Burkett Interprets Steiner Conics through Perspective Drawings
  • Prof. Annalisa Crannell and Ellie Burkett explore the geometry underlying these strange hyperbolas. Prof. Annalisa Crannell and Ellie Burkett explore the geometry underlying these strange hyperbolas. Image Credit: Deb Grove

Lines in the real world often behave differently in perspective space.  

Through their research project, "Interpreting Steiner Conics through Perspective Drawings of the Real World," senior math major Eleanor "Ellie" Burkett and Math Prof.  Annalisa Crannell  gained more knowledge about structures in the real world and the ways in which they work. Given a picture in one-point perspective of a building in the real world, the research duo aimed to determine the meaning of a curve found in the perspective drawing but not immediately apparent in the real world.

 

"Research allows for the best type of learning -- passion driven determination. "

 

"Going into college, research in mathematics always confused me. You couldn’t perform experiments in the same way as you might in chemistry or physics. So, of course, it intrigued me to learn that math research was being carried out. Even further, there are things we don’t know about math! " comments the student researcher. "So, I took a branch of mathematics, Geometry, that I had been studying and and dove deeper.  For example, I had been learning all about projective space and the straight line within it.  So, it occurred to me, what about the curves?  How do curves work in projective space?  These questions led me into my study of Steiner conics and my eventual project."

Says Prof. Crannell, "In our linear perspective class, Ellie got interested in drawing curves, which is hard to do using rulers --- but it's not impossible!   In fact, her question gave me the chance to jump into a related question that I'd been puzzling over:  how do Steiner conics emerge from straight-line perspective drawings?  Steiner conics are like those string-art creations you might see with hyperbolas and other curves, and we know mathematically why they emerge.   But what does a Steiner conic "mean" in the real world?  What does it tell us about our perception of 3D objects?   Thanks to Ellie's work, I have a better grasp on that . . . but in another sense, I have even more questions than before, as well.  There's lots of good room for students to jump in and think deeply about this."

 

"My project didn’t start when I started doing research.  It started when I began going to my professors -- when I realized, in class, that I had so many more questions!  Once I started speaking with professors one-on-one, I found my footing. That is what truly led me to research!"

 

"Doing research  forces you to ask questions, whether they be questions of mathematical nature or questions regarding processes or accuracy of work.  By asking one question, more questions follow, demanding the use of even more mathematics," Ellie says. "In short, one small research question actually required a much wider breadth of knowledge than I had first expected.  Thus, I learned so much more than I first anticipated."

  • At left, image of a building in three-point perspective. A hyperbola, on the right, found in the image of a building in one-point perspective and its real-world circle. At left, image of a building in three-point perspective. A hyperbola, on the right, found in the image of a building in one-point perspective and its real-world circle.

 In Ellie's opinion, working one-on-one with a professor allows for more specific questions, so it allows the student to get to the focus of one’s interests. Those that truly love learning about their subject, she feels, will always benefit from research because it will always teach the student more than he or she expects to find.

 The student researcher gives this advice:  prioritize. "Research is so incredibly doable, " Ellie says, "But, I suggest dedicating a certain amount of time to your research and a certain amount of time to other commitments so that all work gets the time it deserves."