An Analysis of the Mathematical Accuracy of Perspective in Paintings

Art is a great human endeavor and artists, since time immemorial, have tried to capture the beauty of what lies around us in their canvases. But does art accurately depict real-world objects in terms of their sizes and proportions? A particular area where artists have been challenged is in the depiction of the three-dimensional world on a two-dimensional canvas. This is known as perspective. Mathematicians have done a lot of work to understand perspective and this field of mathematics is known as projective geometry.
Our hypothesis was that there are mathematical inaccuracies of perspective in artists’ paintings that we are unable to detect with our naked eyes. Errors of perspective may be more tolerated as the distance from the eye to the object in the painting increases. We wanted to understand the degree of mathematical inaccuracy in several famous paintings and then draw conclusions about the limits of our perception of depth.
We have used a mathematical method from the world of projective geometry, which is called cross-ratio, to analyze paintings for accuracy of perspective. Cross-ratio is measured by using four fixed points on a straight line as reference; this value is always the same irrespective of the viewing angle or the viewing distance. We took structures/buildings in paintings as our four points and compared the painting’s cross-ratio to that of a photograph of the same building. For our research we have taken three famous paintings made by different artists and measured their accuracies. From our analysis we concluded that there are significant errors in perspective in these paintings and the errors increase as the structures in the paintings recede from the viewer.