0.5

1 cm

PERSPECTIVE.

109

To Draw a Sphere in Perspective.

Problem XIII.—Let AB ab (Fig. 227) be the projec-

tions of the picture, c c those of the eye, the circle D that

of the given sphere. If the centre of

the sphere is at the height of the

horizon, the vertical projection of its

centre will be the point of sight 6.

Draw from the eye c the tangents E c

F c. Draw also the chord E F, which

may be regarded as the base of a cone

formed by the visual rays, tangents to

the sphere, and of which the eye c is

the summit. The section of this cone

by the picture will be a circle, since

the cone is cut parallel to its base.

This circle will have for its diameter

ef; and consequently of or c e as its

radius. Then, if with this radius from

the centre e we describe a circle, it will be the perspective

sought.

When the sphere is below the horizon, let D (Fig. 228)

be the horizontal projection of a sphere, c that of the eye,

and L M the picture-line. Draw visual rays tangents to

the sphere, and we obtain on the picture-line e f as the

perspective horizontal diameter of the sphere. At K set

(Fit7.228.) c, Jhrixtn

off the diameter ef so obtained, and from the points e f

raise indefinite perpendiculars. Now, suppose D to be the

vertical projection of the sphere, and c' the vertical pro-

jection of the eye. From c draw visual rays tangents to

the sphere, and we obtain r s as the intersection of the

cone of rays by the picture, and the length r s as the per-

spective vertical diameter of the sphere. But the diameter

r s is greater than the diameter e f, and, therefore, the

perspective representation of a sphere viewed under the

conditions premised, must be an ellipse. When we obtain

the two diameters, we obtain all the measurements neces-

sary for the representation of the sphere when below the

horizon, and in the central plane, viz., the major and

minor axes of the ellipse, and the curve may be tram-

melled by the aid of a slip of paper.

The point g in which the sphere touches the picture is

the projection of its centre, and of its axis D g, and its

perspective representation is 1c. In this there is another

of those apparent contradictions, for it is certain that a

sphere always appears to us to be round on whichever

side we regard it; while in perspective, in every case

except that in which its centre is in the point of sight, it

must be drawn an ellipse with its major axis directed

towards the point of sight. An attentive consideration of

the figures and description will render this evident, and

the reader may also advert to the explanation of this

apparent contradiction, which is given in the text treat-

ing of figures 223-4-5. We shall now proceed to give

some other examples of spheres in perspective.

Let A B ab (Fig. 229) be the projections of the picture,

C C' those of the eye, D that of the given sphere in contact

with the ground plane. Draw from the eye the tangents

E C, F C, through E and F, draw parallel to the picture the

lines E G, F H, and draw also as many parallels to them,

I K, 0 P, L M, &c., as may be considered necessary. These

fi/ (Fig. 229.)

parallels, then, are traces of vertical planes cutting the

sphere parallel to the picture, and all the sections made

by them will be so many circles, which will comprehend

all the visible portion of the sphere, as determined by the

tangents E C, F C. The perspectives of the circles IIF, IK,

&c., which are parallel to the plane of the picture, will

also be circles. If we envelope all these perspective

circles by a curved line, this line will be the perspective

of the sphere. It is easy to find the perspectives of the

circles. First draw the diameter r s, which will be the

horizontal projection of one of the axes of the sphere, pass-

ing through the centres of the circles H F, I K, L M, &c.

Find then the perspective direction of that axis; observe

that the point r is raised above the ground plane by the

height of the radius of the sphere, and that it touches the

picture. Its perspective consequently will be in R, and

(Fig. 227.)

109

To Draw a Sphere in Perspective.

Problem XIII.—Let AB ab (Fig. 227) be the projec-

tions of the picture, c c those of the eye, the circle D that

of the given sphere. If the centre of

the sphere is at the height of the

horizon, the vertical projection of its

centre will be the point of sight 6.

Draw from the eye c the tangents E c

F c. Draw also the chord E F, which

may be regarded as the base of a cone

formed by the visual rays, tangents to

the sphere, and of which the eye c is

the summit. The section of this cone

by the picture will be a circle, since

the cone is cut parallel to its base.

This circle will have for its diameter

ef; and consequently of or c e as its

radius. Then, if with this radius from

the centre e we describe a circle, it will be the perspective

sought.

When the sphere is below the horizon, let D (Fig. 228)

be the horizontal projection of a sphere, c that of the eye,

and L M the picture-line. Draw visual rays tangents to

the sphere, and we obtain on the picture-line e f as the

perspective horizontal diameter of the sphere. At K set

(Fit7.228.) c, Jhrixtn

off the diameter ef so obtained, and from the points e f

raise indefinite perpendiculars. Now, suppose D to be the

vertical projection of the sphere, and c' the vertical pro-

jection of the eye. From c draw visual rays tangents to

the sphere, and we obtain r s as the intersection of the

cone of rays by the picture, and the length r s as the per-

spective vertical diameter of the sphere. But the diameter

r s is greater than the diameter e f, and, therefore, the

perspective representation of a sphere viewed under the

conditions premised, must be an ellipse. When we obtain

the two diameters, we obtain all the measurements neces-

sary for the representation of the sphere when below the

horizon, and in the central plane, viz., the major and

minor axes of the ellipse, and the curve may be tram-

melled by the aid of a slip of paper.

The point g in which the sphere touches the picture is

the projection of its centre, and of its axis D g, and its

perspective representation is 1c. In this there is another

of those apparent contradictions, for it is certain that a

sphere always appears to us to be round on whichever

side we regard it; while in perspective, in every case

except that in which its centre is in the point of sight, it

must be drawn an ellipse with its major axis directed

towards the point of sight. An attentive consideration of

the figures and description will render this evident, and

the reader may also advert to the explanation of this

apparent contradiction, which is given in the text treat-

ing of figures 223-4-5. We shall now proceed to give

some other examples of spheres in perspective.

Let A B ab (Fig. 229) be the projections of the picture,

C C' those of the eye, D that of the given sphere in contact

with the ground plane. Draw from the eye the tangents

E C, F C, through E and F, draw parallel to the picture the

lines E G, F H, and draw also as many parallels to them,

I K, 0 P, L M, &c., as may be considered necessary. These

fi/ (Fig. 229.)

parallels, then, are traces of vertical planes cutting the

sphere parallel to the picture, and all the sections made

by them will be so many circles, which will comprehend

all the visible portion of the sphere, as determined by the

tangents E C, F C. The perspectives of the circles IIF, IK,

&c., which are parallel to the plane of the picture, will

also be circles. If we envelope all these perspective

circles by a curved line, this line will be the perspective

of the sphere. It is easy to find the perspectives of the

circles. First draw the diameter r s, which will be the

horizontal projection of one of the axes of the sphere, pass-

ing through the centres of the circles H F, I K, L M, &c.

Find then the perspective direction of that axis; observe

that the point r is raised above the ground plane by the

height of the radius of the sphere, and that it touches the

picture. Its perspective consequently will be in R, and

(Fig. 227.)