Major Axis
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 fore; shortened 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.
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?
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.
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. )
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.
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.
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, twodimensional 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 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.
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