1. Circles in Perspective - Elliptical Shapes
A circle is a closed curve on a plane whose every point is equidistant from a fixed point called the center. The word circle is taken from the ancient Greek goddess, Circe, who was believed to weave the fates of mankind on her cosmic spinning wheel.

2. In the visual world, circular objects rarely appear as circles.
Major Axis
Minor Axis
An ellipse looks like a circle that has been compressed along a single axis. Ellipses can be made by attaching a length of string to two foci on the major axis that is longer than the distance between the foci. The combined distance from the foci to the elliptical curve is identical for all points on the curvature. The longest diameter is called the major axis, while the shortest is called the minor axis (above left). An ellipse is bilaterally symmetrical (exact correspondence) on both sides of each axis. Although there will be no need to use a mechanical device to construct ellipses as we draw, it might be helpful to understand how they are formed. The sum of two line segments that connect any point on the curve of ellipse with the two foci on the major axis is always the same (above, right). This means that if you attach a length of string to the two foci (longer than the distance between the foci), and then press a pencil against the tightly stretched string, the curvature of the ellipse will be determined by the string’s resistance.
In the visual world, circular objects rarely appear as circles.
Most commonly, an observed circle will appear as an ellipse.

3. An ellipse is a continuous curve, compressed along one axis.
No Flat Spots,
No Football Ends
The two most common errors that occur when drawing an ellipse as a foreshortened circle are the flattening out of longer curving sides and the substituting of what looks like the ends of an American football for the fluid and steep curve that actually occurs at the ends. Both errors can be eliminated by moving your pencil through a series of continuous, sweeping, and delicate gestures that approach the entire ellipse as a unified and seamless entity.
An ellipse is a continuous curve, compressed along one axis.
It is bilaterally symmetrical.

4. A cylinder is a solid with a curved side that is bounded by two parallel circles. A sphere is a three-dimensional surface whose points are all equidistant from a fixed point. The fact that both these solids share essential characteristics with circles allows you to apply the same fundamental elliptical principles governing the appearance of foreshortened circles to all three.

5. In the visual world things that are circular rarely appear as circles
In the visual world things that are circular rarely appear as circles. Most commonly circles in perspective appear as ellipses.

6. At eye level, the circle will appear to be a straight line.
The further the circle from eye level, the more open the ellipse.
When circles are parallel to the ground plane, the major axis of the ellipse is always horizontal to the viewer.
Eye Level
Circles that recede in space and are parallel to the ground plane always have a major axis that is horizontal. At eye level, a circle appears as a straight line. As it moves further up or down from eye level, it appears as an ellipse that is opening wider and wider (its proportion approaches—but will never reach —1 :: 1).

7. Drawing a central axis promotes symmetry in cylindrical forms.
Drawing an ellipse is easier if you do not stop the pencil until you have gone around the ellipse several times.
Drawing an ellipse is easier if you don’t stop the pencil until it has gone completely around at least once. It also helps to maintain a delicate touch with the pencil so that if the ellipse needs multiple adjustments you can continue to gesture without the surface of the drawing becoming too congested. Gesturing the ellipse multiple times generally contributes to a more symmetrical and consistent ellipse. It also helps to indicate the major and minor axes as you create the ellipse because it further encourages bilateral symmetry.

8. Keep your pencil moving until you have completed at least one complete rotation around the entire ellipse. Drawing “through the form” also contributes to the illusion of three-dimensional space because it leaves subliminal traces of receding edges.

9. The amount of decompression (fullness) in the foreshortened circles in cylindrical objects ultimately depends on their relative vertical position on the picture plane. The most compressed ellipse is closest to the top of the page and the fullest ellipse is located closest to the bottom of the page with a full range in between. This change in decompression is a proportional change only and is not affected by the overall size of the foreshortened circles.

10. Circular cross-sections appear to open up (become less compressed ellipses) the further they are from eye-level. As a result, the bottoms of cylindrical forms are always more curved than their tops. This is important to remember given that we conceptually understand the bottom to be flat and are thus predisposed to under-represent the observed curvature.

11. This illustration depicts a series of parallel circles whose central axis is parallel to both the ground plane and the picture plane. In this relationship the foreshortened circles appear as progressively less compressed ellipses as they move out toward the sides. In the current configuration, the widest part of the ellipse (the major axis) is always vertical.

12. Three coffee cups all share a common cylindrical axis
Three coffee cups all share a common cylindrical axis. That axis is, by definition, parallel to the cup’s outside edges and, as such, converges with those edges as they go back in space (railroad track phenomenon). This shared cylindrical axis intersects at 90º the major axis of each of the ellipses that are used to represent the ends of the foreshortened cylinders.
The major axis of a foreshortened cylindrical cross section will intersect the central axis at a precise right angle.

13. In the physical world a central cylindrical axis is, by definition, equidistant from and parallel to the outside edges of the cylinder. In a two-dimensional image, the central axis is always equidistant from the cylinder’s edges, but it can either be parallel to them (when it and the outside edges are parallel to the picture plane) or converge with them at a common point (when the cylinder is receding in space). Any circular cross section of a cylinder that is tilting back in space appears as an ellipse that is compressed along its minor axes and whose major axis is an actual 90° to the cylindrical axis.

14. Foreshortened Circle (spatial illusion)
Ellipse (flat)
Foreshortened Circle (spatial illusion)
The center of an ellipse is located where the major and minor axes of the ellipse intersect. The center of a foreshortened circle (a circle in perspective), however, is slightly more complicated. To locate it mechanically you can inscribe it in what would be a foreshortened square with converging edges and then crossing diagonals from the corners to locate the center of both the foreshortened square and circle. You might, however, find it simpler to just remember that the center of a foreshortened circle is always slightly closer to the back edge of the foreshortened circle than to the front.
The center of a foreshortened circle is always closer to the back edge.

15. In order to prevent distortion, artists need to be sensitive to the
conflict between “what they know” and “what they see”.
The center of an ellipse is found by inscribing it in a rectangle and drawing diagonal lines between each set of corners. The center of the ellipse is located where the lines intersect. The center of a foreshortened circle (a circle in perspective), however, is slightly more complicated. To locate it mechanically you have to inscribe it in what would be a foreshortened square with converging edges. You might find it simpler to just remember that the center of a foreshortened circle is slightly closer to the back edge of the foreshortened circle than to the front.

16. The center of an ellipse is found by inscribing it in a rectangle and drawing diagonal lines between each set of corners. The center of the ellipse is located where the lines intersect. The center of a foreshortened circle (a circle in perspective), however, is slightly more complicated. To locate it mechanically you have to inscribe it in what would be a foreshortened square with converging edges. You might find it simpler to just remember that the center of a foreshortened circle is slightly closer to the back edge of the foreshortened circle than to the front.