8. A vertical section of a square ceiling is shown in Fig. 611, the upper part being of the form of a shallow square pyramid. Put the ceiling into perspective with its horizontal edges inclined

to the picture plane at angles of 30° to the right and 60° to the left, its nearest corner being 12 feet vertically over a point on the ground plane 3 feet to the left of the spectator, and 1 foot within the picture. The eye is to be 12 feet from the picture,

and 5 feet from the ground. Scale inch to a foot.

12

FIG. 611.

[B.E.]

9. Draw, by the method of Art. 265, the perspective projection of the solid shown at Ex. 9, Fig. 612. The edge AB is to be on the ground and parallel to the picture plane. The axis of the solid is to be inclined at 50° to the ground, and the nearest point C is to be in the picture plane, and 14 inches to the right of the spectator. Station point, 4 inches from the picture plane, and 2 inches above the ground.

10. Draw the perspective projection of the solid shown at Ex. 10, Fig. 612. The solid is to stand on the ground with the edge AB in the picture plane, and 1 inches to the right of the spectator. The face ABC is to be inclined at 40°

In reproducing the above diagrams the sides of the small squares are to be

taken equal to half an inch.

to the picture plane. Height of station point 2 inches. The position of the centre of vision, and the distance of the station point from the picture plane is to be determined from the further condition that the distance between the vanishing points for the horizontal edges is to be 7 inches.

11. Draw the perspective of the solid shown at Ex. 11, Fig. 612. The base of the solid is to be on the ground, and the nearest edge AB inch behind the picture plane and 14 inches to the right. The face ABC is to be inclined at 30 to the picture plane. Station point 3 inches from picture plane and 2 inches from ground.

12. A skeleton cube is shown at Ex. 12, Fig. 612. Draw this object in perspective when the nearest vertical edge is in the picture plane, and 1 inch to the left, and a face containing that edge is inclined at 35 to the picture plane. The station point to be 34 inches from the picture plane and 1 inch above the top face of the object.

13. Draw the perspective of the object shown at Ex. 13, Fig. 612. The edge AB to be vertical, and in the picture plane, and inch to the left. The face ABC to be inclined at 30° to the

picture plane. Station point to be 4 inches from the picture plane and 1 inches above the point B.

14. Draw the perspective of the object shown at Ex. 14, Fig. 612. The edge AB to be vertical and in the picture plane, and directly opposite to the station point. The point C also to be in the picture plane. Station point, 14 inches above the level of point C, and 3 inches from the picture plane.

15. A gable cross is shown at Ex. 15, Fig. 612; draw this in perspective. The edge AB to be vertical and in the picture plane and inch to the left. The face ABC to be inclined at 45° to the picture

plane. Station point, 14 inches below the point A, and 4 inches from the picture plane.

16. The plan and elevation of a wheelbarrow standing on the ground are shown in Fig. 613 to the scale, inch to a foot. Represent this object in perspective using the scale, 1 inch to a foot. The line AB is the horizontal trace of the central vertical plane of the wheelbarrow. The point A is to be 1 foot to the left of the spectator and 1 foot from the ground line, and the line AB is to vanish towards the right at 40° to the picture plane. The eye is to be 6 feet, by scale, from the picture plane and 24 feet above the ground plane. [B.E.] 17. Referring to exercise 14, let the station point be moved 14 inches nearer to the object, and let the picture plane be moved parallel to itself a distance of 5 inches, so that the station point comes between the solid and the picture plane.

18. Work the example shown in Fig. 606, p. 307, to the dimensions given. Then take a point 3 inches to the left, 5 inches above the ground, and 3 inches behind the picture plane. Consider this as a luminous point and determine the perspective of the shadow cast by the solid on itself and on the ground; all tho rays of light to come from the luminous point.

CHAPTER XXIII

CURVED SURFACES AND TANGENT PLANES

273. Generation of Surfaces.-Surfaces may be considered as generated by a line, straight or curved, moving in a definite manner. Thus, a plane may be generated by a straight line moving parallel to one fixed straight line and in contact with another fixed straight line. Again, a sphere may be generated by the revolution of a semicircle about its diameter which remains stationary.

The moving line which generates a surface is called the generating line or generatrix of the surface, and a line which serves to constrain or direct the motion of the generatrix is called a directrix.

The same surface may be generated in numerous ways, but generally there are only a few simple ways in which a surface may be generated. Take the case of the surface of a right circular cylinder. There are two simple ways in which this surface may be generated: (1) by a straight line moving in contact with a fixed circle to the plane of which it remains perpendicular, as shown by the oblique projection in Fig. 614, where the moving line is shown in twelve different positions : (2) by a circle moving so that its centre remains on a fixed straight line to which the plane of the circle is always perpendicular as shown by the oblique projection in Fig. 615, where XX is the fixed straight line and 1, 2, 3, 4, and 5 are positions of the moving circle. This surface may however be generated by an ellipse which moves so that all points on it travel along parallel lines, but the ellipse must be such

that its projection on a plane at right angles to the direction of its motion is a circle.

The two simple ways in which the surface of a right circular cone may be generated are: (1) by a straight line which passes through a fixed point (the vertex of the cone) and moves in contact with the circle which is the base of the cone as shown in Fig. 616: (2) by a circle of changing radius which moves with its centre on the axis and its plane perpendicular to the axis, the radius of the circle being proportional to its distance from the vertex of the cone as shown in Fig. 617.

A surface which may be generated by a line, straight or curved, revolving about a fixed straight line is called a surface of revolution. The fixed straight line about which the generating line revolves is called the axis of the surface. Sections of a surface of revolution by planes at right angles to its axis are circles. Sections by planes containing the axis are called meridian sections. All meridian sections are exactly alike.

A surface which may be generated by the motion of a straight line is called a ruled surface. Ruled surfaces may be divided into two classes-developable surfaces and twisted surfaces. A developable surface may be folded back on one plane without tearing or creasing at any point. The generating line of a developable surface moves in such a manner that any two of its consecutive positions are in the same plane. All ruled surfaces which are not developable are twisted surfaces.

274. Plane Sections of Curved Surfaces.-The way in which a curved surface is generated being known the projections of the generating line in any number of positions can be drawn. The intersections of a given plane with the generating line in each of these positions can then be determined. This will give a number of points on the inter

section of the plane with the surface, and a fair curve through them will be the complete intersection required.

When the curved surface can be generated in a number of simple ways, that mode of generation should be made use of which has the projections of its generating line the simplest possible.

EXAMPLE 1. A surface is generated by a horizontal line which moves in contact with the line AB (Fig. 618) and the surface of the

X

FIG. 618.

cone VCD. To find the section of this surface by the plane LMN.

Take a section of the given cone and plane by a horizontal plane cutting ab, a'b' at tt'. The plan of the section of the cone is a circle and tangents to this circle from t are the plans of two positions of the generating line. The points r and 8 in which these tangents cut os the plan of the intersection of the assumed plane of section with the plane LMN are the plans of points on the section required. Projectors from r and s to meet the horizontal through determiner' and '. In a similar manner any number of points on the required section may be found.

EXAMPLE 2. Referring to Fig. 619, ab, a'b' is a horizontal circle 3 inches in diameter. cd, c'd' is another horizontal circle 1 inches

in diameter. The line joining the centres of these circles is vertical and 2 inches long. Two points move, one on each circle, with equal angular velocities in opposite directions. A surface is generated by a straight line which contaius the above mentioned moving points.