1. Two-point Perspective Relationships
by Brian Curtis
© 2002, The McGraw-Hill Companies

2. A PowerPoint lecture series to accompany DRAWING FROM OBSERVATION

3. 2 Point Perspective Relationships
All linear perspective drawings using Brunelleschi’s system are based upon the establishment of a fixation point that is at eye level and on the presence of an imaginary picture plane. The viewer’s line of sight to that fixation point runs parallel with the ground and is the axis upon which the imaginary picture plane glides into the visual field as it searches for parallel edges among the rectilinear objects. When the imaginary picture plane is parallel with only one set of leading edges of a rectilinear object (usually the vertical), the viewer is in a two-point perspective relationship with that object. All of the rectangular solids above are in a two-point perspective relationship with the viewer on the right and, because of the 90° shift in viewing angle of the illustration from that of the depicted viewer, they are also in a two-point perspective relationship for anyone who views the illustration.

4. When your imaginary picture plane aligns with only one set of edges of a particular rectangular form (usually the verticals) as it moves forward along your line of sight toward your chosen fixation point, then you are in a two-point perspective relationship with that object. All vertical edges that are parallel with the picture plane appear straight up and down while the edges that are moving away from the picture plane appear to tilt.

5. All parallel sets of receding edges of objects in two-point perspective relationships converge at a single point.

6. The rectangular boxes to the left of the fixation point are all in a one-point perspective relationship with the viewer and are therefore, by definition, squarely aligned with one another Their receding edges (orthagonals) all converge at the fixation point. The boxes to the right of the fixation point are all in a two-point perspective relationship with the viewer. They all share an identical pair of vanishing points, so they, too, are aligned squarely with one another.

7. A fixation point is essential even when none of the rectilinear objects in the drawing are in a one-point perspective relationship with the viewer. Brunelleschi’s system requires that you draw from a single position and that you look toward a single fixation point throughout the entire drawing. This means you need to establish a fixation point at the very beginning of the drawing and maintain it throughout the entire process. Otherwise there is a tendency to change the direction of your line of sight, and shifting your line of sight (fixation point) even slightly causes marked changes in the angle at which your picture plane aligns with the edges of the objects in your visual field. Changing your line of sight changes the apparent angles of the orthagonals as they converge toward their respective vanishing points. Once you have chosen a fixation point and have calculated the perspectival relationship of the objects in your visual field, you should begin your drawing by gesturing, applying Mondrian lines, checking proportions, and using your straight edge as a clock angle tool to establish the angles at which the receding edges converge. It is also a good idea to pay special attention to your line quality so as to prevent the mechanical character of the system from dominating the expressive character of the drawing. In a two-point perspective relationship your imaginary picture plane is parallel to only one set of leading edges (usually the verticals). Any edges that are parallel to the ground plane are, by definition, receding, and will appear tilted. Sets of parallel receding edges will converge at some point on the horizon other than the fixation point. As can be seen above, one set of these receding parallel edges has a vanishing point to the right of the fixation point and the other to the left. This is always the case with a two-point perspective relationship because vanishing points rarely appear within the drawing surface unless you draw very small or have incredibly large pieces of paper. As a result, Brunellesci’s system of a horizon line with two vanishing points actually turns out to be more helpful as a conceptual aid than as a mechanical tool for determining the angle of convergence. You will generally be much better off if you rely on clock angles to calculate the angular tilt of receding edges than actually looking for vanishing points. Because all the rectilinear objects in this illustration are aligned squarely with one another, there are only two sets of receding parallel edges (orthagonals). Each receding edge converges with all the other edges with which it is parallel.

8. When, as is quite common, the two-point perspective vanishing points are located outside the drawing surface, you should rely on the clock angle tool to calculate the angle of each of the receding edges.

9. Edges of a rectangular solid that is in a two-point perspective relationship that are parallel to the imaginary picture plane are, by definition, perpendicular to the ground and appear as verticals on the drawing surface. The two sets of edges that are parallel to the ground plane and are receding from the imaginary picture plane will appear to converge at two distinct points on the horizon on opposite sides of the fixation point. The position of the two vanishing points on the horizon is determined by the particular characteristics (size, rotation, relationship to eye level) of the rectangular solid. To develop a sense of the relationship between the vanishing points, try imaginatively manipulating the rectangular solid in the illustration above by rotating it on its vertical axis. As the object turns clockwise, the right vanishing point glides outward along the horizon away from the fixation point while the left vanishing point shifts horizontally toward the fixation point. As one point gets closer to the fixation point, the other gets further away. This reciprocal shift continues as you rotate the cube until the left vanishing point reaches the fixation point. At that instant, the left vanishing point becomes the fixation point and the right vanishing point ceases to exist because the rectangular solid is now, by definition, in a one-point perspective relationship.

10. With rectilinear objects in one-point perspective relationships with the observer, all edges receding from the picture plane are parallel and appear to converge at the Fixation Point. For each object in a two-point perspective relationship there are always two sets of receding parallel edges. Each set will converge at a point at eye-level, with one vanishing always on one side of the Fixation Point and one always on the other. In the lower image, the three rectilinear objects are aligned with each other so that their receding edges form two parallel sets—all edges in each of these sets converge at the vanishing point appropriate for their set.

11. When each of the rectangular objects in the visual field is aligned uniquely with the viewer’s picture plane, each object will have either a unique point (if it is in a one-point perspective relationship) or pair of points (if it is in a two-point relationship) on the horizon toward which its receding parallel edges converge. The angle of apparent convergence of any of the receding parallel edges of a rectangular object is dependent on four factors: the object’s rotation relative to the picture plane, its distance up or down from eye level (horizon), its distance from the picture plane, and its distance to the right or left of the fixation point.

12. The angle of apparent convergence of any of the receding parallel edges of a rectangular object is dependent on four factors: the object’s rotation relative to the picture plane, its distance up or down from eye level (horizon), its distance from the picture plane, and its distance to the right or left of the fixation point.

13. It is worth repeating. When, as is quite common, the two-point perspective vanishing points are located outside the drawing surface, you should rely on the clock angle tool to calculate the angle of each of the receding edges.

14. It is not uncommon for a fixation point to appear within the borders of a drawing, but the drawing surface is rarely the home of other vanishing points.

15. For this drawing and several that follow, the students were asked to position their fixation point so that there would be at least one rectilinear object in a one-point perspective relationship to their picture plane and at least two others in unique two-point perspective relationships.

18. This drawing is a precisely rendered depiction of three rectilinear objects in two distinct perspective relationships. The stool and the small box are in two-point perspective relationships with the observer’s picture plane and the box on the top of the stool is in a one-point relationship. As precise and as accurate as the drawing is, however, the tall box on top of the stool is very hard to decipher because it is at eye-level and directly in front of the observer in a one-point perspective relationship. In this relationship, boxes appear identical to flat rectangles and provide no additional information from which to develop a more complete understanding of the specific spatial character of the form being represented. You should be careful to avoid this sort of ambiguity whenever possible.

19. When rectangular furniture in a two-point perspective relationship is flush against the walls of a room, the edges of the objects are aligned with the walls, floor and ceiling, and they all share the same pair of vanishing points on opposite sides of the fixation point on the horizon at eye level.

20. Your ability to use the “clock-angle” tool to measure the tilt of the receding edges of rectangular solids should become progressively more accurate now that you understand the crucial role that the fixation point plays in organizing your perspectival relationships. Although the student who drew the three flags in the drawing above did so long before she was given information about a fixation point, a horizon, or vanishing points, she was so accurate with her clock angles when measuring the receding edges that the parallel edges in all three flags conform to the same vanishing points.

21. In this illustration neither the fixation point nor either pair of vanishing points is located within the borders of the drawing surface. These choices allow the objects in the drawing to be considerably larger than would otherwise be possible. The bottom of the leading surface of the drawing bench has not been included in the drawing because it was outside this student’s 45° cone of vision. Had he included that portion of the drawing bench in the drawing, the receding edges would have formed an exaggeratedly acute angle that would have looked more like the tip of a sword than the bottom corner of a rectilinear solid. The three-dimensional illusion of the drawing has been solidly enhanced by the variation in line quality by drawing through the forms.

22. When the two vanishing points of a rectilinear object are positioned too close to one another on the horizon, the rectilinear planes appear twisted, stretched or even trapezoidal. If the lower corner begins to look as though it is the tip of a sword blade, it’s a good idea to double-check your clock angles.
While mindful of the observations about the distortions in perspective that occur when you position your vanishing points too close to one another on the horizon, it is still possible to strengthen the overall dimensional illusion of your two-point perspective relationship by subtly exaggerating the rate at which receding edges of a rectilinear solid converge. Gently accelerating the convergence can compensate somewhat for a drawing’s inherent lack of binocular information. Delicately tweaking the rate of convergence is identical to the heightened illusionistic depth that one experiences when looking through a wide angle lens. There is a point, however, at which the exaggeration ceases to add to the illusion of space and compromises the integrity of the image. That being said, the general rule of thumb for perspective exaggeration is the same as that of the previous illusionistic drawing techniques: a little too much is generally more effective than a little too little.

23. This drawing has a rich surface and effective depiction of one- and two-point perspective relationships. The sense of spatial recession has been slightly accelerated because the vanishing points for the forward table appear to be slightly closer together than they would normally be. This gives the lower corner of the table a bit of a slightly exaggerated “sword blade” appearance.

24. Exaggeration of perspectival recession (increasing the rate at which parallel edges converge and exaggerating the dimunition of scale of objects as they go back in space) can strenghten the overall feeling of depth in a perspective drawing. Photographers change their lens length when they want to achieve similar results. The above series of photographs were taken with three separate focal length lenses on a 35 mm camera: a 35 mm lens (slightly wide-angle), a 50 mm (normal - 45° cone of vision), and a 70 mm (slightly telephoto). The camera position had to be adjusted for each shot so that the information in each image would remain mostly the same (closer for the wide-angle and further away for the telephoto). The spatial representation in the three images is very different. The spatial recession is much more dramatic with the 35 mm lens and substantially understated with the 70 mm.

25. Two-Point Perspective Relationship
This concludes the lecture
Two-Point Perspective Relationship