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CURVE SURFACES AND TANGENT PLANES.
36. In what has preceded, the required points have been determined by the intersection of planes, and the questions regarding them have been considered as solved when the positions of these planes are determined. We now proceed to treat of a different class of problems, which are dependent upon curve surfaces; it is therefore necessary to make some remarks upon these surfaces.
We know that when a curve is traced upon a plane, it may be considered as a series of points connected together and determined by a law, which furnishes what the geometers call the equation to the curve: thus, the circle furnishes us with an instance of this law, since every point in it is at the same distance from one point, called the centre of the circle. This property gives us both a mechanical method of describing the circle, and a geometrical relation by which its equation may be found. Now, surfaces differ from curves in this respect : that their respective points are in space, and are not limited in position to the same and constant plane, but extend in all directions. Still these points are subject to a law, which particularizes and determines the surface in question.
The curves just spoken of being always in the same plane are called plane curves. There are curves called curves of double curvature, which are not in the same plane : such is the helix, of which the corkscrew is a familiar instance. We might trace such a curve by the motion of a point, and thus say generally, that curves are described by the motion of a point, while surfaces are described by the motion of a line.
37. The surfaces of which we shall treat may be divided into-1, Surfaces of Revolution ; 2, Conical Surfaces; 3, Cylindrical Surfaces.
Surfaces of Revolution are those which may be supposed to be described by the revolution of a curve round its axis. Thus the semicircle revolving round its diameter produces the sphere, and the ellipse round its axes generates the spheroid. From this method of generation, it is plain that every section perpendicular to the axis is a circle ; while every section made by a plane passing through the axis will reproduce the original curve.
Conical surfaces are described by the motion of a line, which is compelled always to pass through a fixed point, and the other extremity moves through a curve traced upon à given plane. If the given curve be a circle, and if the given point be in a line passing through the centre of the circle, and perpendicular to its plane, the surface described is that of the common, or right, cone.
Cylindrical surfaces are described by the motion of a line always parallel to its first direction, but the extremity moving through a given curve in a given plane.
If the given curve be a circle, and the line be perpendicular to the plane of the circle, the surface described is the common, or right, cylinder.
Def. The moveable line which generates the surface is called the generatrix, and the line which directs and limits its motion the directrix.
THE TANGENT PLANE AND THE NORMAL. . 38. We may consider a curve to be a polygon, of which the sides are infinitely small, and one of its tangents to be the prolongation of one of these indefinitely small sides. In the same manner a curve surface may be considered as a polyhedron, whose faces are infinite in number and in-, finitely small, and the tangent plane is the plane of one of these faces. And to have a clear idea of a tangent plane to a point in a surface, we must suppose that round this point there has been taken an infinitely small portion of the surface, which is considered a plane, and this plane, indefinitely extended, is the tangent plane.
39. If we conceive through the point of contact any number of lines whatever to be traced upon the surface, and if from the point under consideration we take infinitely small portions of these curve lines, we may regard these portions as straight lines, which produced will be tangents to the respective curves of which ultimately they form a part; but all these small lines will also be in the tangent plane, and therefore when produced will also be in that plane.
Hence we see that the tangent plane to a surface contains all the tangents, drawn through the point of contact, to all the lines which can be traced through the point upon the surface.
40. This proposition is of great importance in determining the position of a tangent plane, since as it is only necessary to have two lines given in order to find a plane, it follows, that in order to construct the tangent plane to a
given point of a surface, we have merely to draw tangents to two lines traced upon the surface, and to make the plane pass through these tangents. Our choice of the lines on the surface will, of course, be directed to those whose tangents are most readily found.
DÆF. A normal is a perpendicular to the tangent plane, drawn through the point of contact.
Tangent PLANES TO SURFACES.
Given the horizontal trace of a cylinder and the direction of
the generatrixes, to draw a tangent plane to a given point of the cylinder. .
41. Let aecf be the horizontal trace of the cylinder ; and in this instance suppose aecf to be a circle.
Draw ab, cd, tangents to aecf, and parallel to the horizontal projections of the generatrixes; these tangents will be also the limits of the horizontal projection of the cylinder, since all the generatrixes must be projected between these two lines.
Draw the tangents eé', ff', + to xy, and e'g', f'k', parallel to the vertical projection of the generatrixes ; it is clear that these latter lines will be the limits of the vertical projection of the cylinder.
Now, suppose that the horizontal projection m, of a point in the cylinder be given, and let the other projection be required.
Draw through m a vertical plane, parallel to the generatrixes, which can intersect the cylinder only in one or more of
the generatrixes; the intersections of these generatrixes with the vertical, drawn through the point m, give the points of the surface which are projected in m. The traces of this plane are the line mq, drawn parallel to ab, through the point m, and the line 99' 1 to xy. And since the first meets the curve aecf in i and k, it follows that the plane cuts the cylinder in two generatrixes, the horizontal projection of both being in the direction of mq; and their vertical projections are found by drawing ii', kk', – to xy, and i'1' k'" parallel to f'k'.
Draw mm' 1 to xy; and m', m", where mm' meets i' ], and k'l', are the vertical projections of points of the cylinder corresponding to the horizontal projection m.
This being done, next to draw a tangent plane to the point mm'. We see, first, that this plane ought to con