【P51】

Floating House, Lake George, N.Y. D’Amelio & Hohauser, Architects. An example of “one-point” perspective with the point correctly placed at the center of picture. Rendering by Sanford Hohauser

An example of “two-point” perspective with one of the points far to the right and the other correctly falling within the picture. Rendering by Vernon Smith.

【P52】

Distorted and correct One-point Perspective

Many books state categorically that when the vanishing point of one set of horizontal lines of a rectangular subject (such as a railroad track, a cube, etc.) falls within the picture then the other set of lines (at right angles) does not converge and the lines remain parallel and horizontal. The picture above is based on this arbitrary rule.

Note that the rails and the fancy lines converge to their correct vanishing point but that the cross-ties and black lines, which are also oblique to the picture plane (see top view) and should converge to the vanishing point indicated by the observer’s right arm, do not. What about the suitcase? Its receding set of lines correctly vanishes to the point indicated by the central visual ray, while the set parallel to the picture plane remains, also correctly, horizontal and non-convergent. The result is that the front edge of the suitcase comes out parallel to the cross-ties. This surely is wrong! Also, it’s obvious that objects at the far right suffer from distortion. In other words this rule is contrary to basic perspective drawing principles and results in a variety of distortions and inaccuracies.

The reason this rul prevails is that it eliminates the difficulty of working with distant vanishing points. But while this difficulty may complicate T-square and triangle perspective, it surely is no problem in freehand work.

Therefore, when the vanishing point of one set of lines of a rectangular object is placed at the vertical center of a drawing then the other set of lines (at right angles) should appear parallel and horizontal.(E.g., top picture previous page.) But when this one vanishing point shifts away from the center, indicating that the observer is shifting his viewpoint, the other set of lines should begin to converge to a distant vanishing point.(E.g., bottom picture previous page and picture below.)

Long Island saving Bank, D’amelio & Hohauser, Architects. Rendering by Joseph D’Amelio

【P53】

Chapter 8: More on looking up, down, and straight ahead

Glance again at pages 40 and 42, where the cube was viewed by “looking down” and “looking up,” and note that the vertical lines do not actually remain vertical in the picture but instead appear to converge downwards and upwards respectively.

Many books state arbitrarily that such lines should always appear vertical. Although contrary to the “truth” of seeing, this rule is laid down in order to simplify matters. But such simplification is helpful only in mechanical (T-square and triangle) perspective where converging verticals means complicated drafting to establish and word with distant vanishing points, and complicated procedures to determine vertical measurements.

Therefore, when working freehand (without drafting considerations) let the visual truth dictate. If you have difficulty accepting this “truth,” the following will help.

Take a book and hold it horizontally in this manner (left).

What you see (right) are lines converging to a central vanishing point at eye level. Being standard perspective drawing this is readily accepted.

Now hold the book vertically, above your head, in this manner (left), and view it at approximately the same angle.

What you see (right) is exactly the same as before, only now the convergence is upward instead of horizontal.

Therefore, the convergence and hence the picture is identical from both viewpoints. The Reason is Simply that the Relationship Between Eyes (Sight Lines) and Subject (Book) is Identical in Both Cases. (Note angle ~.)

Try this again, from both viewpoints, with the book held almost on a level with the central visual ray. The principle is now more dramatically demonstrated because convergence and foreshortening are almost at a maximum.

【P54】

Things seen by looking straight out and things seen by looking up

But again why is upward and downward convergence so rarely used? The reason is that we usually see things by looking more or less horizontally. Not only is this attitude more natural to the anatomical structure of our neck and head, but so much of what we see exists at or near eye level.

Therefore, most of the time our central visual ray is horizontal, and consequently our imaginary picture plane is vertical (i.e., at right angles to the ground). And under these conditions, vertical elements continue to appear vertical.

Just a few of the infinite number of thins typically seen at eye level. (Note true direction of vertical lines.)

When then would upward or downward convergence be appropriate? For one thing, it could be used when drama or interest was desired. But it probably makes most sense when related to the nature of the subject matter. In Other Words, Things Usually Seen from Below or from Above should be Drawn with Converging Verticals.

Examples of things typically seen by looking up, i.e., objects usually above eye level. (Note upward convergence of vertical lines.)

【P55】

Things seen by looking down

Examples of things typically seen by looking down, i.e. objects usually below eye level. (Note downward convergence of vertical lines.)

【P56】

Review: Looking up, straight out, down

So when we look up or down at an individual element, such as a single cube, each viewing angle results in a different convergence of the vertical lines. At right are the resulting pictures for each viewing angle shown at left.

【P57】

Looking straight out

But – and this is very important – if we were to back away and view all the cubes simultaneously (i.e., all within on cone of vision) then the central visual ray would be approximately horizontal and our face and the picture plane approximately vertical.

This means that all the verticals would still appear vertical. (Note also that the vanishing points must be further apart than in the previous views because the observer is further awy.)

Since looking straight out is so very natural and common, this viewpoint is probably the most frequently used in perspective drawing.

【P58】

Chapter 9: Perspective distortion

Is Related To Spacing Of Vanishing Points And Cone Of Vision

If we were now to add more cubes above and below using the same vanishing points as be fore, these new cubes would appear distorted. Their front corners (as noted) would be less than right angles. A cube would never appear this way.

The reason for this excessive convergence is simply that these new cubes are outside of the observer’s cone of clear vision.

In real life, if the observer stepped back he would see more cubes clearly (i.e., his cone of clear vision would simply include more of them) and the distortion would disappear. (See diagram at right.)

O.K. More Than Right Angle

Vanishing Point

Vanishing Point When Observer “Steps Back”

In a perspective drawing this distortion is eliminated simply by placing the vanishing points further apart.

The diagram at right shows that the observer “points” to increasingly distant vanishing points as he steps back.

Therefore: Placing Vanishing Points Further Apart Eliminates Distortion at Edges of Drawing. It Means Observer has Stepped Back and Sees Mores with His Fixed Cone of Vision.

O.K. More Than Right Angle

Wrong Less Than Right Angle

Top Views

【P59】

Observer-cone of Vision-vanishing points relationship

Picture “In Close”

Vanishing Point “In Close”

Now let’s look at this problem with elements that are placed horizontally. We shall see that the principles and solutions are the same as before.

When the observer stands close to the subject, the vanishing points are relatively close together (see top view) and the cone of vision includes only a few cubes at the center. Cubes outside the cone of vision are excessively distorted and therefore unrealistic (see picture above).

But when the observer steps back, the cone of vision includes more of the subject, the vanishing points spread apart, and the distortion is eliminated (see picture below).

Therefore: If too much distortion appears in one of your drawings, either spread the vanishing points apart (which means you have “stepped back” form the subject) OR show only the undistorted center area (which means you’re respecting a realistic cone of vision).

Vanishing Point Observer “Stepped Back”

Picture When Observer “Steps Back”

【P60】

Vanishing points too far apart

Distortion due to excessively close vanishing points is a common error because close vanishing points in general are easier to handle than distant ones. So don’t let laziness trap you.

But also avoid the opposite extreme. Placing vanishing points too far apart is also wrong because is results in minimal convergence and hence a sense of flatness.

Such is the case in the drawing above (right). The flatness is the result either of viewing the subject from too great a distance, or of limiting the drawing to objects very near the center of the cone of vision (see side view). How is it corrected? Since other objects or foreground or background features (clouds, trees, room details, etc.) would normally be visible all around the subject, these, if drawn, would give the picture a realistic three-dimensional effect. (The other solution is to “move closer” to the subject – i.e., use closer vanishing points and stronger convergence.)

In General: Convergence is minimal at the center of a picture and increases as you approach the circumference of the cone of vision. Beyond this range unrealistic and unacceptable distortion begins to occur. And naturally the further you go, the worse things get (above).

【P61】

Chapter 10: Determining heights and widths

Height lines

Assuming this is a 6 x 6 x 6-ft. cube, then the guide lines to vanishing points make all posts shown dotted also 6 ft. high. The top guide lines could be called the 6=ft. “height lines.”

If we wished to draw a 6-ft. man at point X we would simply extend forward the appropriate bottom guide line and height line.

Suppose the figure were not on an existing guide line but, for instance, at the spot marked ~.

In that case, first draw the ground line to the left vanishing point. Where this intersects the face of the cube draw a vertical line (shown dotted). (This might be still another 6-ft.-high post in perspective.)

From the top of this imaginary post draw another vanishing line. This is the 6-ft. height line for spot ~.

Suppose you wanted to draw something 12 ft. high. Simply double the 6-ft. height and carry around the new 12-ft. height line (lightly dotted).

If the constructions above were imagined as a series of 6-ft.-high picket fences or walls, then the “height line” would be a real thing instead of an imaginary guide line. Here we see more clearly how these lines establish heights as they are “carried around.”

【P62】

Height related to eye level-1: Heights when observer is standing

In this case, those persons (1) of about the same height as the observer and standing on the same ground plane would have their eyes at the same level as the observer’s (i.e., on the horizon line).

Those (2) a few inches shorter (e.g., most women) would have the tops of their heads approximately at eye level.

Children – let’s say 2½ ft. tall, about one-half the height of an adult – would naturally have their head-tops about half way up any standing adult figure. Therefore – no matter where they are placed (3) – the distance from the tops of their heads to eye level must equal their body height.

With eye level about 5 ft. from the floor, a 2-ft.-high wastebasket (4) would stand, wherever it were placed, at the bottom 2/5 of a vertical from ground to eye level.

What about the 5-ft. men on 5-ft.-high stilts? The footrests are at eye level, therefore these 10-unit figures (5) would always appear one-half above eye level and one-half below, regardless of where they stood.

The proportions used above for heights related to eye level are all verified in side view. It should be noted though that these proportions can be worked out “in perspective” without this aid. Reviewing the steps above will show this.

【P63】

2: Heights when observer is elevated position

Assume the observer (1) to be 12 ft. above ground (e.g., a 6-ft.-tall man on a 6-ft. ladder). This means all figures standing on the ground would appear below eye level.

The top of anything 12 ft. high, such a wall would therefor be level with eye level (horizon line) and would appear as shown in the drawing below.

6-ft. figures (3) standing along this wall would always be one-half the wall’s height-i.e., such figures would always stand at the bottom half of a vertical line dropped from eye level. The dotted line is their “height line.”

Therefore 6-ft. figures (4) drawn anywhere on the ground would stand at the bottom half of a vertical dropped from eye level.

4-ft. children (5) would stand at the bottom four-twelfths (one-third) of a vertical from eye level.

Proof of this system – and still another way of determining heights-can be had by first connecting the heads and then the feet of any two figures of similar height. These lines, when brought back, will meet at a vanishing point on the horizon line (see dotted lines).

【P64】

3: Heights when observer is sitting…

Here the observer’s eye level is about 4 ft. above the ground. In such a case, all others who are sitting (1) would also have their eyes at eye level.

Standing figures (2) would always have their heads above eye level. If they were 6 ft. tall then their lower four-sixths (two-thirds) would always be below eye level, and their upper two-sixths (one-third) always above. (I.e., their rib cages would always be at eye level.)

A boy (3) exactly 4 ft. high would always have his head at eye level.

Again, if the heads and feet of any two figures of equal height were connected (see dotted lines) these lines would always converge to one point on the horizon line.

4: Heights when observer is lying down

Here the observer’s eye level is about 1 ft. above the ground. Therefore objects smaller than 1 ft. would appear below eye level (e.g., most beach balls).

All taller objects would have their 1-ft. level at eye level – e.g., the 6-ft.-tall figures (2) would always appear one-sixth below and five-sixths above eye level.

The 5-ft.-high girl (3) would appear one-fifth below and four-fifths above eye level regardless of location.

The 2-ft.-hight dog (4) would always appear one-half above and one-half below eye level.

And the top of the 1-ft.-high sand castle (5) would appear at eye leve.

【P65】

Heights outdoors…and Indoors

From the roof of a 30-story building all taller buildings will appear above the horizon line while those lower than 30 stories will appear below. (This assunes all buildings have similar floor heights.)

Therefore, the eye level-horizon line will cut across the 30-story level of all buildings that have one.

Naturally the 60-story Pan Am Building in the foreground appears taller than the 102-story Empire State Building because it is closer to the observer. But in both cases their 30-story level are on the same horizontal plane as the observer’s eye level.

In sketching a room interior, many heights can be found by relating them to wall heights.

If a room is 8 ft. high then somewhere along the rear wall (1) or along one of the side walls (2) tick off 8 equal divisions. A 7-ft.-high door next to (1) or (2) is easily drawn.

Similarly, a door knob 3 ft. above the floor is located by the 3-ft. height line (3).

Suppose we wanted to draw a 6-ft. figure in the foreground. The dotted line (4) would provide the height.

Find the vanishing point of the receding horizontals and note that the eye level of the drawing is at the 4-ft. level. Therefore all 6-ft. figures would appear four-sixths below and tow-sixths above eye level, no matter where they stood.

Drawing a 3-ft.-high chair at (5) requires carrying the door knob height across to this point. A 3-ft-high boy (6) could be drawn by “carrying” this height line “around” the vertical at (7). (The guide lines shown use the center vanishing point, but any vanishing point along eye level could be used.) Notice how the boy’s height can always be verified by the 3-ft. height line from the left.

The 1-ft.-round lamp hanging over the chair from a 1 ft. chain is drawn by using the guide lines shown as arrows.

【P66】

The Surrender at Appomattox, painted by Ken Riley for Life Magazine. West Point Museum Collections.

The ground is level, so the eye level-horizon line “cuts” across the same level – chest height – of all the figures.

La Grande Jatte, by Georges Seurat. Courtesy of The Art Institute of Chicago.

The ground slopes toward the water. The adults standing on the higher ground have their head-tops approximately at eye level (horizon line). The adults standing along the shoreline have their head-tops generally midway between the ground and the horizon line. Note that this changing relationship of head height to horizon line almost exclusively “explains” that the land is sloping.

【P67】

Determining widths in perspective-width lines

In both pictures we know that the cross-ties are really equal-sized elements. Yet diminution makes them appear successively smaller. The rails converging to a vanishing point act as guide lines for the widths of these cross-ties.

The diminution of other “flat” objects, such as boats, cars people lying on a beach, etc., can be determined by the convergence of similar “width” guide lines.

For Example:

Sketch the first object (1) in a file.

Converging guide lines to eye level will define “widths” of equally long objects in the same file (2).

Objects outside of original guide lines (3) can be drawn by transferring guide line widths to left or right (e.g., dim. X = dim. X or dim. Y = dim. Y).

Suppose (say by means of an hydraulic jack) one of the objects (4) were at a higher or lower level. Its width is easily found by erecting verticals from the appropriate points on the guide lines to the new level. (naturally the same eye level and vanishing point as before would be used when drawing the object.)

To sum up: The widths of equal-sized objects which diminish in perspective can be found by means of width guide lines that converge to a point on the horizon line.

Similarly-sized objects to the left or right can be drawn simply by translating to left or right the width of the guide lines at the point desired.

Similarly-sized elements above or below are drawn with the aid of vertical guide lines constructed at the appropriate points along the width guide lines.

【P68】

Chapter 11: Determining depths

Finding center points by diagonals

The following concept is the basis for most of the aids employed in finding perspective depths:

The diagonals of any square or rectangle (see above) will always intersect at the exact center of the figure – in other words, at a point equidistant from top and bottom and from left and right edges.

Thus, on this ping pong table seen directly from above, the two diagonals will naturally intersect at the net which is equidistant from the ends.

Now, when the table is drawn in perspective, where should the net be placed? If equidistant from the ends, the result is wrong (below).

But if located at the intersection of the diagonals the result remains true. (Imagine the diagonals as actual lines ruled on the table.)

Therefore: To Locate a Midpoint Quickly and Accurately – Use Diagonals.

【P69】

Equal spacing by diagonals

To draw equally-spaced receding elements such as lampposts, first sketch two of them between the desired top and bottom guide lines leading to their vanishing points.

Now let’s develop this further in side view (far right).

Step A: Draw diagonals between (1) and (2) to determine midpoint. A horizontal line through this point gives us midpoint of (1), (2) and all similar verticals.

Step B: Draw diagonals from (1) through midpoint of (2), to locate (3). Since the diagonals place (2) exactly midway between (1) and (3), the location of (3) must be correct.

Step C: Subsequent equidistant verticals are located by similar diagonals. (Note: It isn’t necessary to draw both diagonals. One of them, used with the center line, gives the same result.)

The application of these steps in perspective will assure equally-spaced elements drawn with proper convergence and foreshortening.

Below are Several Examples of This Method. Study The Various Applications.

【P70】

Subdividing a surface by diagonals

Suppose we wanted to divide face A of this object into two equal spaces, face B into four equal spaces, and the top into eight equal spaces.

BELOW is the solution when each face is viewed head-on.

AT Right is the same solution in perspective.

Naturally, this method works only when the number of spaces is 2, 4, 5, 16, 32, 64, etc. Suppose we wished to divide a face into 7 or 10 spaces. Then the following method should be used, for it works for any number of equal spaces.

Dividing A Surface Into Equal Spaces By Using A Measuring Line And A Special Vanishing Point

(Please follow the numbered steps.)

Step 1: From lowest corner of face to be divided draw horizontal line and tick off the number of equal spaces desired (7 in this case).

Special Vanishing Point For Guide Lines

Vanishing Point For Object

Step 2: Connect point 7 to opposite lower corner and continue to horizon line. This gives us a special vanishing point for all guide lines parallel to this one.

Measuring Line

Not: The equal spaces ticked off in step 1 could be at any scale. Those shown are each 3/8” spaces, but could be ¼”, ½”，5/8”, etc. Naturally every spacing will shift the special vanishing point, but the resulting perspective spacing will always be the same.

Why this is so is explained in this top view. Let’s divide the same face in two, using different spacings. From the lowest corner tick off two units of ½” each, two of 1” and tow of 2”. Now, connect each second tick to the far corner (3 lines shown dotted). Then, from the first ticks, draw lines parallel to these. Note that the second lines all intersect at the midpoint of the face. Therefore any of these spacings would work even though each resulting set of parallel-horizontal lines would have it own (special) vanishing point (see across page).

【P71】

Special Van. Pt.-Set C

Special Van. Pt.-Set B

Special Van. Pt.-Set A

Vanishing Point For Object

Measuring Line

This is the previous top view diagram seen in perspective. So remember: The Spacing Used Along The Measuring Line Can Be At Any Scale.

Dividing a surface into unequal spaces with a measuring line and special vanishing point

The “measuring line” method of dividing perspective surfaces may also be used with unequal spaces. Suppose a 2-ft. opening is to be located on a wall, spaced as below.

(Follow numbered steps as before.)

Step 1: Tick off 1 unit, 2 units, and 4 units on measuring line. (As before, the units can be at any scale. The principle is the same as in the case above.)

Step 2: Connect end points to establish special vanishing point.

Step 3: Bring other lines to special vanishing point. This locates opening on wall.

New Special Vanishing Point

Once correct spacing is found on wall, the distances could be extended forward (from left vanishing point) to create, say, a 2-ft.-wide walk or a 4-ft.-long bench.

Or, by carrying guide lines up the wall and over the roof, a 2-ft.-wide chimney could be drawn. But note that to fix the depth of this chimney (1 ft.), we need a new measuring line and special vanishing point, and new guide lines (shown dotted).

【P72】

Determining depths and widths of room interiors by the measuring line method

Here, floor to ceiling screens are placed as shown in top view below. Also at the same one-third points are thin wall lines, e.g., mullions.

- The depth locations of the mullions (and screens) are found as before (see pp. 70, 71) – by ticking off three equal spaces on a measuring line and drawing converging lines to a special vanishing point.
- The left to right locations of the screens are then found by means of another measuring line ten units long (2+2+4+2). (Units may be at any scale. Here, each unit = ½”.)
- The location of this measuring line is determined by sliding a ruler back and forth until the desired number of units fits exactly between projections of the floor lines.

In this top view, the elements receding from observer are unequally spaced. But, as we have seen on the previous page, the same method can be used.

Again the depth locations are found by ticking off the appropriate spaces along the measuring line, connecting the last tick with room corner, and then all other ticks to the special vanishing point. This locates spacing along the left wall base, from which it is carried across the room. The left to right locations are the same as in the case above and are found as above.

【P73】

Another way of getting depths: Te sliding ruler and diagonals method

Suppose we wanted 5 equal vertical divisions in this rectangle (to draw, for instance, 5 equally-thick books).

Step 1: Simply tick off the required spacing on some vertical line by sliding a ruler (as on the previous page) to find a position where 5 equal units fit. (Note that either 5 @ ½” or 5 @ 3/8” would be o.k.)

Step 2: Converge each tick to vanishing point at right.

Step 3: Draw diagonal as shown.

Step 4: Draw vertical lines at each point of intersection. These will correctly demarcate 5 equal divisions in perspective.

Why this is so s explained by these front views of various rectangles. The diagonals always divide the adjacent sides proportionately. In other words, by means of the diagonal the spacing along a vertical edge is transferred proportionately to a horizontal edge.

Suppose, instead of being divided into equal spaces, the rectangle were to be divided unequally. For example, on the same 5-unit-long wall let’s draw a 2-unit door located 1 unit away from the front end. The drawing at right show that the same method can be used for unequal spacing.

(Note: always start vertical spacing ticks from the same top or bottom edge as diagonal. E.g., at far right the diagonal starts at bottom, therefore the 1-unit tick also starts there.)

this method is also applicable to horizontal planes such as a floor. Again, equal or unequal spacing can be determined.

For instance, let’s divide both the depth and width of this plane into 5 equal spaces.

Step 1: 5 spaces @ ½” fit here and so can be used to divide the width. (Or use 5 units @ ¾”, below.)

Step 2: Draw guide lines to vanishing point, then draw diagonal.

Step 3: Draw horizontal lines at intersection points. These are the required 5 equal spaces in depth.

【P74】

Drawing equal-sized but unequally-spaced elements-vanishing point of diagonals method

Suppose we drew one shape, such as rectangle A, and wished to repeat it (for instance, in order to draw a line of cars on a road). If the rectangles were touching, the method of diagonals shown on page 69 could be used, but since the are not, another method is needed.

Step 1: Draw the diagonal of the first rectangle and extend it to the horizon line. This locates the vanishing point for this and all other lines parallel to it.

Step 2: Extend the sides of rectangle A to their vanishing point. These are the “width guide lines” for all rectangles in line with the first.

Step 3: Draw front line of next shape (shown dark). Then from point 1 draw line to diagonals’ vanishing point. Intersection at point 2 locates the back line and thereby creates a second rectangle equal to the first.

Step 4: For other rectangles, follow the same procedure. Identical diagonals will produce identical rectangles.

This method will also work for vertical planes, such as a row of building facades, sides of trucks, etc.

The procedure is exactly as above and the diagram is identical. (Revolve this book 90 degrees and see.)

Note that the horizon line of the first case now becomes a vertical line. But like the horizon line, this vertical receives all lines on ,or parallel to, the wall plane. The diagonals, therefore, converge to a point on this line as shown. (If the other set of diagonals (shown dotted) were used, their vanishing point would be above eye level but on the same vertical line.)

Diagonals’ Vanishing Point

An examination of the top view of the first example (left) and the side view of the second (above) will show why this works.

Note that once the width or height lines are drawn, any set of parallel lines will strike off equally-long rectangles by becoming their diagonals. And since they are parallel these lines naturally converge to the same point in perspective (the diagonals’ vanishing point).

【P75】

Diagonals as and aid in drawing concentric and symmetrical patterns on rectangles and squares

Within a square or rectangle, a multitude of concentric patterns can be drawn in correct perspective by bringing horizontals to their vanishing point, drawing verticals, and “turning” the pattern at the diagonals.

Essentially, the diagonals allow you to “carry the pattern around,” thereby maintaining symmetry. Studying the side view of the design at right will help explain how this works in perspective.

Suppose you had determined one point on a rectangle (such as one of the knobs of a radio) and wished to locate anther symmetrically.

(The following procedure applies both to top view and perspective drawing.) 1^{st}: Draw diagonals. 2^{nd}: Carry around guide lines (arrows) as shown. Basically, this creates a concentric rectangle. 3^{rd}: Draw line parallel to side of rectangle (shown dotted) to locate the desired point.

Many symmetrical patterns can be drawn accurately and quickly by using diagonals in this manner.

【P76】

And design or pattern can be reproduced in perspective by means of a grid that locates its important points

For example, in the drawing at right, grid lines (light lines) have been drawn through the design’s key points. This grid “transfers” the spacing of the points to the edges of the surrounding rectangle, thus creating measuring lines 1 and 2.

Measuring Line No.1

Special Vanishing Point No.1

Special Vanishing Point No.2

To locate the design in perspective, we simply draw the rectangle by approximation and then lay out measuring lines 1 and 2 from point A as shown. By using the special vanishing points of these lines, we can then transfer the edge measurements to the perspective rectangle. This, in turn, allows us to draw the grid in perspective, and the grid intersections enable us to reconstruct the design.

Measuring Line No.1

Meas. Line No.2

Here again a series of key points has been located on a grid, which has then been drawn in perspective.

The spacing of the points was transferred to the perspective view by using measuring lines “A to E” and “0 to 6”.

(measuring line “0 to 6” was located simply by sliding a paper with ticked-off spacings back and forth until it fit exactly between the proper guide lines.)

Vanishing Point For Measuring Line A.E

Measuring Line

【P77】

Chapter 12: Inclined planes – Introduction

Since the bottom of this box is horizontal, its converging lines always vanish to eye level. An observer pointing in the direction of the box (horizontally) therefore points to its “vanishing line” (first drawing).

So it is with the pivoting box top. An observer pointing in the same direction as this variously-inclined plane points to its successive vanishing lines.

Box and box top are parallel, therefore only one vanishing line (eye level).

Here, the box top “points” somewhat below eye level, therefore it converges to a point slightly below eye level.

(Note That The Vanishing Points For Box And Box Top are Always On This Vertical Center Line. This Is True Regardless Of The Top’s Inclination.)

Here, the box top again points below eye level, but because it is nearly vertical, its vanishing point is far away.

Here, the box top, like the box front, is parallel to the observer’s face (picture plane), therefore it does not converge. (Vanishing points are at infinity.)

Here, the box top points far above eye level to its distant vanishing point.

Here, the box top’s vanishing point is still above eye level, but closer.

Finally, the box and box top are again parallel, therefore they point to and use same vanishing point.

【P78】

Vertical vanishing line and horizon line are based on same theory and serve similar purposes

First let’s review: The top and bottom edges of each page of the books at right lie on horizontal planes (see side view). Therefore, they must all converge to the horizon line (eye level), which we recall is always on a horizontal plane through the observer’s eyes – i.e., parallel to the planes in question. (See pages 28 and 29 for further review.)

Now suppose we held the book this way (if it were possible) so that both the “top” and “bottom” (here, left and right) edges of the pages are now bounded by vertical, rather than horizontal, parallel planes. As above, these two parallel planes, and all the lines that lie on them, will converge to one vanishing line. But in this case the vanishing line is vertical. Where is it located? As in the case above, it is situated on a plane through the observer’s eyes and parallel to the planes in question (see top views).

In the diagram at far right, the left and right edges of the pages, and hence the plane through observer’s eye, point to the right, therefore the vertical vanishing line shifts to the right.

So it is important to realize the vertical vanishing lines and the horizon line (eye level) and are similar in concept. Both “contain” the vanishing points for parallel sets of lines which lie on parallel planes. The horizon lines serves all sets of lines on horizontal planes while a vertical vanishing line serves all sets of lines on parallel vertical planes. Both are found in the same manner; in fact the drawings above and below are identical except for being turned 90 degrees from each other.

The diagrams at the bottom of this page and on the previous page also show that the vanishing point for parallel lines on inclined planes (e.g., the edges of the book pages or the sides of the box top) is always directly above or below the vanishing point those lines would have if they were pivoted to a horizontal position.

【P79】

Uphill and downhill (Inclined planes)

The building lines, the windows, the brick coursing, and the steps, are all horizontal, therefore they converge to a vanishing point on the horizon line (eye level).

The sidewalk, street, curb baby carriage, auto, and line of intersection between sidewalk and houses, are all inclined upwards, therefore they converge to a point directly above the first.

Now note the ice cream vendor’s wagon and the pushcart. The wagon is on the level part of the street, so its lines naturally converge to vanishing points on the horizon line. The lines of the pushcart on the inclined plane, on the other hand, converge to points along the vanishing line of the inclined plane. But since both vehicles are turned at the exact same angle from the sidewalk, their vanishing points (left and right) lie on the same vertical vanishing lines, directly above and below one another.

Here again (foreground) lines that are horizontal converge toward a horizon line vanishing point (near center), and those that are inclined (downwards) converge to a point directly below.

Now, when the road turns to the right, what happens to the horizontal lines of the houses? Their vanishing point remains on the horizon line but simply shifts accordingly to the right.

And if the road also goes downhill its vanishing point is directly below this new horizon point. (Note that both downhill vanishing points are on the same horizontal vanishing line because the angle of incline is the same.)

To Repeat: The Vanishing Point For Parallel Lines On An Inclined Plane Is Always Directly Above Or Below The Vanishing Point Those Lines Would Have If They Were Pivoted To A Horizontal Position. (See p. 77.)

【P80】

Some applications of inclined plane perspective

【P81】

Chapter 13: Circles, Cylinders and Cones

Circles and ellipses: Circles, Except When They Are Parallel To Observer’s Face, Will Foreshorten And Appear As Ellipses

This silver dollar appears as a perfect circle only when seen “front face.” When pivoted around a diameter line, it changes from a ground to a “skinny” ellipse, till finally it appears as a thin line.

Stood upright and pivoted around a vertical line, this circle appears much the same as above.

Upright circles parallel to one another and at right angles to observer (i.e., to picture plane) will appear as increasingly rounder ellipses the further they are from center line of observer’s vision. (Above.)

Horizontal circles stacked one above the other and at right angles to observer (picture plane) will appear as increasingly rounder ellipses the further they are from observer’s eye level-horizon line. (Right.)

Horizontal circles seen simultaneously, even the ones far to the left and right of center line of observer’s vision, should all be drawn as true ellipses – despite the fact that in rigorous mechanical perspective circles at far left and right would come out as distorted ellipses.

【P82】

Drawing the ellipse

The two-dimensional circles on the previous page could represent coins, phonograph records, pancakes, lenses, etc. (right). But circles are also key parts of three-dimensional objects such as cylinders and cones, and as such have wide application in representational drawing. Cylinders are the bases of an endless number of things such as cigarettes, oil tanks, threadspools, chimneys, etc. Cones are the bases of ice cream cones, hour glasses, martini glasses, funnels, etc. Therefore, the importance of learning to draw circles in perspective – i.e., ellipses-can hardly be overestimated.

What Is An Ellipse And How Can We Learn To Draw It?

An ellipse is an oval figure with two unequal axes (major and minor) which are always at right angles to one another. These axes connect, respectively, the figure’s longest and shortest dimensions, and about each of them the curve of the ellipse is absolutely symmetrical. This means four identical quadrants, with each axis dividing the other exactly in half (x=x, y=y).

One should learn to draw an ellipse freehand with a loose, free-swinging stroke. Ellipses A and B are attempts at this. Anyone familiar with ellipses can visualize the major and minor axes and see that A is good while B lacks the necessary symmetry. (If we draw B’s two axes, we can see the errors much more clearly. Notice how each quadrant differs.)

You might find it helpful to sketch a rectangle around the tick marks. This creates four more guide lines within which the shapes can be judged and compared (above).

So a good way to begin learning to draw (and to visualize) ellipses is to start by sketching these exes. Tick off equal dimensions on either side of the center to locate extremities.

Another good technique is to draw the curve carefully in one quadrant. Then transfer (tracing paper works well) this shape to the other three quadrants, using the axes as reference lines (above).

Then try drawing four equal quadrants. Note: the ends are always rounded, never pointed.

【P83】

The center of a circle drawn in perspective dose not line on the corresponding ellipse’s major axis – It is always further away (From observer) than major axis

This astonishing fact is often a cause of great difficulty (even in books on the subject). What is the relationship between the circle’s center and the ellipse’s axes?

A true circle can always be surrounded by a true square. The center of the square (found by drawing two diagonals) is also the center of the circle (left).

The circle in perspective (right) can also be surrounded by a foreshortened square. Drawing the diagonals will therefore give the center of both square and circle. We know from page 68 that this point is not midway between top and bottom lines. So the circle’s diameter drawn through this center point is also not midway between top and bottom.

Yet we know (right) that the major axis of an ellipse must be midway between top and bottom lines.

So, combining the two drawings (right) we see that the circle’s diameter falls slightly behind the ellipse’s major axis. (Note, too, that the minor axis is always identical with the most foreshortened diameter of the circle.)

The top view (left) explains this seeming paradox. The widest part of the circle (seen or projected onto the picture plane) is not a diameter but simply a chord (shown dotted). It is this chord which becomes the major axis of the ellipse, while the circle’s true diameter, lying beyond, appears and “projects” smaller.

Note: points of tangency between ellipse and square (1, 2, 3, 4) are exactly at diameter lines, just as in the true top view (upper left).

This is true regardless of the angle or position of the ellipse.

So don’t make the mistake of drawing a foreshortened square and using its center to locate the major axis of an ellipse. The resulting figure would look something like this (right).

Also, if you wish to draw half a circle (or cylinder) you cannot draw an ellipse and consider either side of the major axis to be half of a foreshortened circle. E.g., the figure at left is not half but less.

The two at right, however, are each proper halves, because the circle’s diameter is used as the dividing line.

【P84】

Cylinders

Regardless of the position or angle of an ellipse, its major and minor axes always appear at right angles.

When Drawing A Cylinder – Its Center Line Must Always Be Drawn As An Extension Of The Related Ellipse’s Minor Axis. Therefore, this center line (the axle of wheels, the crossbar of bar bells, the shaft of a gyroscope, etc.) always appears at right angles to the major axis of the ellipse associated with it.

But note that this center line connects to the ellipse at the center point of the circle and not to the center of the ellipse. (Otherwise the shaft would be eccentric – literally “off center.” See previous page.)

By redrawing two of the objects above, we can see here that foreshortened squares in any direction can be constructed as guides around a circle. But in every case The Opposite Points Of Tangency (dots) will Terminate Diameter Lines Through The Circle’s Center. (In reality these lines are at right angles.)

The ellipse’s major axis (dotted) has nothing to do with this – it is merely a guide line for drawing the ellipse. (Note again that the ellipse’s center is closer to the observer than the circle’s center.)

Below are some applications of these principles.

【P85】

Cones

Drawing cones is similar to drawing cylinders. The center line of a cone is also an extension of the related ellipse’s minor axis … it lies at right angles to the ellipse’s major axis … and it connects to the ellipse not at the ellipse’s center point, but behind it. Study these various principles in the drawings above.

The cone within the cylinder (right) naturally has its center line parallel to the table top. Therefore the cone’s apex is in the air. To draw the cone resting on the table its apex must drop (arrows) so that its center line falls approximately to the dotted line.

The cone at far right is drawn with this dropped center line. (This motion slightly foreshortens the length and makes the ellipse “rounder.”)

Therefore, Cones Lying On Their Sides Have Center Lines Inclined To The Planes On Which They Rest.

The similarity of the ellipses at right indicates that these cones are similarly oriented but of different lengths.

While here the varying ellipses and foreshortened lengths suggest that the cones are pointed in various directions and are approximately similar. (Note that the sides of the cone always connect to the ellipse tangentially.)

【P86】

Circles, ellipses, cones, cylinders and spheres applied to a “Space Age” drawing.

【P87】

Chapter 14: Shade and Shadow

First, let’s clarify our terms: SHADE exists when a surface is turned away from the light source. SHADOW exists when a surface is facing the light source but is prevented from receiving light by some intervening object.

For example: This suspended cube has several surfaces in light and several in shade (those turned away from the light). The table top is turned toward the light and would be entirely “in light,” except for the shadow “cast” on it by the cube above. We might say that the intervening object’s shaded surface has “cast” a shadow on the lighted surface.

The Shade Line is that line which separates those portions of an object that are “in shade” from those that are “in light.” In other words, it is the boundary line between shade and light. This shade line is important because it essentially casts, shapes, and determines the shadow. (Right.)

Note that the shadow line of a flat, two-dimensional object is its continuous edge line. (One side of the object is in light, the other in shade.)

Shade and shadow naturally exist only when there is light. Light generally is of two types, depending on its source. One type produces a pattern of parallel light rays, the other a radial pattern.

The sun, of course, radiates light in all directions, but the rays reaching the earth, being 93 million miles from their source, are essentially a small handful of single rays virtually parallel to one another. Therefore, when drawing with sunlight the rays of light should be considered parallel.

The other type of light originates from a local point source such as a bulb or candle. Here, the closeness of the light source means that objects are receiving light rays that radiate outward from a single point. Therefore, when drawing with local point sources the rays of light should be radial.

【P88】

Parallel light rays (Sunlight) parallel to observer’s face (And picture plane) – The simplest case of shade and shadow drawing

The top view at left shows observer looking at a table which has a pencil (small circle) stuck into it. The parallel light rays arriving from the left are parallel to the observer’s face and to the picture plane. Therefore, the pencil’s shadow must lie along the light ray shown dotted.

The shadow’s length depends on the angle of the light ray, but this can only be seen in perspective (right). All angles are possible. Here, we use 45 degrees, which makes the shadow’s length equal to the pencil’s length. The light ray from eraser to dotted line locates the eraser shadow and hence fixes the length of the pencil’s entire shadow.

Similar pencils will cast similar shadows. The shadows at left are all parallel to one another and to the picture plane. Therefore in perspective they remain parallel. Note individual light rays “casting” eraser shadows on table.

Here, the two pencils at loser left are “bridged” by a third pencil which casts a new shadow line connecting the eraser shadows of the first two. This new shadow must be parallel to its shade line (bridging pencil), therefore, in perspective, both shade line and shadow use the same vanishing point. The other pencils in this drawing are “filled in” to form an opaque, two-dimensional plane. Its shadow is outlined exactly as before, so in both cases the outline pencils are the shade lines which determine the shadow’s shape.

Now let’s build two cubes using the existing two planes as sides. This creates new shade points at 1 and 2 which cast shadow points 1^{s} and 2^{s}. These points help locate the shadow lines (shown dotted in top view) of the new top and vertical shade lines. Note the two vertical shade lines of the above drawing that have ceased to be shade lines here, since they are no longer boundaries between light and shade.

【P89】

The Following Application Sketches All Have Shadows Cast By Parallel Light Rays Parallel To The Observer’s Face

Therefore, the principles developed on the previous page will be evident. Note: Arrowed lines are light rays used to locate important shadow points. Dotted lines are temporary guide lines required to locate shadow.

【P90】

Parallel light rays (Sunlight) oblique to observer’s face (And Picture Plane)

- When the rays are not parallel to observer’s face (and picture plane) then they must appear to converge. This means more complex drawing but it does enable us to represent shadows the way we usually see them in sunlight.

In the top view below, light rays arrive at the angle shown by arrows. Therefore the pencil’s shadows must lie along the dotted lines. As before, equal pencils cast equal and parallel shadows. In this case, though, the shadows are oblique to picture plane, so in perspective (right) they converge to a point on the horizon line. - How do we draw the light rays that determine that shadow lengths? Previously, when the rays were parallel to the picture plane, we simply drew them parallel to one another. But now, being oblique, they must converge.

their vanishing point, furthermore, must lie on the same vertical vanishing line as the shadows’ vanishing point. Why? Because both the rays and the shadows lie on parallel planes (see top view. For a review of vertical vanishing lines, see pages 77-80).

Once this point is fixed, light rays passing through the erasers locate the correct shadow lengths.

Light Rays’ Van. Point

Shadows’ Van. Point

Light Rays’ Van. Point

Though difficult to visualize, this vanishing point is actually the SUN many millions of miles away radiating a handful of parallel (converging because of perspective) light rays. - Now bridge a pencil across the eraser ends of the two lower left pencils below. Its shadow on the table will naturally connect the shadows of the eraser ends. Since the upright pencils cast shadows of equal length this new shadow must be parallel to the bridging pencil (shade line).

The other two pencils are “filled in” to create an opaque, two-dimensional plane. Again the new top shade line casts a parallel shadow line on the table top. - In perspective, these two new sets of parallel horizontal lines will each converge to a vanishing point. Note, thought, that these new vanishing points are not essential to the drawing, since all key shadow points can be determined by the same light rays and shadow lines as above. (The new vanishing points can help, however, to verify the results.)

Shadows’ Van. Point - 【P91】Now let’s again build two cubes using the existing planes as one side of each. This creates two new shade points 1 and 2 which cast shadow points 1
^{s}and 2^{s}.

The two new top shade lines again cast shadows parallel to themselves, thus forming two new sets of parallel, horizontal lines, each of which converges to its own vanishing point on the horizon. - Notice that each cube has a total of only two vertical shade lines. The other verticals that had been shade lines in previous steps are now “buried” in shade, i.e., are no longer boundary lines between light and shade; and therefore do not cast shadows. Only their shade points (*) remain to cast shadow points (*s).
- In all examples so far the sun has been in front of the observer, and therefore the vanishing point of the parallel light rays has been in fact the sun itself.

what vanishing point is used for the rays when the sun is behind the observer (top view below)? The theory is exactly the same. In perspective (right) the inclined parallel rays converge to a point, and, as before, this point is located on the vertical vanishing line that passes through the vanishing point of the pencils’ shadows (since rays and shadows lie on parallel planes). - But two important differences do exist: 1.) This vanishing point is not the sun itself but simply a vanishing point for a set of inclined parallel lines. 2.) It is always below the horizon line. (If above, it would mean the sun is in front of and visible to the observer.)

The side views below, picturing the observer pointing to vanishing points, should explain why this is so.

Sun From Behind

Sun From In Front

【P92】

The Following Application Sketches All Have Shadow Cast By Parallel light rays(Sunlight) oblique to observer’s face (And to picture plane)

Therefore, the principles developed on the previous two pages will be evident. Note that when that vanishing point for light rays is close to the horizon the sun is low in the sky and the shadows are long. As the vanishing point increases in distance from the horizon – either above or below – the sun rises higher and the shadows become shorter.

Note: Arrowed lines are light rays used to locate important shadow points. Double-arrowed lines are shadow directions of uprights on ground plane. Dotted lines are temporary guide lines required to locate shadows.

The key to drawing the shadow of an inclined object (above) is locating the shadow of a point such as 1. This is done by constructing an upright (i.e., dropping a vertical) from point 1 to ground point 2 (located on ground-line B-C, which goes to right vanishing point and is directly below the oblique pole A-B.) Then, if we cast the shadow of the upright 1-2 (first along the ground and then up the wall ) we can fix point 1^{s}. Thereafter, by extending a line from A thru 1^{s} and by connecting B with the ground intersection point of this line, we arrive at the correct shadow directions.

【P93】

Two unique examples of shade and shadow resulting from sunlight.

Ice Glare, by Charles Burchfield. Collection of Whitney Museum of American Art, New York

Under the Swings, by Robert Vickrey. Collection of Dr. and Mrs. John M. Shuey, Detroit. Courtesy of Midtown Galleries, New York

【P94】

Shade and Shadow created by local point sources of light

A light bulb hangs above a table at the point shown in top view. The diverging light rays must cast shadows along the dotted lines. In perspective, these same shadow lines are found by drawing the same diverging lines from the same table point (asterisk) through the same pencil points. These lines, naturally, are not light rays, for the actual source of light is somewhere above the table. To determine the exact lengths of the shadow along them, actual rays are drawn (from the light source) through erasers.

Rule: A local point source radiates diverging light rays which locate exact shadow points. But the shadow direction of upright lines on a plane is determined by lines diverging from the point on the plane that is directly below the light source.

Placing a “bridging” pencil over the lower left pencils and “filling in” the other two to create an opaque plane (below) results in tow new top shade lines. These lines cast shadows parallel to themselves. The parallel set at lower left, being parallel to picture plane, remain parallel in perspective. The parallel set at right converge to the horizon. RULE: As in the case of parallel rays (sunlight) a straight line casting a shadow on a parallel surface casts a shadow parallel to itself.

Now let us gain build tow cubes using the existing planes as one side of each (below). This creates three new shade points (1,2,3) which cast shadows points 1^{s}, 2^{s}, 3^{s}. The three new top shade lines again cast shadows parallel to themselves. Therefore, in perspective, these shadows will converge to the vanishing points of their shade lines. (Note how the location of the light creates three top shade lines on one cube but only two on the other.)

【P95】

The Following Application Sketches Have Shadow Cast By Local Light Sources

Therefore, the principles developed on the previous page will be evident.

Note: Single-arrowed lines are diverging light rays used to locate shadows.

Dotted lines are guidelines which “project” the light source onto the various surfaces (walls, floor, table top, etc.).

They originate from the light source and are drawn perpendicular to the surfaces. (The projected light sources are marked by asterisks.)

Double-arrowed lines (originating from the asterisks) are the guidelines that determine the directions of shadows cast by shade lines perpendicular to a given surface.

【P96】

18th-century drawing, artist unknown. Courtesy of the Cooper Union Museum

## 0条评论

## 评论此文：