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Home , Anamorphosis, Leonardo da Vinci

2014-06-27

Plan

} Definition and examples } Construction of perspectival with deformation grids } Descriptive } Conical anamorphosis } Cylindrical anamorphosis Mathematicsin - anamorphosis } Application } More examples

The Faculty of Mathematics and Computer Science Ada Palka } Summary Jagiellonian University [email protected]

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Definition and examples Definition and examples } To give the images a normal , the artists began } Thus a square becomes a trapezoid in the perspective image. By creating their work by the horizon line h at the level of moving the observation point above the main point, keeping the the eye. Then at the center of the line they marked the main point same distance from the focal point we get the opposite effect, the A, which is the point of intersection of all the parallel lines. At a trapezoid begins to be seen as the square, returning to its distance equal to the distance of the eye to a central point, there original form. This peculiar perspective was initially used to was point B is marked there. This is a point of the intersection of validate the perspective projection by contrast, however after diagonals. To be read properly,the image should be viewed from some time gave the beginning to the formation of anamorphic a particular viewpoint. In this way a square of the grid of the images . ground after projecting appears to be a trapezoidal checkerboard, on which it is enough to put other items in proportion to the decreasing dimensions of the boxes.

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Definition and examples Definition and examples

} Anamorphosis is like a deviation from the norm. Although the word first appears in the seventeenth century, but is referring to the images known much } Anamorphosis -kind of intentional earlier. picture deformation } In summary anamorphosis (Gr. ana - back, morphe-form) is an extreme consequence of linear perspective, which involves the deformation of the image by placing the of the of vision away from the main point and observation point close to the plane of the work. Hans Holbein, The Ambassadors, 1533,

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Definition and examples Definition and examples Erha rd Schön (1535) The idea of anamorphosis appeared as a byproduct on the investigation of oblique images and wide-angle views by and Leonardo .

Leonardo da Vinci, www.exploramuseum.de 7 2014-06-27 8 2014-06-27

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Definition and examples

Profet Jonah

Erhard Schön 9 www.exploramuseum.de 2014-06-27 10 2014-06-27

Definition and examples -R. Paprocki- Krosno Definition and examples-R. Paprocki- The Salt World

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Definition and examples -R. Paprocki- Niedzica Definition and examples -R. Paprocki- Niedzica

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Construction of perspectival anamorphosis with Construction of perspectival anamorphosis with deformation grids deformation grids } In the R3 space we have a plane a (called wall) and a vertical } To project the vertical lines upon the wall we use the diagonal method. plane ß, perpendicular to a. Let us assume that we are given an So we intend to construct the images of the diagonal AB of the grid. This eye point O, and also a squares grid in a plane ß. This grid will be image of this line on the wall intersects the deformation grids in the point B. Its vanishing point is point P which we obtain at the projected from O upon a wall a. At the beginning we consider the ? images of the horizontal lines of the grid. The orthogonal intersection of the line passing O and parallel to AB. The ? POO’ = 45 so we construct the point P on the vertical line through O’ in such a way projection of O upon the wall gives us the point O’ which is the that ’=O’O. Due to one of the most fundamental theorems of the vanishing point of this set of parallel lines. Thus the images of the perspective theory PB is an image of AB. The images of vertical lines of horizontal lines are lines radiating out from O’. the grid we obtain as vertical lines at the point of intersection of images of horizontal lines with lines PB.

P

B

O’ O’

O O A

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Descriptivegeometry A cone and a cylinder in Monge projection

} We wish to represent three-dimensional figures in a plane. This figures represent a cone and a cylinder } We choose two mutually perpendicular planes: one is called the ground (or horizontal) plane a while the other is called the vertical plane ß. Sv } Their intersection is the ground line t. } We project a point A in a space orthogonally on those two planes. Bv Av } This gives Av (in ß) and Ah (in a ).. ß Bv Av Av t Bh A Av

h A h Ah S Ah Bh Ah a (Av,Ah) represents A t

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Definition and examples – conical anamorphosis Conical anamorphosis

} Imagine a right circular cone standing on the ground plane and an O eye point O directly above the tip of the cone. } Let P be a point in the ground plane. It is required to construct a point P’ in the ground plane so that reflected in U the mirror – seen from O – it appears to be P. } P’ In other words, satisfies the P following condition: P’ Let OP intersect the cone at U. A ray from P’ striking the cone at U is being reflected along UO.

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Ov Conical anamorphosis Definition and examples– cylindrical anamorphosis Sv } Let p be the plane through the cone’s axis parallel to the vertical plane. In case P

is in p, the problem is v trivial. We are going to U make use of this special case to solve the general Qv Rv problem. Av } Rotate the ground plane around the (vertical) axis of the cone till P coincides with some point Q in p and Oh Qh Rh solve the problem Q. This Ah gives R. Apply the inverse Ph rotation to R to find P’. P’ h

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Ov Ov Cylindrical anamorphosis Cylindrical anamorphosis } An eye point O and a } So we need to solve the cylindrical mirror standing problem: Let P be a point in p on the horizontal (ground) Pv and let C be the point where Pv plane are given. the segment OP cuts the cylinder. Determine the point } Let p be the polar plane of P’ in the ground plane so that O with respect to the the ray P’C reflects along CO. v v cylinder (i. e. the plane Q } We let Q be the point of Q determined by the lines of intersection of OP and the contact of the tangent ground plane. The laws of planes through O). p reflection and some p } elementary geometry show Because of the eye does not h h h h Oh h that P’ C and C O make the Oh h distinguish between points C Ph C Ph on the same visual ray, we same angle with the tangent Qh line at Ch to the circular base Qh assume that the light seen and that ChP’h=ChQh. from O is coming from points in p. } This solves the problem.

h h 23 P’ 24 P’

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Application Application

} The most popular form of } Proper flat and reflective anamorphosis are used in interior design, for example as a form of on the walls anamorphosis are plane which, depending on the point of the observation, show anamorphosis. We can another image. include the entire group of } They also can be seen in the architecture, where the mirror columns are not only the support of architecture, but also horizontal road signs to this an additional element of the design, in which different group. Disproportionately figures made on the floor specifically for this purpose are stretched signs painted on reflected. the street, from the view of the road users take the natural proportions.

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Emmanuel Maignan (1642) „St. Francis of Paola”, Summary } Anamorphosis I?GiIi?HiH??IiH?IHigIhHkiiIiG?IIihIi both art historians and mathematicians. It is unknown how many artists wanting to show their talent created anamorphic works, which still may never be found out or correctly classified. Contemporary analytic and , easily copes with the design of deformation grids, which was a huge problem for mathematicians and painters at the turn of the centuries. This allows them to spread anamorphosis and thanks to that we can meet it more often in the everyday life in the world around us.

E. Maignan „St. Francis of Paola” 27 2014-06-27 28 2014-06-27

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Reference

} Andersen K., History of Mathematics: states of the art,The Mathematical Treatment of Anamorphosis from Piero Della Francesca to Niceron, Academic Press, 1996. } Baltrusaitis J., Anamorfozy, Gdansk, 2009. } Drabbe J., -Randour Ch. Descriptive geometry and anamorphosis, (users.skynet.be/ mathema/ en g.htm) . } Kemp M., The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat, Yale, 1990. } Massey L., Picturing Space, Displacing Bodies, Anamorphosis in Early Modern Theories of Perspective, Pennsylvania, 2007.

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