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and touching the latter in point a, which becomes the centre of vision. Project this point to a line drawn parallel
^ £ J distant 12 ft., to scale, locating the stationpoint thereon. From this point draw lines parallel with the sides of the frame a~d and a-b, intersecting the H.L., which locate thereon the left-hand and right-hand vanishing points respectively. Draw the ground line G.L. 5 ft. below the ILL. and parallel therewith ; this locates the picture plane, upon which we can now produce the projection. From each corner of the frame a-b-c-d draw projectors to the S.P., stopping them when they reach the horizontal line, upon the picture plane, as shown at a -b'-c'-d'. Drop perpendiculars from these points to the ground line. It will be remembered that the angle a is to be in the line of sight and upon the ground, therefore, where the line drawn from S.P. to a intersects the ground will be the position of a'. Next draw, from point a', lines to the two vanishing points, and the intersections of the projectors with these; in points b2. d'1 and c2 locate the corners of the frame upon the ground. The respective heights or verticals at these points are obtained by setting up the height or' thickness at a' and drawing lines tc the vanishing points as shown. In like manner any other point in the plan is first brought to the. horizontal line, then projected to the ground and its position located upon the vanishing planes.
A Stone Pedestal in three tiers, with its sides making angles of 34° and 56° respectively, with the picture plane and its nearest corner 2 ft. to the right of the spectator and 13 ft. away from him, is represented in perspective projection on page i2i. An elevation is given in Fig. 1 to indicate dimensions.
Commence by drawing the ILL. in a convenient position near top of sheet and the G.I.. 5 ft. distant therefrom, and a parallel at 13 ft. distant, on which mark the station-point S.P. Draw a perpendicular from this to the H.J.., locating the C.F., and mark oft along the ILL. point 1, two feet to the right. From this point set out the sides of the lower block, making an angle of 340 and 56° with the II.L., and complete the plan as shown. Locate the vanishing points as before by lines drawn from S.P. parallel to the sides of the plan. Draw projectors from each angle of the plan towards the station-point, but stop at H.I,. Drop a projector from point 1 to the ground ; this locates the salient angle of the lower slab, and on it measure up the height of point II. (see Fig. 1). Draw lines towards the vanishing points and intercept them by projectors from points 2 ft. and 4 ft. upon the horizontal line, thus locating the position of the distant corners of the block in the picture. To obtain the heights of the second block we must mark off the heights as given in Fig. 1 upon the perpendicular 1 -II., which lies in the picture plane, and therefore shows real heights : then draw lines to the left and right vanishing points which, intercepted by the vertical projectors from the said points on ILL., will show the heights those points reach in the distance. A few of the projectors from the plan are taken across to the S.P. to indicate the direction, but these, of course, are not dealt with below the ILL. It will be noticed that the sides of the second block which stand within the lower one are produced by means of dotted lines to the face of the lower block, and thence projected to the ILL. ; this is necessary in all cases where parts of the object stand back from the main face. When they are located upon this face in the perspective projection they must be projected back again into their relative
122 PERSPECTIVE PROJECTION
position to the face by means of projectors taken from the points to the other vanishing point, as shown on the drawing.
The Perspective Projection of a large Chest (page 123).—This simple object has been selected to illustrate a method of dispensing with the use of one or both vanishing points when the angle is so low that it takes these beyond the drawing sheet. Fig. 1 is the end elevation of the chest from which the necessary dimensions are obtained, Fig. 2 is the plan of the chest in its relative position to the observer, and Fig. 3 is the perspective projection. To simplify the explanation one vanishing point is shown on the right hand; the other is outside the picture.
We will assume that the projection has been made :n accordance with previous instructions up to the point where position of the two vanishing points has to be located. The plan of the right-hand vanishing plane is drawn from the station-point parallel to the right-hand side of the plan, and its intersection with the H.L. gives the R. V.P. A line is also drawn from S.P. parallel to the left-hand side of the plan, so far as the limits of the paper allow. At any convenient point in this line, as A, erect a perpendicular to the H.I.., and at an equal distance from the line of sight on the right vanishing plane, erect a similar perpendicular, as at B. Now the angles on either side of the line P. V. or S.C. are alike, and any divisions of the base line P.V., drawn to the apex of the triangleR.V.P., divides the parallel B in the same proportion that the whole length of B is to P.V. It follows that, as the angles on the left side of P.V. are similar and equal to those on the right, any proportions upon the parallel A will be similar and equal to those upon B, if drawn from the same points in the base; therefore, al! that is necessary to obtain corresponding reductions ir the left vanishing plane to those in the right is to make the divisions on A equal to those upon />', and draw lines to those points from the heights marked upon the vertical angle of the object
which touches the picture plane. For example, take the point i in line B. Where the projector from the top edge of the chest passes through it, draw a horizontal line from this point to line A, intersecting it at point i', and a line drawn from the front top angle of the chest through this point will give the angle of vanishing projector as correctly as if it were drawn to a vanishing point. All the other points found on B can be projected in like manner to A.
The Pedestal Table shown in plan and elevation in Figs, i and 2, page 125, and in perspective in Fig. 3, is an example of the method of making the projection when the vanishing point is too distant to find and the angles of the plan are unequal. In this case the plan makes angles of 30° and 60r with the horizontal line, and the object is 13 ft. from the observer. In the previous case, where one V.P. was missing, the position of the plan enabled us to form two proportionate triangles alike on each side of the common " base " line. Here that method cannot be followed, but we can proceed to locate the left vanishing point which lies within the picture, and so obtain a triangle in which we can arrange a smaller one, as at A-a, LA'.P.
We can, with this as a base, construct a similar triangle on the other side, which shall bear the same relation to the hypotenuse of that triangle as A-a does to the line S.P.-L.V.P. Then all reductions obtai ied upon A-a, if transferred to the proportionate parallel B -b, will give proper reductions for that side ot the picture, and we shall obtain exactly the same lines as if we had taken them directly to a vanishing point upon thai side.
Proceed to lay down the plan, the II.L and G.L., and the station-point, to given data and draw the plans of the vanishing planes as far as possible, and parallel the respective sides of the plan. Anywhere on L.V.P.-S.P. erect a perpendicular to the ILL., as A-a. Draw a parallel to the ILL. from A, intersecting the right vanishing plane in B ; erect a perpendicular to ILL., and transfer all reduced
points on A-a to B-b, as shown, and thus obtain direction of vanishing lines upon that side.
The completion of the perspective view will now proceed as described in previous cases, the heights of the drawers, plinths and door rails being obtained from the elevation, Fig. 2. A scale of feet is provided by aid of which the sizes may be read off.
Chaplek IX
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