Arapin Reflects on Mirrors and Projective Geometry 
  • Anton's research involved three types of mirrors that were convex and rotationally symmetric. Anton's research involved three types of mirrors that were convex and rotationally symmetric.

“Research is very different from classwork in terms of the approach, goals and overall flow of the progress,” says Anton Arapin, a math and computer science major. “Working one-on-one with a professor provides a more thorough understanding of the material, as well as helps foster the relationship in general. Also it provides a unique opportunity to learn from the professor, especially in approaching the research objectives, facing dead ends and overcoming conceptual obstacles,” he adds. 

 

Partaking in research provides experience in approaching problems from different sides, aiming at coming up with something new and innovative; and, research skills are very useful even in daily life.”

 

Anton’s research with Prof. Annalisa Crannell focused on the exploration of the geometric projective properties of mirrors. The goal was to determine the shape of the mirror given the general type of mirror (spherical, conical, parabolic) from a photograph of objects reflected in that mirror.  Anton defined functions that map a point from the photograph (image plane) onto the world plane,  and explored special ratios of a mirror’s unique features.

“Determining the exact shape of the mirror, one can get information about the distance of the reflected object to the mirror, its size and other geometric properties,” comments the student researcher.  “Also, the mapping of the objects from the image reflected in the mirror to the real-world dimensions can enable one to reconstruct the objects as well as their environment in a panoramic setting. This is useful for the systems that require a wide angle of view, and can be used to imitate a 360-degree camera, for instance.” 

  • In the study, Anton used two-dimensional examples to investigate the two pictured mirror phenomenons:  edge of mirror and vanishing ring. In the study, Anton used two-dimensional examples to investigate the two pictured mirror phenomenons: edge of mirror and vanishing ring.

“The fields that are concerned with geometric representation and overall interpretation of three-dimensional world interest me. I am planning to pursue my further education in the field of computer vision, which entails a lot of mathematical and research aspects,” Anton explains. The student took a Projective Geometry course with Professor Crannell, which Anton found extremely interesting.

Says Anton, “Projective geometry is very closely tied to the field of computer vision.  Furthermore, different types of curved mirrors are widely used in the robotics systems and navigation, which is relevant to my work outside of the College and my future.”