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What is a good book/article explaining the mathematics behind perspective painting? I have already looked at the wikipedia article on the topic, so I am looking for something more advanced than this.

I am a research mathematician of limited artistic ability and knowledge.

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The fascinating case about Mathematics, Murder and Art behind it's (re)inventionn in the Renaissance: perlentaucher.de/buch/25202.html AMS Notices on the painting: ams.org/notices/200703/comm-cass.pdf –  Thomas Riepe May 9 '10 at 8:30
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6 Answers 6

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The geometry of an art by Kirsti Andersen (amazon)

Mathematics for the non-mathematician by Morris Kline (See Chapter 10- math and painting in the renaissance)

Mathematics and its history by John Stillwell (See chapter 8 on Projective Geometry)

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George Francis' "A Topological Picturebook" provides somewhat of an overview about perspective, and is very helpful for learning to draw. It's very easy to ogle his diagrams.

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Google Books (option full view) has many books on perspective, free for downloading or perusing. Perspective being a rather ancient subject should be very adequately covered in a number of these books (among those that you can actually download). The quality and completeness of the scans should be verified as they are often defective.

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More on the artistic side I appreciated

The Invention of Infinity: Mathematics and Art in the Renaissance by J.V. Field

Its theme is the interaction of mathematical and artistic inquiries as characteristic of Western art in the Renaissance, with perspective and precise description of geometrical forms (such as polyhedra) as turning point, and embodied in several key artists such as Piero della Francesca, Leonardo da Vinci, Albrecht Dürer.

The same author has written a book dedicated to Piero della Francesca:

Piero Della Francesca: A Mathematician's Art

Another source are books about the camera obscura and pinhole photography.

More contemporary: the techniques used to enhance digital images and their perspective with mathematical models of camera lens deformation have given birth to relatively sophisticated applied mathematics. There are a few private companies such as DxO selling software to correct (among other things) perspective in image files by reversing the nonlinear effects of multiple lens systems found in camera.

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You might look at:

http://www.springer.com/mathematics/geometry/book/978-0-387-97486-6

which provides the contributions of Brook Taylor and places his contribution in historical pesrpective.

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According to Leonardo da Vinci, the best way to learn perspective painting is to get a framed pane of glass (an empty pictureframe) and something you can mark it with. Hold it up to the perspective you want to draw and trace the lines (as quoted in Ruskin, The Elements of Drawing).

Ruskin himself says that most of the 'great artists' had a minimal grasp of perspective.

Another great book on this topic is Secret Knowledge by David Hockney.

My opinion is that you should never let your judgment of a drawn object be deternmined solely by the underlying geometry. We have two eyes, not one -- that means that, in order to appear 3d, objects drawn on a flat piece of paper must somehow appear more than what they are.

The only way to do this is to distort the perspective/dimensions of the object and 'harmonize' it between what the left eye sees and what the right eye sees. Some enterprising mathematician could come and formalize this someday, but for now it's enough to simply 'eyeball' as well as using geometry. Your eyes and hands will naturally choose the shape that appears more 3d in binocular vision.

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