Looking Through the Glass

  • Annalisa CrannellEmail author
Living reference work entry


Projective geometry allows us, as its name suggests, to project a three-dimensional world onto a two-dimensional canvas. A perspective projection often includes objects called vanishing points, which are the images of projective ideal points; the geometry of these points frequently allows us to either create images or to reconstruct scenes from existing images. We give a particular example of using a pair of vanishing points to locate the position of the artist Canaletto as he painted the Clock Tower in the Piazza San Marco. However, because mappings from three-dimensional space to a two-dimensional plane are not invertible, we can also use perspective and projective techniques to create and analyze illusions (e.g., anamorphic art, impossible figures, the dolly zoom, and the Ames room). Moving beyond constructive (e.g., ruler and compass) projective geometry into analytical projective geometry via homogeneous coordinates allows us to create and analyze digital perspective images. The ubiquity of digital images in the present day allows us to ask whether we can use two (or many) images of the same object to reconstruct that object in part or in entirety. Such a question leads us into the emerging field of multiple view geometry, straddling projective geometry, algebraic geometry, and computer vision.


Linear perspective Multiple view geometry Projective geometry Anamorphism 


  1. Agarwal S, Furukawa Y, Snavely N, Simon I, Curless B, Seitz SM, Szeliski R (2011) Building Rome in a day. Commun ACM 54(10):105–112. With a Technical Perspective by Prof. Carlo TomasiCrossRefGoogle Scholar
  2. Andersen K (2006) The geometry of art: the history of the mathematical theory of perspective from Alberti to Monge. Springer, New YorkGoogle Scholar
  3. Andrevruas (2011) Português: Casa construída de forma a fazer a pessoa parecer grande ou pequena dependendo da perspectiva, na cidade do Rio de Janeiro, 24 Jan 2011., from Wikimedia Commons
  4. Beever J (2019) Julian Beever’s website.
  5. Boing Boing (2015) Watch 23 of the best dolly zooms in cinematic history, 26 Jan 2015. Google Scholar
  6. Bosse A (1648) Manière universelle de Mr. Desargues, pour pratiquer la perspective par petit-pied, comme le Geometral, ParisGoogle Scholar
  7. Byers K, Henle J (2004) Where the camera was. Math Mag 77:4:251–259MathSciNetCrossRefGoogle Scholar
  8. Canaletto GA (circa 1730) The Clock Tower in the Piazza San Marco., oil on canvas, 69.22 × 86.36 cm, current location at the Nelson-Atkins Museum of Art
  9. Carroll L (1871) Through the looking-glass. Macmillan & Co, LondonGoogle Scholar
  10. Crannell A (2006) Where the camera was, take two. Math Mag 79:4:306–308CrossRefGoogle Scholar
  11. Desargues G (1987) Exemple de l’une des manieres universelles du s.g.d.l. touchant la pratique de la perspective sans emploier aucun tiers point, de distance ny d’autre nature, qui soit hors du champ de l’ouvrage. In: The geometrical work of Girard Desargues. Springer, New York, p 1636Google Scholar
  12. Deutsches-Technikmuseum (2008) Penrose triangle sculpture., images from Wikimedia Commons
  13. Futamura F, Lehr R (2017) A new perspective on finding the viewpoint. Math Mag 90(4):267–277MathSciNetCrossRefGoogle Scholar
  14. Hartley R, Zisserman A (2003) Multiple view geometry in computer vision, 2nd edn. Cambridge University Press, New YorkzbMATHGoogle Scholar
  15. Holbein H (1533) The Ambassadors., oil on oak, 209.5 cm× 207 cm
  16. Robin AC (1978) Photomeasurement. Math Gaz 62:77–85CrossRefGoogle Scholar
  17. Taylor B (1719) New principles of linear perspective: or the art of designing on a plane the representations of all sorts of objects, in a more general and simple method than has been done before, LondonGoogle Scholar
  18. Tripp C (1987) Where is the camera? The use of a theorem in projective geometry to find from a photograph the location of a camera. Math Gaz 71:8–14CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Franklin & Marshall CollegeLancasterUSA

Section editors and affiliations

  • Bharath Sriraman
    • 1
  • Kyeong-Hwa Lee
    • 2
  1. 1.Department of Mathematical SciencesThe University of MontanaMissoulaUSA
  2. 2.Department of Mathematics Education, College of EducationSeoul National UniversitySeoulSouth Korea

Personalised recommendations