Projective Geometry / Perspective Drawingpeople.math.umass.edu/~tevelev/391A_2015/projective.pdf · 2015-04-01 · mathematicians to make discoveries in projective geometry •Perspective

Projective Geometry /

Perspective Drawing

Presented by: Rune Percy & Nate Mok

Perspective Drawing vs Projective Geometry

• Connection can be drawn between art and science• Brunelleschi’s algebraic formulae in his artwork• Advancements in perspective drawing in the 15th century pushed

mathematicians to make discoveries in projective geometry

• Perspective drawing is merely a visual representation of projective geometry (3-dimensional object being projected onto a 2-dimensional plane)

History (Perspective Drawing)• First attempts developing system of perspective in art

mentioned by Plato around 5th century BC

Pageant of Orestes2 A.D.

-constructed lines shows different converging points

• By late 13th century, Giotto begins developing an algebraic method to determine where his perspective lines should be

Christ before CaiaphasGiotto

1305

• In 1413, Filippo Brunelleschi perfected a geometrical method of perspective consisting of vanishing points & a horizon line

• Since then, all artists have used perspective correctly and usefully in their work

-Brunelleschi’s experiment tested how consistent his drawing of the Florentine baptistery was compared to actual observation of the building in person

History (Projective Geometry)

• First geometrical properties of projective nature ~ 3rd century AD by Pappus of Alexandria

• Brunelleschi’s study of perspective in art in the early 1400’s launched development of projective geometry (vanishing point)

• Johannes Kepler and Girard Desargues independently developed concept of “point at infinity” in the mid 1600’s

•Projective geometry acted as an intermediate step from traditional analytic (Cartesian) geometry to modern algebraic geometry (using solutions of systems of polynomial equations to define curves/shapes)

•Homogeneous coordinates developedto simplify calculations in projective geometry

Homogeneous Coordinates (Non-Cartesian)

• Allowed for reduction of “special cases” and for infinity to be described as a finite point

• Proving the visualization in perspective drawing (vanishing point)

● Addition of coordinate W serves as “depth” from perspective point

Projective Geometry – Desargue’s Theorem

• “The theorem states that if two triangles ABC and A′B′C′, situated in three-dimensional space, are related to each other in such a way that they can be seen perspectively from one point (i.e., the lines AA′, BB′, and CC′ all intersect in one point), then the points of intersection of corresponding sides all lie on one line”

• Even if triangles are parallel, by modifying

Euclidian space (projective space), the two points

would theoretically meet at infinity, and therefore

still satisfy the theorem

Parallels to Perspective Drawing

• When viewed from a specific point, the image appears to be congruous

• All “lines” and curves connect

seamlessly

Parallels to Projective Geometry (Cont.)

• When viewed away from any other point, it is clear there are multiple, separate images.

Bibliography

• http://www.britannica.com/EBchecked/topic/158758/Desarguess-theorem• http://www.webexhibits.org/sciartperspective/perspective1.html• http://www.songho.ca/math/homogeneous/homogeneous.html• http://www.tomdalling.com/blog/modern-opengl/explaining-homogenous-coordinates-and-projective-

geometry/• http://www.huffingtonpost.com/2012/07/12/street-artist-julian-beev_n_1668616.html• https://maitaly.wordpress.com/2011/04/28/brunelleschi-and-the-re-discovery-of-linear-perspective/• http://www.geni.com/people/Euclid/6000000011138332887• http://pixshark.com/filippo-brunelleschi-sculptures.htm• http://www.brighthub.com/science/space/articles/40406.aspx• http://www-history.mcs.st-and.ac.uk/Posters2/Desargues.html