elementos de la geometría de sólidos-en inglés

[...]... also 2 ft for determining the position of any plane area with reference to three rectangular coordi For since any two planes make with each other the nate planes is made by two lines perpendicular to them respec the angles made by a plane with the rectangular coordinate planes are the angles made by a perpendicular to the plane with the coordinate axes respectively Thus if be the perpendicular to a same... = 2 r o in rectangular coordinates is a y* a r* is a wiih circular base in plane xy (since jv right cylinder circle in plane xy) and its axis coincident with the axis of z 4- +y = = c o is a plane parallel to the axis of g, intersect And ax + by c ing the plane xy in the line ax + by o represents either a cylindrical surface with Similarly F (x, z) = = axis parallel tojy or a plane parallel o represents... plane parallel to this axis An equation containing a single variable represents a plane or planes parallel to one of the coordinate planes a represents a plane parallel to the planejyz Thus x 23 = And as _/"(.#) values of x, as = o when solved will x= x = a, x I, give a determinate c, etc., so it number of represents several planes parallel to the coordinate planers o represents a number of planes... cosine right line on any plane finite of which the angle plane PQ Let be the given finite straight line, xOy draw PM, QN perpendicular to it then tion of PQ on the plane Now the angle made by jection is to is makes with it ; ; the angle MN = the PQR made by PQ meeting angle PM made with MN Through QR = MN, R, then by PQ with MN in Now the plane of pro is the projec PQ with the plane MN Q draw QR parallel... which tively, OP plane, the angle with xz ft, made by the plane with the plane xy and with yz is the angle a is the angle ft cos y, are called also the direction cosines of a plane is ; the angle So cos That y ; a, cos is, the ON SOLID GEOMETRY iVOTES g direction cosines of a plane with of a the direction cosines The 14 line relation cos 2 reference to rectangular coordinates are perpendicular a + cos 2... represent points determined by the intersection of a sphere, cone and plane CHAPTER III EQUATION OF A PLANE COORDINATES OBLIQUE OR RECTANGULAR To find equation of a plane 29 origin and Let terms of the perpendicular from the in OD and let it its direction cosines p be the perpendicular from the origin on the plane, make with the axes O,r, Oy and Qz the angles a, ft and y Let respectively plane ; The OP... between the planes x+ Show 2 2y + 32 = and $x 5 4- z 4y = 10 that the planes x + 3y 20 anc* 2X 5Z + V + I0 are perpen z dicular to each other Write the equation representing planes parallel to the plane 3* 3 + = 2y 6z 34 To find ii the expression for the distance from a point P (x y z ) to a plane (coordinates rectangular) i Let the equation of the plane be of the form x cos a +y cos Pass a plane through... the plane 36 To find Let OP = +A A^v Ax Ay + A^ , y y the theorem the polar equation = POS r, P 6, and A x we may write the equa , , = 3V of a plane OM = cp = OD = POD = and jff, Then yyp - = (22) be the polar coordinates of a (Fig 12.) point P of the plane a Let perpendicular on plane = Hence for base true angle ; DOS D OM a, GO cos POD = cos = or GO, cos GO Now in order to express GJ in polar coordinates... represents ihe two plane bisectors of the supplementary angles made by the given planes That is to find the equations to the plane angles made by two given planes, put their and then add and subtract them Example bisectors the supplementary of equations in the normal form Find the two planes which bisect the supplementary the planes 2.v + 3.y Vz = 5 and 3^ + 4^23 = 4- angles made by Result, A/14... projection on any plane of any bounded plane area is equal to thut area multiplied by the cosine of the angle between the planes i We begin with a triangle of which one side shall to the plane of projection area of the projection A D = AD planes ABC = area of ABC = - - BC xA D BC But is parallel x AD, and the BC = BC and ADM Moreover ADM = the angle between the A B C= ABC x cos angle between the planes (Fig . the same angle which is made by two lines perpendicular to them respec tively, the angles made by a plane with the rectangular coordinate planes are the angles made by a perpendicular to the plane with the coordinate axes respectively. Thus. each pair of axes determines a plane, CXr and Oy determining the plane xOy ; O.v and O2 the plane xOz ; Oy and Oz the plane yOz. And the posi tion of the point P with reference to the origin O is determined by its. cosines also for determining the position of any plane area with reference to three rectangular coordi nate planes. For since any two planes make with each other the same angle which is made by two