Perspective Drawings: Using Geometry to Make Sense of a 3D World

Materials Needed:

• Drawing Surface
• Pencil
• Eraser
• Ruler

 

Connections Between Optometry, Neurology, Art and History:

Figure 1. Railroad tracks

For historical reasons and the hegemony of the western world, Euclidean geometry continues to be the main focus of high school geometry (NCTM, 2018). All high school students are taught Euclid’s fifth postulate which states two parallel lines never intersect. However, what about when we look at railroad tracks and they seem to meet at a point in the distance, or the trees in the distance that seem smaller than they really are (Figure 1)?

In a recent conversation with one of my best friends who is an Optometrist, I asked her about the ways in which our eyes and brain connect and perceive the 3-dimensional world. Our brain knows those railway tracks in a 3-dimensional world are parallel, yet on a 2-dimensional surface we see them as meeting at a distant point on the horizon. Our brain also knows that those buildings in the distance are much larger than they appear, yet, on a 2-dimensional projection they are proportionally drawn smaller. Our eyes and brain are constantly connecting the ways in which we 2-dimensionally view and perceive the 3-dimensional world around us, yet we rarely pause to notice and wonder about such a mathematical wonder.

Figure 2. Raphael’s School of Athens

This idea of understanding the 3-dimensional world through a 2-dimensional portrayal comes from projective geometry and is most commonly used in perspective drawings to describe an observed reality. The origins of this type of geometry comes from Renaissance artists and architects attempting to achieve a sense of depth and a realistic portrayal when projecting 3-dimensional objects on a 2-dimensional plane (Figure 2).

Note: While the Renaissance period is most cited as the origin, like much of mathematics and art history, it is quite possible that others in different parts of the world had independently discovered or written about similar ideas.

Figure 3. Photograph in one-point perspective highlighting the Washington Monument

In perspective drawing, parallel lines meet at points(s) called the vanishing point(s). Renaissance painters often used one vanishing point as a narrative tool in guiding observers’ attention to the important part of the painting, a practice that continues to be common in modern photography and city planning (Figure 3).

 

Thoughts for Classroom Implementation:

High school geometry has traditionally focused on the study of postulates, measurements, properties of 2- and 3-dimensional figures (NCTM, 2018) and on proving axioms that have no real connection to us, our students, or the world around us. Geometry offers a useful and meaningful lens for understanding the visual world, and while projective and hyperbolic geometries tell us so much more about the world around us, school mathematics continues to emphasize Euclidean geometry.

The process of conjecturing and proving/justifying is a major part of high school geometry. Perspective drawings naturally connect to the six axioms of projective geometry which offer a useful and meaningful lens for understanding the visual world and would be worth exploring in a geometry class.

Axioms: If A and B are distinct points on a plane, there is at least one line containing both A and B. If A and B are distinct points on a plane, there is not more than one line containing both A and B. Any two lines in a plane have at least one point of the plane in common. There is at least one line on a plane Every line contains at least three points on the plane. All the points of the plane do not belong to the same line. (Veblen and Young 1938, Kasner and Newman 1989)

In addition to proofs and justifications, projective geometry also connects to many mathematical concepts such as:

• drawing 2- and 3-dimensional objects
• 2D <–> 3D projections
• angles
• conics
• non-Euclidean geometry
• parallel lines
• proportions
• vectors

While perspective drawings were initially created as a method of creating 2-dimensional drawings to portray a 3-dimensional space, modern uses also go from 2-dimensional drawings to plan and 3-dimensionally model and create new architecture. Projective geometry and perspective drawings also present opportunities to integrate other content areas including computer design, photography, art, and architecture.

 

Example and Additional Resources:

My first introduction to perspective drawings was in 6th and 7th grade art classes. These drawings opened up a whole new possibility of projecting a three-dimensional scene onto a two-dimensional surface (Figure 4 and 5), and also gave me lots of practice in realistically drawing 3-dimensional solids! Yes I still have my portfolios and sketchbooks from middle school 🙂

Figure 4. One-point perspective drawing from 6th grade

Figure 5. Two-point perspective drawing from 7th grade

 

The video below is a very quick and general intro to making perspective drawings.

In-depth videos for one, two, and three-point perspective.

While linear perspective drawings (1-3 point perspective) are the most commonly used in art, photography and architecture, this activity can also be extended to curvilinear perspective drawings (4-6 point perspective). More on 1-6 point perspectives

 

Some more about connections between neuroscience, optometry, and perception: How your eyes trick your mind, How do our brains reconstruct the visual world?

 

For more #MathArtChallenges take a look at Annie Perkin’s blog!

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