´´ Artists now make amazing images using computers ( e.g., visit PIXAR.). But taking into account the complexity of all the physical laws of of development light requires extensive computation.

We'll focus on mathematical concepts and derive the basic formulas that are at the heart of all rendering machines. Management! We shall concentrate on the geometry of ericksons stages of development drawing objects, which can be described by points in space. We'll calculate the perspective transformations that locate the Our Moral Obligation Animals Essay points on the drawing. Then we'll use MAPLE's capability to draw polygons and lines to ericksons of development, show the effect of transformation. We'll apply this to show some mathematical / artistic concepts. (This note is a slightly expanded version of lecture notes [TA].) A mathematical theory of perspective drawing could only be developed when the Renaissance freed painters to depict nature in a way closer to what they observed [IW]. Tintern! The biographer Vasari (1511-74) says that the Florentine architect Filipo Brunelleschi (1337-1446) studied Greek geometry, developed a theory of perspective and undertook painting just to apply his geometry [KM]. Stages Of Development! The first treatise, Della pittura (1435) by Similarities Iraqi Constitution Essay Leone Battista Alberti (1404-72) furnished most of the rules. Of Development! Our diagram of the perspective view of the circle occurs in behaviour, his text. A complete mathematical treatment De prospectiva pingendi (1478) was given by the Italian fresco painter Piero della Francesca (1410-1492).

Leonardo da Vinci (1452-1519) incorporated geometry in his painting and wrote a now lost text on perspective Tratto della pittura. Albrecht Dürer (1471-1528) also wrote a text on the practice of geometry Underweysung der Messung mid dem Zyrkel und Rychtscheyd (1525) which was important in passing on to the Germans the Italian knowledge of perspective drawing. In it, Dürer invented several drawing machines to teach perspective. Alberti was first to ericksons, ask if two drawing screens are interposed between the viewer and analyse coca, the object, and the object is projected onto ericksons of development, both resulting in two different pictures of the Similarities and Differences US and Constitution Essay same scene, what properties do the two pictures have in common [KM2]. Alberti's Problem: What do the two projections have in common? This question prompted the development of a new subject, projective geometry whose exponent was Girard Desargues (1591-1661). Desargues studied perspective geometry from ericksons, a synthetic point of Our Moral to Protect Animals Essay view, meaning he built up the geometry from axioms about points, lines and planes. A sampling is stages given in the section on projective geometry. Our Moral Obligation To Protect! There we address the question why the perspective image of a circle necessarily the ellipse.

It can also be answered using analytic geometry methods, such as in our chapter on analytic geometry, where first, points and lines are reduced to equations. A modern deductive footing for perspective drawing was given later by ericksons stages Brook Taylor (1685-1731) and J. H. Creativity! Lambert (1728-77). A competing point of view has held by mathematicians such as René Decartes (1596-1650), Pierre de Fermat (1601-1665) and Julius Plücker (1801-1868) who studied these question algebraically. Their work spurred the of development development of algebraic geometry. Mathematical issues and history are more completely covered in [BP], [DH], [FP], [M], [OW], [PD], [SD] and [WC].

Some current popular drawing texts such as Edwards' [EB] de-emphasize the analytical approach in favor of an coca intuitive sighting method for perspective. However, complicated drawing situations require more analysis [EB], [EM]. If a student wishes to pursue linear perspective in the history and art see, e.g. [CA]. There are a number of web sites worth perusing, [DJ], [EM], [MS] as are sites discussing perspective and the internet [KJ], [XX]. The computer science of graphics is stages discussed in [BP], [FP], [PP]. Many of our illustrations were generated by the MAPLE symbolic algebra, graphics and computation system. Our code in the MAPLE programming language is available at coca, http://www.math.utah.edu/ treiberg/HS.html Differential geometric graphic applications of MAPLE are described in Oprea [OJ] and stages, Rovenski [RV] and and Differences of the US and Essay, of MATHEMATICA in Gray [GA]. Ericksons! Other graphics packages to render differential geometric objects, e.g., are Richard Palais' 3D-Filmstrip or Konrad Polthier's JavaView.

Parallel transformation of points. The perspective transformations that describe how a point in three space is mapped to the drawing plane can be simply explained using elementary geometry. We begin by setting up coordinates. A projection involves two coordinate systems. A point in the coordinate system of an object to be drawn is given by X=(x,y,z) and the corresponding in the imaging system (on the and imagination drawing plane) is P=(u,v) . If we use the standard right handed. Projecting an ericksons stages object to the drawing plane.

system, then x and y correspond to width and depth and z corresponds to height. On the drawing plane, we let u be the horizontal variable and wordsworth, v the vertical. We can measure the distances between pairs of points in the usual way using the ericksons stages Euclidean metric. If. The projection from X to wordsworth abbey, P is ericksons stages of development called a parallel projection if all sets of parallel lines in the object are mapped to parallel lines on the drawing.

Such a mapping is given by an affine transformation, which is of the form. where T is a fixed vector in the plane and A is a 3 x 2 constant matrix. Parallel projection has the further property that ratios are preserved. That is if X 1 , X 2 , X 3 and sonnet 11, X 4 are collinear points in the object, then the ratio of distances is preserved under parallel projection. Of course denominators are assumed to be nonzero. To illustrate, let's begin with an stages of development object in three space, say a simplified house. It consists of the wordsworth tintern abbey points [0,0,0] , [0,0,3] , [3.5,0,5] , [7,0,3] , [7,0,0] , [0,9,3] , [0,9,0] , [7,9,3] , [7,9,0] , [3.5,9,5] which define eight corners of a box and two gable points and [3,9.1,0] , [5,9.1,0] , [5,9.1,8] , [3,9.1,8] , [3,10.2,0] , [5,10.2,0] , [5,10.2,8] , [5,10.2,8] which define the chimney. MAPLE generated 3d plot of house.

The most frequent parallel projections are called elevations , oblique projections and isometric projections. Of Development! The elevations are just the front, top and side views of the object. Thus the projections are given by the functions. Applied to the house object, we get three views. MAPLE generated elevations. In oblique projection, which is also called Cavalier projection, the front view is undistorted, but the sections of the object corresponding to y = y 0 constant are drawn up and to analyse swot, the right depending on how far back y 0 is. If we write w = ( 2 -1/2 , 2 -1/2 ) T , the unit vector in the plane at 45 , then we may write. The vectors have been written in column form to ericksons stages of development, here to facilitate matrix multiplication, but we'll not fuss about whether a vector is a row or column and use both forms interchangeably.

Note that since w is a unit vector, lengths in the y directions are mapped to sonnet 11, equal lengths along the of development 45 line in the drawing. Indeed, putting c = 2 -1/2 , Let R denote a rotation in creativity and imagination, the plane which moves points (x,y) about the origin an angle h 0 . If a vector is written in polar coordinates (x,y)=(r cos h,r sin h) where r= 1/2 is the distance of the point to the origin and h is the angle from the positive x -axis, dist((0,0),(x,y)), and h is the angle, then using trigonometric identities the rotated point. = (r cos h 0 cos h - r sin h 0 sin h, r sin h 0 cos h + r cos h 0 sin h) cherk/ccli/bob/Rotation/Rotation.html ] and we present a modification of his approach. Without knowing that sines and ericksons stages of development, cosines are involved, it's possible to write down the sonnet 11 rotation transformation just knowing what vectors are rotated into. For example, if we are given a vector (a,b) and wish to rotate it into the vector (0,r) , then we know that r= 1/2 is the length of the vector, and the angle of rotation has to be h 0 where sin h 0 =a/r and cos h 0 = b/r . Thus. Hence R(a,b)=(0,r) as desired. A variant of oblique projection is called military projection. In this case the horizontal sections are isometrically drawn so that the floor plans are not distorted and the verticals are drawn at stages, an angle.

The military projection is given by behaviour management rotation in stages of development, the x - y plane and a vertical translation an amount z . Thus. F military (x,y,z) = ( c x - s y, s x + c y + z). We chose h 0 =-53.1 so that c=0.6 and s=-0.8 . Note that the floorplan is drawn rotated but without distortion. MAPLE generated military projection. The isometric projections are that class or parallel projections for which a round sphere projects to a round circle. Sonnet 11! The most common case is when measurements along the x -axis are plotted at ericksons stages, 30 , those along the y axis at sonnet 11, +150 and the vertical axis.

Thus if w 1 =(3 1/2 /2,1/2) , w 2 =(-3 1/2 /2,1/2) and w 3 =(0,1) are unit vectors at -30 , +30 and 90 (vertical), the isometric projection is. MAPLE generated standard isometric projection. The general parallel projection is obtained by applying a general affine transformation of the form F(X) = AX + T . If we choose T=0 and. and measure the depth according to GH(x,y,z) = x + 2y - z and then the stages picture is distorted so that none of the directions measure actual length. As mentioned before, parallel lines and proportions are preserved. MAPLE generated projection by the affine function F(X) = AX + T . We now describe the creativity perspective transformation. It is the composition of a rigid motion followed by the perspective transformation that reduces distant objects. The rigid motion moves the object in front of the drawing plane in ericksons stages of development, such a way that the eye point ep=(x e , y e , z e ) is moved to the origin, so that the vector from Our Moral Animals, eyepoint to centerpoint cp=(x c , y c ,z c ) toward which the stages eye is and Differences of the US and looking is moved to the positive y -axis and so that the vertical line through the centerpoint is drawn vertical. We shall accomplish the rigid motion by first translating the object to move the eyepoint to the origin using. Let the new vector eye to center be the displacement dp:=T(cp) . Then we rotate the stages of development object around the theory origin.

Every rotation is the ericksons composition of a rotation around the tintern abbey z -axis by an angle h , around the stages of development new x -axis by an angle k and around the y -axis by an angle l . The three angles h , k , k are called the Euler angles. We only need the Obligation Animals Essay first two rotations, and we can compute the stages of development cosines and sines involved using only the eyepoint and centerpoint coordinates. First we rotate dp around the z -axis so that (dp 1 , dp 2 ) moves to (0, r 1 ) where. Let rdp= R(dp)=(0, rdp 2 , rdp 3 ) be the rotated dp . Wordsworth Tintern! The length of rdp is the same as the length of dp which is r 2 = 1 2 + dp 2 2 + dp 3 2 1/2 = 2 2 + rdp 3 2 1/2 . The second rotation takes rdp to ericksons, (0, r 2 , 0) . Setting c 2 = rdp 2 /r 2 , s 2 = rdp 3 /r 2 , the rotation around the x -axis becomes. The composite SRT(X) = S(R(T(X))) is the desired rigid motion. The perpendicular projection is the front view or (x,z) part of the rotated object. and the depth is computed by GPpP(x, y, z)= s 1 c 2 (x - x e ) + c 1 c 2 (y - y e ) + s 2 (z - z e ) . For example, taking the eyepoint ep=(11.0,-15.0,2.0) and centerpoint cp=(3.5,5.0,3.0) projects the house so: MAPLE generated orthogonal projection.

Because light reflecting off the object travels in analyse swot coca, straight lines, the object point is seen on the drawing plane at the point where the line from the eyepoint to the object point intersects the ericksons stages drawing plane. Wordsworth! The perspective transformation is simply to of development, deduce the coordinates (u,v) on the drawing plane, which is behaviour a distance d from the origin, from the ericksons stages point X=(x,y,z) using triangles. The triangles (0,0):(0,d):(u,d) and (0,0):(0,y):(x,y) in the x-y- plane and the triangles (0,0):(d,0):(d,v) and (0,0):(y,0):(y,z) in the y-z- plane are similar. It follows that. Similar triangles used in computing perspective projection. We have been using d=1 from which the perspective transformation may be calculated.

This is just the x-z- coordinates of the perpendicular transformation divided by the depth ( y -coordinate.) Using the same eyepoint and centerpoint as for the perpendicular transformation, we plot the house by perspective transformation. MAPLE generated perspective projection. Perspective transformations have the property that parallel lines on the object are mapped to pencils of of the US and Iraqi Essay lines passing through a fixed point in the drawing plane. To see this, note that each line in the rotated object lies in the plane passing through the line and through the eyepoint. Of Development! This plane intersects the drawing plane in a line hence the Obligation Essay image of a line in space is a line in of development, the drawing.

Any parallel lines in the object are parallel to the drawing plane or not. If the lines are parallel to the drawing plane (the y -coordinates on the line are constant) then the division by the depth (the y coordinate of the wordsworth rotated object) is division by constant. Thus the formula reduces to a constant multiple of the numerator which is an affine transformation that maps parallel lines to of development, parallel lines. If the parallel lines are not parallel with the creativity and imagination drawing plane, then their image on the drawing plane passes through a fixed point, called the stages of development vanishing point. The easiest way to see this is to consider a pair of points on two parallel lines that travel together away from the drawing plane. Imagine that a wire of fixed length connects the points. Because the pair can get farther and farther from the drawing plane without letting go the behaviour wire, their perspective images get closer and closer in the drawing since the denominators are getting large whereas the difference in their (x,z) directions are bounded. Imagine the cliché of two rails of a track converging at of development, infinity. For general choices of the eyepoint and sonnet 11, centerpoint, the ericksons parallel lines originally in the x , y and z -axis directions are not rotated to a position parallel to the drawing plane. Thus these three directions each have their own vanishing points.

This is called three-point perspective. The three points may not so easily seen since they may not be within the US and cone of vision that limits the width of our view. To illustrate one and two point perspective we change our eye and center points to guarantee some parallel lines parallel to the drawing plane. Let us consider specific choices of eyepoint and centerpoint for which some of the stages of development objects axes are parallel to the drawing plane. Let the eyepoint ep=[6.0,-15.0,2.0] and the centerpoint cp=[6.0,5.0,2.0] . Because dp=[0,20,0] no rotation is necessary. The x and z -axes are parallel to the y=1 plane.

The perpendicular projection is just the front elevation and the perspective view has one vanishing point corresponding to the y -axis direction. The vanishing point is indicated (it is the behaviour management theory position of the centerpoint.) MAPLE generated front elevation and one point perspective projection. Let the eyepoint ep=[16.0,-15.0,2.0] and ericksons, the centerpoint cp=[6.0,5.0,2.0] . This time dp=[-10,20,0] so that the sonnet 11 only rotation is of development about the z axis. The z -axis is parallel to Our Moral Obligation to Protect, the y=1 plane. The perpendicular projection is of development now a corner elevation and the perspective view has two vanishing points corresponding to the x - and y -axis directions.

The centerpoint is indicated. MAPLE generated orthogonal and two-point perspective projection. Another pair of views come by taking the eyepoint ep=[-6.0,5.0,9.0] and creativity, the centerpoint cp=[6.0,5.0,2.0] . This time the horizontal lines are parallel to the drawing plane but the vertical and receding lines are not. Therefore the vanishing points correspond to the vertical and receding directions. Another MAPLE generated orthogonal and two-point perspective projection. Using vanishing points and measuring points. The maximum view that the eye can take in is a cone of about 30 about its axis (the cone of vision. ) It is possible for of development, the computer to plot points outside the cone of coca vision, but such a drawing has a distortion like a fisheye camera photo. Thus usually both vanishing points aren't visible in the same scene, as in stages of development, this computer-generated view of a cube with parallel lines. MAPLE generated perspective view of Obligation to Protect unit cube showing vanishing points. How do we locate the ericksons stages of development vanishing points in the drawing?

The vanishing points for tintern, the x -axis and y axis parallels are always on the horizon line. If d is the distance from ericksons stages of development, eye to drawing, then the two vanishing points in the drawing for x -axis and analyse, y -axis lines are on lines which meet at the eyepoint at 90 . This is easiest to see by imagining the top view. MAPLE generated perspective view and construction of vanishing points from top view. The drawing plane is a distance d from the eyepoint E . The rays emanating from the ericksons stages of development eyepoint at analyse swot coca, right angles parallel to stages of development, the y and x -axes are the line segments EA and EB . To Protect Animals Essay! A is the ericksons stages u -coordinate of the y -axis vanishing point V1 and B is the u -coordinate of the x -axis vanishing point V2 . The v -coordinates are v=0 which corresponds to the eyelevel and horizon line. A circle whose center is on Similarities Iraqi, the drawing line and passes through the eyepoint intersects the drawings line at two points, say A and B for which AEB is a right angle. Ericksons Of Development! This is the geometric fact that a diameter AB subtends an angle 90 from tintern abbey, any point E on the arc AEB . MAPLE generated perspective and top view of vanishing points and of development, their construction.

How do we measure distances in the receding direction? The idea is to figure out sets of parallel lines which transfer measurements along the baseline, a line parallel to behaviour management, the drawing plane, to the receding line. The projective transformation may scale but not distort distances along the baseline. To see how this works, consider the top view of a 3 x 3 square. MAPLE generated parallel sets of ericksons stages of development measuring lines. The baseline is the Similarities and Differences Iraqi Constitution Essay line af . The baseline has equally spaced points a , b , c , o , d , e , f in order. Ericksons Stages Of Development! The spacing is the Obligation to Protect Animals Essay same as along the square o , c' , b' a' and o d' , e' , f' . The square has been rotated an angle foX . The parallels to oX and the parallels to oY are along the two sides of the square. Their perspective images converge to two vanishing points.

The other two sets of lines are called measuring lines. One family are the parallels oP , dd' , ee' , ff' measure the oX side of the square and the other set of parallels oQ , cc' , bb' , aa' measure the oY side of the square. This is what it looks like in perspective. MAPLE generated top and perspective views showing parallel measuring lines, vanishing and measuring points. Because the lines connect equally spaced points, the triangles fof' and aoa' are isosceles. Ericksons! This means that if the line oW is Our Moral Obligation to Protect Animals chosen so that the angle foW bisects the angle foX , then the lines oP and oW are perpendicular and the angle Similarly, the triangle aoa' is isosceles so the angle But since the total angle of a triangle is and since they are supplementary, It follows that so These angles may be easily constructed on ericksons of development, the circle. MAPLE generated top view for constructing vanishing points and creativity, measuring points. As before, we locate the eyepoint E and centerpoint O on the drawing and let line EF be parallel to AB . Ericksons Stages Of Development! The sides of the box from the previous diagram are along the rays EA and analyse swot, EB so that the vanishing points in the drawing are located at A and B . Since a line intersects parallel lines so that opposite angles are equal, Draw a circular arc with center B and radius BE until it meets the drawing plane line AB at M1 . EBM1 is ericksons stages of development a similar triangle to fof' so Thus M1 is the point where the eye views the Similarities US and Iraqi Constitution first family of measuring parallels; thus M1 is the vanishing point for this set of stages parallels. Similarly, so that if one draws a circle with center A and radius AE then this circle intersects the picture plane line at M2 . Now the angle so that the analyse coca point M2 is the vanishing point for the second family of measuring lines. Now we can use the measuring lines to mark off equispaced points on ericksons, the perspectively receding lines.

MAPLE generated measuring lines viewed in perspective and their construction viewed from the abbey top. We mark off equally spaced points a - f on the baseline as before. The lines oV1 and oV2 correspond to the bottom edges of the box. Ericksons Of Development! Moving up one unit from o gives the upper corner of the coca box and the rays to V1 and ericksons stages of development, V2 give the upper front edges. Now, the first measuring family was chosen so that the intersections with the right front edge were points spaced the same distance apart as on the baseline. Thus, where the lines dM1 , eM1 and fM1 intersect oV1 are the equally spaced points d' , e' f' in Similarities and Differences of the US and Iraqi Constitution Essay, the perspective drawing. Similarly, the measuring family of parallel lines for the left side of the box have a vanishing point at M2 . The intersection of aM2 , bM2 and cM2 with oV2 correspond to the equally spaced points a' , b' , c' on the line oV2 . The rest of the ericksons box is management theory constructed by extending the ericksons vertical lines up from a' - f' . If one is using two-point perspective, these are truly vertical in the u-v- plane. Otherwise you have to use the vanishing point corresponding to analyse, the vertical family (which is probably way below the picture.)

MAPLE generated measuring lines viewed in of development, perspective and their construction viewed from the top. Analytic Treatment of the Perspective View of a Circle. One is taught in drawing class, that circular objects in three-dimensional Euclidean Space are drawn in behaviour, perspective as ellipses. Ericksons Stages Of Development! The usual construction is to draw a square around the circle, and then project the perspective view of the US and Iraqi Essay square by finding its edges using the vanishing points and measuring points, the center by drawing the diagonals, and stages of development, then sketching the tintern abbey projected circle by drawing it tangent to the projected square. A beginner will sometimes make the mistake of trying to make the tangency points the same as the endpoints of the axes of the ellipse, but they are not the ericksons stages same as seen in the p. 17 figure. But why is the creativity and imagination image exactly the ellipse and not some other closed curve? We shall answer this question by figuring out the equation of the stages of development image of the management theory circle on the perspective drawing. We'll be using the methods of analytic geometry, where curves are represented by equations. Thus we shall describe a circle in three space by of development describing it as the locus of points satisfying certain equations. We then compute the behaviour theory corresponding perspective locus in terms of the Cartesian coordinates of the drawing plane. Ericksons Stages! Finally, after some simplification, we will be able to recognize the curve as an ellipse.

The conic sections in the plane are given as the locus, that is the set of all points (u,v) in E 2 which satisfy a quadratic equation of the form. au 2 + 2buv + cv 2 + eu + fv + g = 0, where a , b , c , d , e , f , g are constants. This can be deduced from the geometric description of the conic section as the intersection in three space of a plane with a right circular cone. All possible conic sections arise this way including degenerate ones such as lines and points and creativity, the empty set. For example if a=b=c=0 then.

is the of development equation of creativity a line and if a=c=1 , b=0 , d=-2u 0 , e=-2v 0 , g=-u 0 2 -v 0 2 then. is satisfied only by one point (u,v)=(u 0 ,v 0 ) whereas. has no real solution at all. On the other hand if the discriminant. is negative, then the conic is a hyperbola, if D=0 the ericksons conic is a parabola and if D is wordsworth tintern positive the conic is an of development ellipse. The easiest to see are the canonical conic curves given by the formulae. Of course if a=b the ellipse is a circle. Now let's see what a projective transformation looks like analytically.

For simplicity, we assume that the set is creativity and imagination located in ericksons, front of the observer (all points of the circle satisfy y0 .) Then the horizontal and Similarities US and, vertical coordinates of the drawing plane (points which satisfy y=1 ) are. where (x, y, z) runs through all points of the original set. Now suppose that we consider a circle in space with center (x 0 , y 0 , z 0 ) and radius r and which lies on a plane not parallel to the drawing plane. By a rotation around the y -axis, we may arrange that the intersection line of the ericksons stages of development circle plane and the drawing plane is horizontal. In other words, the equation of the plane through the center of the circle sloping away from the drawing plane with slope m is given by. To be able to tintern abbey, see the circle, we require that the eyepoint (0, 0, 0) is not on the plane of the circle, which means z 0 does not equal m y 0 . The circle also lies on the sphere of radius r centered at (x 0 , y 0 , z 0 ) , which has the equation.

The circle is the collection of points satisfying both (3.) and (4.) These are projected using (2.) to the drawing plane. By substituting (3.) into (4.), We are trying to see how these equations relate u to v . Using (2.), we substitute in the equations (3.) and (5.) Substituting into equation (5.) and multiplying by (v - m) 2 yields. Multiplying out and ericksons stages of development, collecting factors of u 2 , uv , . yields. Thus (u,v) satisfy a quadratic equation in the plane. The discriminant is. Since the eyepoint is not on the plane of the circle z 0 - m y 0 0. Since the circle is in front of the y=0 plane, the point (x 0 , 0, z 0 + m y 0 ) which is both in the y=0 plane and on the circle plane is can't be on the circle, in fact it is farther from the center than any point of the circle, hence. Thus D 0 and the locus is an tintern abbey ellipse.

Perspective view of the circle. Here is a diagram from Alberti's treatise. The square that surrounds the stages circle projects to a trapezoid. The circle itself projects to an ellipse which is tangent to all four sides of the trapezoid. Behaviour Management! Observe that the left and right endpoints of the axes of the stages of development ellipse where the ellipse is widest occur below the tangency points. But be careful when drawing the ellipse which is not centered on the eyepoint to centerpoint line! The original impetus to projective geometry came from perspective drawing. Alberti's textbook Della Pittura (1435) formulated new questions that tempted mathematicians to study new questions beyond those addressed by the Greeks. Sonnet 11! If two artists make perspective drawings of the same object, their drawings will not be the same, for example because different parts of the ericksons stages of development object will be closer to each of the the two artists. Creativity! But what properties of the drawings remain the same? (Diagram of Alberti's question.) The perspective projection, which takes points X of the object which are in three space and plots them as points P on the drawing plane.

Let us write this. It has the property that points are mapped to points and lines to ericksons, lines. However, parallel lines in three space which are not parallel to the drawing plane must be drawn to converge at their vanishing points. Thus the Similarities Iraqi Constitution Essay correspondence between lines and points in three space and lines and stages of development, points on sonnet 11, the drawing isn't perfect. Thus if L 1 and ericksons of development, L 2 are parallel lines in three space then f(L 1 ) and f(L 1 ) are lines which intersect at V their vanishing point. However L 1 and L 2 don't intersect at any point. Management! In the ericksons stages of development diagram, lines AB, CD and E'V' are parallel. Their projections A'B', C'D' intersect at a point V' which is called the sonnet 11 vanishing point since it has no corresponding point in three space. The solution was proposed by Girard Desargues (1591-1661) a self educated man who worked as an architect after leaving the army.

His opus with the of development ponderous name, Broullion project d`une atteinte aux événemens des renconteres du cône avec un plan, (1693) which describes projective methods in geometry went unnoticed. Jean-Victor Poncelet (1788-1867), an engineer in Napoleon's army reworked the theory in Traité des proprietiés projectives des Figures (1822) while a prisoner of war in Russia in 1813 [KF]. Similarities Of The Iraqi Essay! This elevated Desargues work in projective geometry to one of the ericksons success stories of synthetic geometry, whose merits versus analytic geometry were being debated at the time. We sketch two theorems from projective geometry. For a more rigorous treatment, the reader should consult any of a number of texts, such as O'Hara Ward [OW] or Wylie [WC]. To complete the correspondence, Desargues introduced ideal points, called points at infinity one for each set of parallel lines. The points at creativity and imagination, infinity don't contradict any axioms. They function as a convenience since now every pair of lines intersects at one point, the case of parallel lines does not have to be treated as an exceptional case. The following is now called Desargues' Theorem of Homologous Triangles. Theorem.

Suppose there is a point O and triangles ABC and A'B'C' in the plane or three space. If they are projectively related from the point O , that is, the triples , and ericksons stages of development, are all collinear. Then the points of creativity and imagination intersections of the ericksons corresponding sides AB and A'B' , AC and sonnet 11, A'C' and BC and B'C' (or their prolongations) are collinear. Conversely, if the three pairs of ericksons of development corresponding sides meet in three points which lie on one straight line, then the creativity and imagination lines joining corresponding vertices meet at one point (are projectively related.) The proof is easier for the case that the triangles are not coplanar. Ericksons Stages! See Dörrie [HD] or Meserve [MB] for proofs. Diagram of Our Moral Obligation Animals Desargues' Theorem of Homologous Triangles. To see how we may use projective geometry directly to argue that the perspective image of a circle is an ellipse, we use a theorem due to Blaise Pascal (1623-1662).

Pascal, who was urged to investigate the relationship between projectivities and conics by Desargues, published his Essai sur les Coniques when he was sixteen. Stages! Although he didn't prove the tintern converse part, the theorem is known as Pascal's Hexagon Theorem. Theorem. Let a hexagon be inscribed in a (nonsingular point-) conic. Then the three points of intersection of pairs of opposite sides are collinear. Conversely, if the opposite sides of a hexagon, (of which no three vertices lie on a straight line) intersect on a straight line, the six vertices lie on a non-singular point-conic. Diagram for Pascal's Hexagon Theorem. Pascal's Theorem may be used to stages, deduce that the perspective image of a circle is an ellipse. Thus if c is the circle and sonnet 11, f(c) is its image in a perspective drawing relative to ericksons stages, the eyepoint O , then we have to show that if any six points A, B, C, D, E, F are chosen on f(c) so that no three of them lie on a straight line then the pairs of opposite sides intersect in tintern abbey, collinear points. Then by the converse of Pascal's Theorem, the six points lie on a nonsingular point-conic.

But since five points determine a conic, the sixth point which may be any general point of f(c) must be on the same on ericksons of development, the conic. It follows that no matter which six points are chosen, they lie on the same conic, thus f(C) is (part of) a single point-conic. One argues that f(c) is bounded and nondegenerate thus can only sonnet 11 be the ellipse. But the six points are in perspective correspondence to points A', B', C', D', E', F' on c which is a circle, hence a point-conic. Therefore, by Pascal's Theorem, the stages pairs of opposite sides ( A'B' and and imagination, E'D' ), ( B'C' and F'E' ), and ( C'D' and A'F' ) intersect at points P', Q', R' respectively, which are collinear in the plane of c . Their perspective images P, Q, R in the plane of F(c) must also be collinear since the perspective image of stages of development a line not containing O is a line. Moreover the planes OA'B', OE'D' contain the edges AB, ED , resp., since they are perspective to each other, and thus the planes intersect along the creativity line OP'P . In other words, the point P is the intersection of the edges AB and ED . Of Development! Similarly Q is the intersection of the edges BC and FE and R is the intersection of the edges CD and BF . Similarities Of The US And! Thus P, Q, R are collinear and stages of development, we are done.

Without going very deep into computer science complications, we explain something about the Similarities of the US and Iraqi mathematics behind computer drawing. Computer science issues are treated, e.g. in stages, [PP]. One of the ways that a computer renders three-dimensional object is to build up the image from little constituent pieces. The object is regarded as a collection of polygons. The visual position of and imagination each little piece is computed and the polygons are drawn one polygon at a time.

The computer screen is given a Cartesian (horizontal and vertical axis) coordinate system and the polygon is drawn specifying the position of each P=(u,v) of its vertices. For example, as the three triangles red (0,0),(4,-1),(1,.5) , green (4,2),(1,-.5),(4,1) and blue (4,-2),(3,1.5),(3,-1.5) are drawn, each one covers the previous ones. If for example, we wish to draw the front elevation of the an object in space consisting of the three triangles [(0,1,0),(4,0,-1),(1,1,.5)] , [(4,1,2),(1,0,-.5),(4,1,1)] and [(4,1,-2),(3,0,1.5),(3,1,-1.5)] viewed toward the +y -axis, we have to ericksons stages of development, draw the triangles as before, because the projection is given by. This would result in an incorrect picture because one of tips of each triangle is closer to the viewer than some one of the other triangles. e.g. , the base of the first triangle (0,1,0),(1,1,.5) at y=1 is creativity farther from the viewer than the remaining vertex (4,0,-1) at of development, y=0 . Another source of error would be if polygons in the object actually intersected. Analyse Swot! To correctly render the front elevation, the triangles have to be subdivided further into ericksons of development parts and the parts in front have to be drawn on top of parts in back. The most fundamental way to depict depth in analyse coca, a picture is overlapping closer objects over farther ones. In general it is quite involved to decide if some part of the object can be seen or not.

The simple way to deal with this is to draw all polygons of the object back to front. Some of the polygons which are in back of the object eventually get completely covered up. This is called the painter's algorithm. The way it works in our MAPLE program, first we compute the distances of each point to the eye. Then a typical distance is stages of development given for each polygon, which in our case is the distance to abbey, the nearest point. Then the polygons are sorted according to their typical distances, and are rendered back to front. Our program does not try to account for complicated overlaps or intersections so will sometimes render objects incorrectly. To illustrate the painter's algorithm, suppose we render a cube.

The faces are drawn back to front, depending on the distance of each side to the viewer. Ericksons Of Development! In the example, the sky being farthest is drawn first, followed by the earth, the back face, the creativity and imagination base, the sides, the top and finally the front, eventually covering up all but two sides. Another way of rendering a three-dimensional object is called ray tracing. In ray tracing, the computer follows light rays back from the eye to a point on the object from where it figures out how intense the light is and what its color is by following back the rays which illuminate that point. This can continue for of development, several stages. At each stage the computation accounts for surface properties like shine and color and creativity, body properties such as refractive index and transmittivity. An example of ray tracing is a rendering of the same house made by the program POV-Ray TM (Persistence of Vision TM Ray-Tracer Version 3.1.) We have specified that the ericksons stages body be made of gray glass and be positioned on a chessboard. To get a sense of management theory what is ericksons stages of development state of the art in ray-tracing, visit Steven Parker's website Interactive Ray Tracing -- MPEG demo at the Scientific Computing and Imaging Institute http://www.sci.utah.edu/index.html in the Graphics and Visualization group http://www.cs.utah.edu/research/areas/graphics/ in the School of Computing at the University of Utah.

Design an object to test MAPLE's 3d capability. Be sure that your object doesn't have any symmetries, so that you can tell front from coca, back, left handed from right handed. Explore the projection for stages of development, differing values of projection . (We had projection=0.7. in the runs.) A right circular cone C whose center is the origin and whose axis is the z -axis satisfies the management equation. F(x,y,z) = x 2 + y 2 - c 2 z 2 = 0. where c0 is constant. Suppose T: E 3 -- E 3 is a rigid motion, and T -1 is the ericksons of development inverse motion. Show that the equation of the general right circular cone T(C) is. where X = (x, y, z) . Using this fact, show that the points of the intersection of the cone T(C) with the plane z = 0 also satisfy equation (1.) Our rigid motions were constructed by composing rotations around the vertical axis and horizontal axes. The resulting motion maps the vertical axis to a vertical line.

More generally, a rotation may occur around any axis. Find an expression for the rotation of an angle µ around an arbitrary unit vector (u 1 , u 2 , u 3 ) . A projection whose rays are perpendicular to the image plane is called an orthogonal projection. It has the management property that a sphere projects to the round circle. The general projection does not have this property. Suppose a test cube with side length a is projected by orthogonal projection. Consider the images of ericksons stages of development three sides incident to a corner of the cube and denote their lengths a 1 , a 2 and a 3 and let µ 1 , µ 2 and µ 3 be the angles as shown on the figure.

Show that [BP p. 35] This problem requires a little calculus. Show that if (x(t), y(t), z(t)) = (a t + x 0 , b t + y 0 , c t + z 0 ) are points on a line that recede from the management theory drawing plane ( b 0 ) as t goes to infinity then the of development perspective transformation F persp. (x(t), y(t), z(t)) converges to a point depending on the direction of the line (a, b, c) and creativity, not on which line (not on (x 0 ,y 0 ,z 0 ) .) The limit point is the vanishing point for all parallel lines going this direction and it corresponds to the intersection of the line (a t, b t, c t) through the eyepoint and the drawing plane. For one point perspective, explain why the measuring points are 45 as in the perspective view of the ericksons of development circle figure. If a perspective drawing is made of a circle on the floor, which is not centered on analyse coca, the eyepoint-centerpoint line, which direction will it tip? Can you find a graphic construction for the major and minor axes? [Answer [EM], p. 93]

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A Few General Facts about ericksons How To Write 5 Paragraph Essay ? In general, the 5-paragraph essay is behaviour regarded as the typical essay writing task. This type of essay is utilized in ericksons stages, the majority of sonnet 11, well-established examinations, like TOEFL, IELTS or SAT. Seeing as in the majority of such examinations you need to stick to a time limit when it comes to finishing the “Writing” part, it’s advisable to learn the ericksons stages of development, structure of the 5-paragraph essay by heart. This way, you’ll be able to creativity, complete the exam swiftly and efficiently. The best feature of stages of development, this format is the fact that it can be used for sonnet 11, a large variety of essays, such as Expository, Narrative, Persuasive, Cause and ericksons of development Effect or Persuasive essays. 5-Paragraph Essay Subject Examples. Here are a few of the analyse, most recurrent subjects on which students write 5-paragraph essays: Is an individual able to ericksons stages of development, memorize a life lesson from an event they weren’t part of? Is one able to learn from the errors of other individuals?

Is it moral to creativity and imagination, conduct experiments on ericksons stages of development animals? Should homosexual matrimony be legalized? Should the legislation on firearms become harsher? Should the capital punishment be fully eliminated? Should cannabis become legal? Should all students benefit from free-of-charge education? No doubt, you can write 5-paragraph essays on many other topics in addition to these examples. The Structure of a 5-Paragraph Essay. Analyse Swot Coca? The introduction is the section which lays down the outline of the entire essay. The initial phase represents the HOOK sentence. The Hook Sentence has the purpose of catching the ericksons stages of development, reader’s interest.

In general, the behaviour, Hook Sentence is a rhetorical one. Of Development? Additionally, it could also constitute a life example or an outstanding piece of information. For instance: Let’s say that your 5-paragraph essay approaches the topic of environment protection. In that case, you can come up with a sentence like: “Is it normal to live in a world of barren lands and waste?” The question above is sonnet 11 a rhetorical one. This means that no one expects a response, as the answer is evident. Short Introduction of Substantiated Arguments (1 to stages of development, 3) In this section, you should concisely present your substantiated arguments. The key is to avoid disclosing an excessive amount of information. As a piece of analyse swot coca, advice, picture this short introduction as the trailer of a film, meaning that it ought to be captivating, but it must not reveal the “STORY.” For instance: Environmental protection is ericksons essential to Our Moral Essay, preserving the well-being of ericksons stages, our planet. This is the most essential part of the whole essay; it represents your argument.

The argument will serve as the analyse swot, premise of the entire paper. Ericksons? Seeing as your essay deals with environmental conservation, your thesis could be something like: “Environmental protection is essential to averting huge natural calamities.” A small piece of advice: if you believe that the body paragraphs are not related to the thesis you’ve chosen, the sonnet 11, best solution would be to modify the thesis. Stages Of Development? The Three Body Paragraphs: 5 to 7 Phrases. This represents the “bulk” of your paper. In this part, you need to justify the perspective you’re supporting (Thesis Assertion). In general, the three body paragraphs have the following outline: Introductory Phrase (1), Substantiated Argument/Justification (3-5), Conclusion Phrase (1). The Introductory Phrase must concisely present your argument.

It shouldn’t disclose too much. Similarities Iraqi? For instance, you could say something like: “Disforestation and atmosphere contamination affect the characteristics of the atmosphere and intensify the probability of illness in addition to damaging our planet!” Substantiated Argument and Justification: This section involves particularizing the ericksons, subject, while still, most significantly, SUPPORTING THE THESIS! For instance: Materialism and egoism represent important factors which harm our surroundings, as they are responsible for analyse swot, destroying our forests and polluting our air. While initially a couple of people may benefit from this, in the long run, these factors are dangerous to the entire population. For instance, in Beijing, the atmosphere quality is so low that people are obliged to use masks to be allowed to move around the city. Stages Of Development? The Conclusion Phrase ought to and Differences of the US and Iraqi Constitution Essay, represent the contrary of the introductory one. Rather than presenting your argument, you should concisely conclude it, moving on to the subsequent one. For instance: To sum up, the deterioration of our natural assets, as well as the ericksons stages, quality of our atmosphere, does not only creativity, affect the Earth’s health but the entire humankind. KEEP IN MIND THAT ALL 3 BODY PARAGRAPHS MUST HAVE THE SAME STRUCTURE! The arguments you offer ought to be presented in the following order: The first body paragraph ought to include your second most powerful argument The second body paragraph ought to describe your poorest argument The third body paragraph ought to present your most powerful argument. To have a better idea of the structure of a 5-paragraph essay, take a look at the following table: Conclusion (3 to of development, 5 Phrases): This must reflect your introduction.

Reiterate Your Thesis (Phrase 1): You must reiterate your primary argument (thesis) in a straightforward manner. Sonnet 11? To demonstrate that your perspective is valid, you should show confidence when you rephrase the thesis. For instance: The security and endurance of our planet are highly reliant on ericksons the manner in which we choose to sonnet 11, behave towards it, and the more attentively we stimulate the procedure, the more we will profit from stages it. Providing conclusions for your substantiated arguments (1 to 3 Phrases): This part involves paraphrasing the and imagination, central ideas of your arguments in a single phrase per paragraph. In case part of your substantiated arguments are alike, you can simply merge them into a single phrase. This way, you’ll preserve an adequate organization. For instance, let’s say that one of the arguments you provided deals with restricting the utilization of resources. In that case, you can write something like “Restricting the utilization of our natural assets and enhancing their performance represent essential methods of strengthening the health of the Earth.” Drafting a Conclusion for the Hook Phrase (Facultative) A great manner of finalizing a paper is by offering something unpredicted, which may amaze the reader.

A great idea would be to devise a second hook, one which summarizes your essay in only a couple of ericksons stages of development, words. Ideally, create a rhetorical question. For instance: “The soundness of the behaviour, Earth is highly significant, and in the end, we don't want to turn our planet into a desert, don’t we?” This way, your essay will present a certain level of excitement towards the end, and the reader will ponder over ericksons of development, your assertion. Different educational establishments from all over the globe utilize different rules. Nevertheless, one of the analyse swot, most well-established criteria types is the 5 point type. This involves five different sections, namely Focus, Organization, Conventions, Style and Content. Stages? Focus: Did the student take enough time to demonstrate their hypothesis? Did they meet this target? Organization: Did the essay have a fluent style? Did the student move from one paragraph to the other in sonnet 11, a steady manner?

Did they stick to the adequate structure without deviating from it? Conventions: Was the paper written using good grammar? Were the phrases too long? Style: Did the ericksons stages of development, student utilize high-quality lexicon? Did they repeat words too often? Were the phrase structures original enough? Content: Did the student succeed in behaviour, demonstrating their argument?

Did they include coherent and accurate assertions? Did they come up with powerful arguments? How Do I Proceed If I Have No Idea How to Write the Paper? In case you still feel like your essay writing skills aren’t good enough, there is absolutely no reason to be scared! Many students encounter such issues.

Fortunately for you, Elite Essay Writers is the most reliable essay writing service you could ever hope to find! All of our essay writers are experts in drafting 5-paragraph essays. Whether you need a full essay written from stages of development scratch or simply a bit of proofreading, we’re always here to help! If you want to purchase a professional essay because you don’t feel ready to write one yourself, take a look at our review section. You’ll be amazed by of the US and Constitution, how simply and rapidly we can offer solutions to your college issues, for the right price!

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Free Essays on Pakistan Independence Day. then began to rebel against Great Britain and ericksons stages of development fight for their freedom. After the analyse Revolutionary War, America won its independence . The American flag is stages, a tangible representation of independence , another reason why people feel so patriotic towards it. Many people would argue that there are things that are. poem on further. When Key mention’s the “dawn’s early light” he is referring to an early morning; a new day and a new beginning. The “twilight’s last gleaming” suggests a very special time of day ; right before the sun is setting and management theory there is a tiny streak of light that appears in the sky. These two.

The Independence Day Independence Day is annually celebrated on July 4 and is often known as the Fourth of July. It is the anniversary of the publication of the declaration of of development, independence from Great Britain in 1776. Sonnet 11. Patriotic displays and family events are organized throughout the United States. Personal Britain “But I don’t wanna Mommy” “Mother I would greatly appreciate 5 more minutes in ericksons front of the TV because I have had a long hard day ”. Words are so interesting. They can say the same things but one set of Our Moral to Protect Animals, words can get you what you want while the of development another set can get a slap in abbey the. significant celebrations around the world. Fireworks play an important role in the Independence Day Celebrations in the US.

Though United States became independent in the year 1776, the fireworks were used for Independence Day Celebration in the US only on the 4th of July 1777. Facts about Fireworks Display. Government after getting Independence from UK. On the other hand New Zealand is following the ericksons stages of development system of Westminster model, which also remained colony of creativity, UK. Similarly, Nigeria also adopted the Presidential form of Government after getting independence from France. Of Development. Like this, Pakistan and India had also adopted.

KASHMIR CRISIS April 7, 2003 In 1947, the British Indian Empire was divided into two countries: India and Pakistan . “The partition of British India into an independent India and sonnet 11 Pakistan was a very complicated problem. The British had made no preparation for such things as the ericksons division of analyse, financial. Introduction to Pakistan and Proton. my opinion, Proton is stages, likely to achieve great success on their products when launched in analyse swot coca Pakistan . Ericksons Stages. Pakistan is situated in sonnet 11 the southern part of Asia and shares borders with China, Afghanistan and ericksons stages of development India. Pakistan is said to be located in Our Moral Obligation Essay a very globally important location and becomes the main route.

manual Corporate Govrnance Pakistan. Manual of Corporate Governance SEC Securities and Exchange Commission of Pakistan This manual is for reference only and does not constitute any legal requirement on companies, their officers, directors or auditors. This manual may be used for guidance and compliance must be ensured with. Thomas Jefferson's Impact on America. remembered in stages history, not only for he held the and Differences of the Constitution offices, it also for his belief in the natural rights of man that he expressed in the Declaration of Independence and his faith in the people’s ability to govern themselves. Jefferson left an impact on his times equal by few others in American history. From. terrorism for over half a century. Since independence in 1947, the eastern states have been under the ericksons stages of development spell of insurgents, abetted by foreign countries.

The disputed region of Kashmir is a powder keg, threatening to sonnet 11 explode into conventional and possibly nuclear war any day . India has fought four major wars. DEVELOPMENT OF PAKISTAN , AND MANAGEMENT IS A KEY PROBLEM Pakistan from its birth has faced so many problems infect every type of problem those problem can be inequality, war, economic, terrorism means we can say that Pakistan has never seen a constant peace in its time. As already discussed that Pakistan has faced. Entrepreneurship in Developing Countries Like Pakistan. but they are not ready to make changes in their existing production methods. ENTREPRENEURSHIP IN DEVELOPING COUNTRIES LIKE PAKISTAN . Ericksons Stages Of Development. If we analyze the countries like Pakistan we find that the entrepreneurial activities are more or less confined to business, trade, commerce, road transport , entertainment. L9/26/12 Thomas Jefferson Declaration of Independence Virginia, 1776 Dear Thomas Jefferson: I am writing to you, to express my opinion on the Declaration of sonnet 11, Independence . It is something that our country is based off of, so I feel as if I should be giving you my feedback on the matter. The. Pakistan dayIndependence Day (Urdu: ??? ?????; Yaum-e Azadi), observed annually on August 14, is a national holiday in Pakistan , commemorating the day when Pakistan achieved independence and was declared a sovereign nation, following the end of the British Raj in 1947. Pakistan came into stages, existence as.

wife symbolized the struggle that much of the community was facing with their government and he only found peace with his dog. Rip went hunting one day , met a man, was drinking and fell asleep for 20 years; only to management wake up to a very different America after the Revolution. Rip endured an emotional rollercoaster. in many other fields, Pakistan is lagging behind in the realm of architecture. And it has its reasons IN the ericksons stages of development third week of January 2006, the results of the CDA Headquarters design competition in Islamabad were announced by a jury that included eminent architects of the sonnet 11 Pakistan Council of Architects. What world event had the greatest impact on stages of development your life as a teenager? The India Pakistan war During the India Pakistan war families were extremely terrified as the tintern abbey places which were going to be attacked next were never known . Ericksons. For Lucy Correa being. Future of behaviour management, Democracy in Pakistan I. Quaid’s vision of Democracy II. United Nations and democracy · United Nations Democratic Fund · UN charter endorses democracy · Article 21 of Universal Declaration of Human Rights highlights democratic values . ? The Declaration of stages of development, Independence What is the declaration of independence ? It’s the tintern abbey first text to ericksons stages of development use the name of the United States of America it was signed on July 4, 1776. It was the birth certificate of the sonnet 11 American nation.

It came to of development be seen as the most succinct and memorable statement of. Objectives Resolution: 1st Prime Minister of Pakistan. Objectives Resolution 1st Prime Minister of Pakistan In office: 14 August 1947 – 16 October 1951 Preceded by Office Created Succeeded by Khawaja Nazimuddin 1st Finance Minister of British India In office: 17 August 1946 – 14 August 1947 Preceded by Office Created Born: 2 October 1896 . back-channel intelligence exchanges” between India and wordsworth tintern Pakistan , a report in The Washington Post claimed Monday. “In the aftermath of the Mumbai terrorist attacks, the CIA orchestrated back-channel intelligence exchanges between India and Pakistan , allowing the two former enemies to quietly share highly. Rhetorical Analysis of the Declaration of Independence. Thomas Jefferson, former president of the United States, in his document, the Declaration of Independence , establishes U.S. independence . Jefferson’s purpose is to declare that the stages of development American people were not going to stand for despotism. He adopts a professional tone in order to …in the U.S. Congress and. Pakistan Political Stability Since March 2007, tensions in Pakistan have been rising: the political instability surrounding both the presidential and parliamentary elections is commingling with the increase in of the militant activity within Pakistan proper, which led to around 60 suicide attacks in Pakistan. Political Party At partition in 1947 there was practically one overwhelming party in East Pakistan , the muslim league. In the elections of 1996 in Bangladesh, eighty one political parties participated.

In spite of the stages of development rise of so many political parties over coca, time, the public is essentially grouped under. plausible reason for becoming a national and very important holiday is represented by the fact that in 4 July, 1776 “congress adopts the Declaration of Independence drafted by Thomas Jefferson“(Urdang, 1996: 138) who was the third President of the United States (1801–9). PARTEA ISOTRICA It is well known that. together. First of all, both national holidays in Thailand and the United States celebrate tradition. Songkran festival is the traditional Thai New Year's Day and is celebrated from 13 April to 15 April.

It is also known as the “Water Festival” as people believe that water will wash away bad luck. Many Thais. ?Population Report Pakistan Done By: Risatul Haque Chowdhury (Risatt) ID: 1307C2**11 To: Mr. Daniel Layng Introduction Pakistan is stages, located in Southern Asia, bordering the Our Moral Obligation Animals Arabian Sea, between Iran and Afghanistan on the west, India on ericksons the east and China on the north.1 Islamabad is the new. India Muslim League who founded Pakistan and served as its first Governor-General. He is officially known in Pakistan as Quaid-e-Azam (Urdu: ???? ???? — Great Leader) and Baba-e-Qaum (???? ???) (Father of the Nation).

His birthday is a national holiday in Pakistan . Jinnah rose to prominence in. Pakistan India Relations As seen in behaviour “The Nation” newspaper. Link http://nation.com.pk/archives/May-20-2008 Pakistan -India relations since independence have been the victim of oscillations between short-lived periods of euphoria and raised expectations followed by long intervals marked by disappointment. Rhetorical Analysis of the Declaration of Independence. Thomas Jefferson, Founding Father and the third president of the ericksons stages of development United States of America, in his letter “The Declaration of Independence ” (1776) argues that the thirteen colonies must demand freedom from Great Britain. To support his conclusion, Jefferson makes changes in wordsworth between his rough draft. to arrest them in time.

For example- hundreds of terrorist are being trained at the Pakistan occupied Kashmir (POK). Stages. And in Punjab, there is no doubt that the terrorist are receiving training and weapons from Pakistan . Assam has been the most unstable state in and Differences Essay the last few decades. Assam remains the only. to the approach of warriors of the Cimbritribe in 105BC. The French National Convention declared in ericksons September 1793 that terror is the wordsworth abbey order of the day . The period 1793–94 is referred to as La Terreur (Reign of Terror).

Maximilien Robespierre, a leader in the French revolution proclaimed in 1794 that. ?1930 Gandhi leads civil disobedience On March 12, 1930, Indian independence leader Mohandas Gandhi begins a defiant march to stages the sea in protest of the British monopoly on salt, his boldest act of tintern, civil disobedience yet against British rule in India. Britain’s Salt Acts prohibited Indians from. The history of formulation of the constitution of ericksons stages of development, Pakistan begins with the Lahore Resolution in 1940. It was here that the idea of Pakistan , a separate homeland for the Muslims of India, was first outlined. It came to be known as the Pakistan Resolution. Sonnet 11. On June 3, 1947, the ericksons stages British Government accepted. Following his release Mahatma Ghandi returned to his homeland of India. Not long after his return, he began taking the lead in the fight for India’s independence from Britain.

He never wavered in his belief in non-violent protest and religious tolerance. When Muslim and creativity Hindu peoples committed acts of violence. John Locke and ericksons of development the American Political System. how significant Locke’s beliefs were, and still are. His ideas were even clearly put in the American Declaration of Independence . In Jefferson’s words, the Declaration of Independence states: “We hold these truths to sonnet 11 be self-evident, that all men are created equal, that they are endowed by their Creator. information: http://www.tandfonline.com/loi/csas20 Pakistan , the BJP, and the politics of identity Howard Brasted Adeel Khan a a a University of ericksons, New England Available online: 08 May 2007 To cite this article: Howard Brasted Adeel Khan (2002): Pakistan , the BJP, and the politics of identity, South. Pledge of Allegiance every day at school. I sometimes sing The National Anthem. These things are patriotic because I am pledging to management the American Flag and stages of development singing America's song. Also, I celebrate many of America's holidays such as: The Fourth of July, or Independence Day . An American, like myself.

poem by Mohammed Iqbal, the idea of Pakistan would become the defining force in the life of wordsworth tintern, Mohammed Ali Jinnah. Stages Of Development. Likewise, Mohammed Ali Jinnah would become the defining force in the creation of, and early life of the nation of Pakistan . Jinnah was born on Christmas day of 1876 to an upper class family. Pakistan relations with neighbor coutnries. ?Essay On Pakistan And Her Neighbours. Pakistan is situated in Southern Asia. Its geographical position has made her very important for Similarities and Differences US and Constitution big powers of the world. That is why they have always tired to influence Pakistan's relations with her neighbours. We have two big powers as our immediate neighbours.

Poverty in Pakistan. Causes and ericksons Remedies to Overcome Poverty in Similarities and Differences Iraqi Constitution Pakistan. POVERTY IN PAKISTAN CONTENTS 1. Introduction 2. Poverty Condition in Pakistan 3. Definition 4. Causes of Poverty in Pakistan * Government Policies * Poor Governance * Corruption * Unemployment * Overpopulation * Education 5. Remedies to Overcome Poverty. Thomas Jefferson’s “Declaration of ericksons of development, Independence ” and Martin Luther King’s “I Have a Dream” are texts that have a common denominator: the pursuit of liberty. Although both documents do not focus their context in the same historical moment, both have greatly influenced the creativity and imagination history of the United States. rippling in the wind. Clothing is of development, even impacted by the flag, plastered on shirts, socks, shoes, and more. Most people dwelling in America cannot go a day without seeing the symbolic “Stars and Stripes” somewhere. It has become a thing that we have culturally adopted as a whole, and even foreigners know. 80t gas steam boiler set in Pakistan.

steam boiler set in Pakistan One of the most important items of equipment aboard ships, boilers are used for numerous purposes. This article describes the behaviour management theory construction and working of a 80t gas steam boilers. ? Gas steam boiler set in Pakistan As we know in those days Pakistan faces energy crises. Educational Leadership in Pakistan. BOOK REVIEW REFLECTION A book on leadership “Educational Leadership in Pakistan : Ideals and Realities” was reviewed in group of ericksons stages, six course participants as a part of our Pedagogical Leadership (PL) course. This paper discusses the rationale of selecting the book, introduction of book and authors. USE OF CELL PHONES AND INTERNET Now days the biggest problem among teens is the use of mobile phhones and internet.Most of the sonnet 11 time we find them using internet or cell phones .Many problems rises like talking with strangers on internet through any source like facebook is stages of development, very famous these and other.

Anti-Colonial Nationalismi in British Colonial India. India. Although independence was a huge victory, it led to wordsworth tintern the emergence of rivals between different religious groups within the country both wanting control. The leader of this revolution, Mohandas Karamchand Gandhi, inspired people across the nation to fight for their independence in ways other than. Stages of Ethnicity to East Pakistan Crisis (1971) Introduction; Pakistan got independence on 14th August 1947.

The separate homeland was demanded by ericksons Muslims to live their lives according to their own way or parameters. After separation the Our Moral Obligation Essay task of ericksons stages of development, nation building in Pakistan was constrained by many. Bread for Poor or Bombs for Barrels. in 9/11. Even after more than six decades of independence , one out of every two Pakistanis is short on food, one out of every two Pakistanis is Our Moral Obligation to Protect Animals Essay, food-insecure and one out of every two Pakistanis is managing to subsist on of development less than 2,350 calories per day . In Mar 2007, there were 60 million Pakistanis. Rhetorical Analysis of the Declaration of Independence. PART I The Declaration of Independence is Our Moral to Protect Animals, considered by many to ericksons of development be the Our Moral Obligation to Protect finest piece of political prose ever written. It can be seen as a document in five parts: the introduction, the preamble, the denunciation of George III, the denunciation of the British people, and the conclusion.

We are going. Freedom at ericksons Midnight (1975) is a book by Obligation to Protect Essay Dominique Lapierre and Larry Collins. Stages Of Development. It describes the events in the Indian independence movement in 1947-48, beginning with the appointment of Lord Mountbatten as the last viceroy of British India, and ending with the death and funeral of Mahatma Gandhi. The. SIR SYED UNIVERSITY OF ENGINEERING TECHNOLOGY.( ELECTRONIC ENGINEERING DEPARTMENT CONSTITUTION OF PAKISTAn [pic] NAME :MUHAMMAD SHAHROZ AFZAL CLASS :EE-2B ROLLNO :EE-2008-094 SUBJECT :ENGLISH TABLE OF CONTENT |S NO. |TOPICS . The country attained independence after hundreds of leaders including Gandhiji pledged their life and involved in various movements and the Quit India Movement held in 1942 forced the Britishers to quit the country. Today’s generation should understand the amount of patriotism involved in such movements. India- Pakistan Relationship India and Pakistan share geographic location and are linked by political and religious issues. Since the partition of the subcontinent the Our Moral Obligation Animals two neighbors have found it difficult to stages of development have friendly relationship and any improvement seen has only been very short lived. There.

Pakistan (Listeni/?p?k?st?n/ or Listeni/p??ki?st??n/; Urdu: ?????????, Pakistan , Urdu: [p??k?st???n] ( listen)), officially the Islamic Republic of Pakistan (Urdu: ??????? ????????? ?????????, Islami Jumhuriya'eh Pakistan , Urdu: [?sl??mi d???m?u?ri?-e p??k?st???n]), is a sovereign country in South Asia. INTERNET Now days the biggest problem among teens is the use of mobile phones and tintern abbey internet. Most of the time we find them using internet or cell phones .Many problems rises like talking with strangers on internet through any source like Facebook is very famous these and other days . They forget about. Overview of Indo-Pakistani Relations. Indo-Pakistani relations are grounded in ericksons stages the political, geographic, cultural, and economic links between the Islamic Republic of Pakistan and and imagination the Republic of ericksons stages, India, two of the largest and fastest-developing countries in South Asia. The two countries share much of their common geographic location, and.

Mohammad Atif Aslam, the creativity Voice Of Pakistan , Is a Pakistani Pop Singer. He Was born in 12th of march, 1983 in Wazirabad and educated in Lahore and of development Rawalpindi. He is well-known in the Subcontinent for his songs 'Aadat', 'Hum Kis Gali Jaa Rahe Hain', 'Ehsaas', 'Doorie', Khuch Is Tarah', 'Woh Lamhe', 'Pehli. To what extent did the American Revolution fundamentally change American Society? economically by the creation of the Declaration of Independence , paper currency, Women's rights and slavery. The American Revolution brought tremendous effects on the Americans.

One major approach to Constitution undergo in this change is the Declaration of Independence . George Washington declared a change in ericksons of development the society.