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PROPOSITION VI. To find the intersections of a given plane with a given cone or cylinder.
The sections of the cone may be either ellipses, parabolas, or hyperbolas, including all their varieties (Conic Sections, p. 221). Those of the cylinder only the ellipse and its varieties, and straight lines. Generally, for the determination of these, five elements are required. Thus far with respect to the sections themselves; but in the present case it is moreover required to find their orthographic projections on the coordinate planes, which, however, does not alter the species of the figures, the ellipse still being projected into an ellipse, a parabola into a parabola, and an hyperbola into an hyperbola.
The most obvious mode of proceeding is—to take any five convenient generatrices of the cone or cylinder, and find the projections of their intersections with the given plane. The conic sections traced through the five points thus obtained on the coordinate planes will be the projections required. The figure, however, which contains the entire process is too much crowded with lines to be given here; but it is easily drawn, with proper care, upon a larger scale, and the whole process is too intelligible to need further remark.
PROPOSITION VII. To find the intersection of a given line with a given cone or cylinder.
Find av B, the traces of the given line; take any point a, a, in it, and find the traces Yu, ê, of the line through a, a, and v, vg. The plane whose traces are a, Y. and B, è, passes through the given line and the vertex of the cone, and cuts the cone in those two of its edges which pass through the intersections of the given line and given cone.
Let ri, si, be the intersections of the traces of this plane and the cone. Find as usual (and as indicated by the lines in the figure) the vertical projections rv2, sv, of these edges, and let them meet the vertical projection of the given line in P2, 92. Draw P2 Pu 929 perpendicular to the axis, meeting the horizontal projection of the given line in pug. Then pa po and q. 9e are the points in which the line meets the cone.
In the case of the cylinder, the only difference is—that in finding the line through the assumed point a, a, in the given line, drawn to a given point, v, vg, that line is to be drawn parallel to the given axis (or a generatrix) of the given cylinder. The work is slightly lessened, but the principle is the same.
Through a given line to draw a tangent plane to a given sphere. On account of the multiplicity of the lines which enter into all the steps of the construction, the process can only be described here; but the student will be required to perform them in detail.
(1.) Through the centre of the given sphere draw a plane perpendicular to the given line, finding the projections of the point of intersection.
(2.) Find the distance of this point from the centre.
(3.) Construct (separately) a right-angled triangle having the distance in (2) for its hypothenuse and the radius of the given sphere for one of its sides, so as to find the angle opposite to this radius.
(4.) Through the given line and centre of the sphere describe a plane.
(5.) Through the given line describe a plane to make with that in (4) an angle whose profile is equal to that determined in (3).
This will be the plane required; or, rather, there will be two, one on each side of the plane (4).
To find the intersection of a given sphere with a given plane.
· Generally the projections will be ellipses ; one projection being a circle only when the given plane is parallel to one of the coordinate planes, and the other a limited straight line. This case is so simple that it may be left without further instruction for the student's exercise.
(1.) When the given plane is oblique to both coordinate planes, find its traces.
(2.) From the centre of the sphere draw a line perpendicular to the given plane. The projections of the point of intersection of the line and plane will be those of the centres of the ellipses into which the sectional circle is projected.
(3.) The major axis of each ellipse is parallel to the trace of the given plane on the plane of projection, and equal to the radius of the sectional circle.
(4.) The minor axis perpendicular to this is the same radius reduced in a given ratio. (See Conic Sections, p. 221.) (5.) The radius of the sectional circle is thus found :
Find the length of the perpendicular from the centre of the given sphere on the given plane : form a right-angled triangle having the given radius of the sphere for its hypothenuse and the perpendicular for one of its sides; the other side will be the sec
tional radius. SCHOLIUM. The problem of finding the intersection of two given spheres is at once reducible to this, since that section is a circle whose plane is readily determinable.
PROPOSITION X. To find the intersections of a given straight line with a given sphere.
(1.) Draw a perpendicular from the given centre to the given line; and assign the point of intersection, and the distance of the intersection from the centre.
(2.) Construct a right-angled triangle (separately) whose hypothenuse is the radius of the given sphere and one of its sides the distance just found. The other side is the semi-length of the segment of the intercepted line.
(3.) Describe on this line as hypothenuse another right-angled triangle having one of its angles equal to the inclination of the given line to either plane of projection. The side adjacent to this angle is the semi-length of the projection of the intercepted line on that plané.
(4.) The remaining part of the construction (one single step) is obvious.
SHADOWS. This is one of the simplest applications of Descriptive Geometry, at least as far as the subject is of any use in engineering and architectural drawings. If, however, pursued into all its conceivable applications, it becomes extremely complex. Only the simplest and most elementary problems have been deemed necessary here, but a full comprehension of these will be fully adequate to enable the draughtsman to project any shadow that can occur in his practice.
A few general but brief remarks may be usefully made at the outset.
1. Although the method of determining the shadows of objects here given is perfectly general, and adapted to all circumstances of the relative position of the object, the light, and the surface upon which the shadow is thrown ; yet in fixing upon them in the case of sun-light, it is the universal English practice to select one specific position of the sun with respect to the principal face of the object, that face being also parallel to the vertical plane of projection. I know, indeed, of no merely professional work published in England in which the general method is given, and only one, and that only semi-professional, in our language:* though such treatises, illustrated with ample examples, are frequent on the Continent. It is, however, always preferable to understand the general principles that run through an entire system, rather than to be confined to the special cases of them which the study of mere convenience may render it advisable to adopt in practice.
2. This special rule to which reference has been made is that wherein the luminous rays are said “ to make angles of 450" — a phrase, certainly, which in itself does not contain a very definite meaning. It is, however, this:
The direction of the sun's ray is such that if the line be projected in the horizontal and vertical planes, its projections will make, with the axis, half a right angle; the right angle, Trig., Vol. I., p. 213, being designated as 90%, and hence 45° denotes half a right angle.
3. There are, indeed, good reasons offered for this choice as a general rule, especially where the simplicity of the work and the effect of the drawings are any way objects of consideration. The simplification will be best seen by a few trials, which make a stronger effect on the mind than mere verbal description. They belong, too, rather to the province of the practical instructor than to the development of fundamental principles.
4. The effect is two-fold. It gives a breadth of shadow agreeable to the eye, midway between the “ meagre” and the “ heavy.” It also tells to the professional eye the prominence of the object which casts the shadow, without reference to the projection on the other plane, since the shadow's breadth is equal to that prominence. A shadowed elevation is, therefore, tantamount to an unshadowed plane and elevation together.
5. Although direct constructions admitting the feasibility as well as possibility of constructing the conic sections from a suitable number of elements are theoretically and practically elegant, yet in actual drawings they are seldom resorted to, except in the simplest cases. For the shadows on pillars, niches, as well as for shadows cast by them on planes, the best method in practice is to find the separate shadows of points at convenient distances, and then trace a curve through each of these. In this case it is always advisable to draw planes through the points of the original, parallel to the ray and perpendicular to the horizontal plane of projection.
6. Other suggestions might be added were they not in some degree foreign to a mere sketch like the present.
7. Of the degrees of intensity of the shadow in different regions within its limits, nothing can be said that would be of the least practical use. The problem, indeed, is a mixed one-partly physical and partly geometrical ; and though capable of a perfect mathematical solution, we are yet left without any means of constructing the solution ; for we have no means practically of making a shadow of any specified intensity. We trust to the eye after all. Even could we by mechanical means “lay on” a shadow of given intensity at a given point, our modes of shading are still incompatible with any approach
*“A Treatise on Shades and Shadows, and Linear Perspective.” By Charles Davies, Professor of Mathematics in the United States Military Academy. New York, second edition, royal 8vo. 1838. This work deserves my most cordial recommendation to the English reader who wishes to enter fully into the subjects treated in it.
towards perfect gradation of intensity. We cannot shade by a point at a time, or even by a line at a time.
DEFINITIONS. 1. The portion of any figure which is exposed to the direct light of any luminary is called the illuminated part, or the light part of the figure.
2. The portion of the figure which does not receive direct light from the luminary is called the shaded part of the figure, or the shade of the figure.
3. The portion of any other surface from which the direct light is intercepted by a given figure is called the shadow of that figure.
4. The boundary of the light and shade upon any figure is called the line of shade.
5. When the light, instead of emanating from a single point, comes from all the points of a luminous body to which the figure is exposed, and produces a variety of shadows of different intensities, the band between the absolute shadow and absolute enlightenment is called the penumbra.
PROPOSITION I. To find the shadow of a point, the rays of light being parallel to a given line.
Let a, a, be the given point, and R, R, the line to which the light is parallel ; then the line through a parallel to R will pierce one, and only one, of the coordinate planes in the visible or first region of space. We have, therefore, only to construct the visible trace of that line.
This is the same problem as that in Descriptive Geometry: to draw through a given point a, a line parallel to a given line R.
In Fig. 1, the shadow falls on the vertical, and in Fig. 2 on the horizontal coordinate plane.
SCHOLIUM. This is the foundation of the entire system of shadowing in architectural and mechanical drawings.
PROPOSITION II. To find the shadow of a point, the light emanating from a given point.
Let s be the luminous origin, and a the given point: then the problem is identical with that in Descriptive Geometry, which requires us to find the visible trace of a line passing through the two given points s and a.