1. Circles in Perspective
(ellipses)
by Brian Curtis
© 2002, The McGraw-Hill Companies
© 2002, The McGraw-Hill Companies

2. A PowerPoint lecture series to accompany DRAWING FROM OBSERVATION
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3. 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.
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4. Ellipses
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.
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5. 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.
© 2002, The McGraw-Hill Companies

6. 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).
© 2002, The McGraw-Hill Companies

7. It takes practice to move your pencil fluidly around the contour of an elliptically shaped foreshortened circle. It is a challenge to maintaining the continuous curve, preserve the bilateral symmetry around both axes, and establish the proportion of the ellipse so that it appropriately reflects its relative distance from the eye level.
© 2002, The McGraw-Hill Companies

8. 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.
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9. Even though you probably won’t ever respond quite as inaccurately as is depicted in this manipulated photo, it is important to recognize how powerful your rational tendency is to substitute what you know for what you see. Pre-conceived notions that suggest the mouth of the cup is circular and the bottom is perfectly flat will lead to substantial underestimation in the progression of elliptical proportion.
© 2002, The McGraw-Hill Companies

10. To capture the apparent increased curvature that occurs when foreshortened circles (ellipse) are positioned progressively further away from our eye level, we need to make a concerted effort to make each foreshortened circle slightly fuller than the one preceding it.
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11. 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.
© 2002, The McGraw-Hill Companies

12. Foreshortened circles appear fuller (less compressed) when they move down a vertical axis from eye level. You see this difference in the tops and bottoms of individual cylinders (a). Foreshortened circles also become fuller when they move toward the picture plane. You can see this by comparing the ellipses from the two identical glass cylinders positioned at different distances from the observer (b). Although the two types of movement (down vs. out) are quite unrelated in terms of real space, they produce identical changes in elliptical decompression. Any two ellipses whose major axes are horizontally aligned on the picture plane, regardless of whether that position is determined by physical height or distance from the picture plane, share identical proportions. This can be seen in image c where the more distant cylindrical object, by being repositioned directly above the closer object and with the addition of some unifying chiaroscuro is proportioned appropriately to appear as the cap of a bottle in the foreground.
© 2002, The McGraw-Hill Companies

13. 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.
© 2002, The McGraw-Hill Companies

14. 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.
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15. 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.
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16. When you establish a secondary axis by extending a line out from the center of a cross-section of a core cylinder through any point on the outer ring of that cross section, this new axis can be conceptually understood to penetrate the side of the cylinder at a 90° angle. This is a conceptual 90° and not a literal one. It actually measures out on the surface of the illustration to be more like 105°.
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17. The tilt of the major axis of the ellipse that represents the foreshortened circle, however, does intersect the secondary axis at a very real and measurable 90°.
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18. The foreshortened circular ends of a cylinder tilting back in space become compressed along one axis. Their proportion is measured as the relationship between their major and minor axes. The major axis of an ellipse will sometimes be vertical, sometimes horizontal, or it can tilt anywhere in between, but it will always be perpendicular (90°) to its central axis.
© 2002, The McGraw-Hill Companies

19. 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.
© 2002, The McGraw-Hill Companies

20. In this illustration there are nine cylinders that are aligned with their circular ends parallel to the picture plane. Only the circular end of the cylinder that is at eye level and is directly in front of the viewer appears perfectly round.
© 2002, The McGraw-Hill Companies

21. Curvilinear Perspective
Around 1850, photographers began experimenting with cameras that rotated in one direction while the film was being moved past the lens opening in the opposite direction. These “cirkutcameras” capture panoramas ranging from 100° to 360° by creating images with curvilinear perspective where objects at the edges of the picture plane appear to curve back in space. Imaging programs for computers can stitch several sequential photographs together into similar panoramas. Strangely, in curvilinear perspective, edges that are straight often appear curved and curved edges can appear straight.
© 2002, The McGraw-Hill Companies

22. 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.
© 2002, The McGraw-Hill Companies

23. This illustration represents three levels of exaggeration that can be used when locating the center of a foreshortened circle. Accurately positioned, the center of a foreshortened circle is slightly behind where the major axis crosses the minor axis (top). It is possible, however, to apply varying degrees of exaggeration with regard to the placement of the center of the foreshortened circle as a way to compensate for the loss of binocular, stereoscopic information and increase the overall illusion of depth. In the middle image, the only adjustment was to nudge the center of the foreshortened circle a little extra toward the back edge. In the bottom image, the center was pushed even further back and the front edge of the large black ring was enlarged to bring it forward (bottom).
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24. To attach a circle to the surface of a cylinder you must first locate the central axis of the core cylinder. Because the central cylindrical axis is, by definition, equidistant from and parallel to the outer edges of the cylinder, it always mimics the minor axes of any cylindrical cross-section (below).
© 2002, The McGraw-Hill Companies

25. Mark the point on the surface of the cylinder where you intend to position the circle and draw a cylindrical cross-section that passes through that point (d). Take care to proportion this cross-section relative to its vertical location on the cylinder.
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26. From the center of this latest cylindrical cross-section (remember that it’s always on the cylindrical axis slightly behind the center of the ellipse that represents the foreshortened cross-section), draw a line out through the point you have marked. This line becomes a secondary axis (understood conceptually to intersect the surface of the core cylinder at a 90° angle). A circle attached at this point will appear as an ellipse whose major axis intersects this secondary axis at an actual 90° angle.
© 2002, The McGraw-Hill Companies

27. To create a horizontal ring of equally sized circular shapes on the surface of a cylinder, start by establishing three equally spaced cylindrical cross sections (ellipses) through the core cylinder. The middle cross section is used to align the centers of the circles in the row. The upper and lower cross sections determine the height of each circle relative to its position on the curved surface of the cylinder. Please note that the secondary central axis for each new circle penetrates the new circle at a point slightly behind the major axis of the ellipse that represents it.
© 2002, The McGraw-Hill Companies

28. As foreshortened circles move away from eye level they appear as progressively less compressed ellipses. As a result, the distance between matching edge points constantly decreases as the edge moves away from the observer. This delicate progressive relationship between the ellipses that are used to represent foreshortened circles requires smooth, symmetrical curvature and a highly developed sensitivity to proportion.
© 2002, The McGraw-Hill Companies

29. Identically sized circular elements that are attached in a vertical column to the side of a cylinder have their apparent centers (the center of the foreshortened circle, not the center of the actual ellipse) aligned directly above one another.
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30. The outside contour of a sphere always appears circular and does not become visually compressed like circles do when they are foreshortened. The center of a sphere is always located in the center of the sphere’s circular contour. Spherical cross-sections, however, because they are foreshortened circles do appear as ellipses compressed along the minor axis. When we bisect the sphere, the cross section’s center coincides perfectly with the center of the sphere. However, the center of the ellipse that represents this cross-section will always be located slightly off-center.
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31. When a sphere is below eye level the poles appear to shift
When a sphere is below eye level the poles appear to shift. The top pole moves forward and down while the bottom pole disappears up and back behind the lower edge. Any line drawn out from the sphere’s center is understood to penetrate any given point on the sphere’s surface at a 90° angle. The tilt of an attached foreshortened circle is then determined by positioning the widest part (major axis) of the foreshortened circle at a 90° angle to the axis emerging from the center of the sphere.
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32. Standardizing the size and alignment of multiple circular attachments on the surface of a sphere requires three equally spaced spherical cross-sections as guidelines. Keep in mind, however, when attaching circles on cross-sections that do not bisect the sphere, the secondary axis that determines the tilt of the foreshortened circle always originates in the center of the sphere and not from the center of the non-bisecting cross-section.
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33. For those cross sections that do not bisect the sphere, the center is located at a point on the spherical axis but not in the center of the sphere. Axial lines along this type of cross section do not penetrate the surface at 90° and cannot be used to establish the tilt of an attached ellipse. Cross sections that do not bisect the sphere are centered on an axis that passes through the center of the sphere.
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34. When bisecting a sphere through its vertical axis the spherical cross-section must pass through the two poles and make contact twice with the outside edge of the sphere. The center of each of these cross sections is, as you have seen, located at the center of the sphere but the center of the actual ellipse that is used to represent the cross section is positioned a little to the side. This shift is necessary so that the center of the foreshortened spherical cross section will be positioned closer to the back edge of the cross-section than it is to the front edge.
© 2002, The McGraw-Hill Companies

35. An axis penetrates two sides of a rectilinear solid at a 90° angle when it is parallel to edges that are perpendicular to the plane through which it passes. Since we know that receding parallel lines converge, we can establish the apparent tilt of the central axis by having it converge with the edges with which it is parallel. As expected, when attaching a circle/cylinder to the surface of a rectilinear solid, the widest part (major axis) of the foreshortened circle will cross its central axis at an actual 90° angle.
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36. Circles in Perspective/ Foreshortened Circles (ellipses)
This concludes the lecture
Circles in Perspective/
Foreshortened Circles
(ellipses)
© 2002, The McGraw-Hill Companies