2014-06-27
Plan
Definition and examples
Construction of perspectival anamorphosis with
deformation grids
Descriptive geometry
Conical anamorphosis
Cylindrical anamorphosis
Application
More examples
Summary
}
}
}
}
}
Mathematics in art - anamorphosis
}
}
The Faculty of Mathematics and Computer Science
Jagiellonian University
Ada Palka
[email protected]
}
2
20 1 4 -0 6 -2 7
Definition and examples
Definition and examples
}
}
To give the images a nor mal per spective, the ar tists began
cr eating their wor k by dr awing the hor izon line h at the level of
the eye. Then at the center of the line they mar ked the main point
A, which is the point of inter section of all the par allel lines. At a
distance equal to the distance of the eye to a centr al point, ther e
was point B is mar ked ther e. This is a point of the inter section of
diagonals. To be r ead pr oper ly, the image should be viewed fr om
a par ticular viewpoint. In this way a squar e of the gr id of the
gr ound after pr ojecting appear s to be a tr apezoidal
checker boar d, on which it is enough to put other items in
pr opor tion to the decr easing dimensions of the boxes.
3
2 0 1 4-0 6-2 7
Thus a squar e becomes a tr apezoid in the per spective image. By
moving the obser vation point above the main point, keeping the
same distance fr om the focal point we get the opposite effect, the
tr apezoid begins to be seen as the squar e, r etur ning to its
or iginal for m. This peculiar per spective was initially used to
validate the per spective pr ojection by contr ast, however after
some time gave the beginning to the for mation of anamor phic
images .
4
20 1 4 -0 6 -2 7
1
2014-06-27
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
earlier.
} 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
vanishing point of the pyramid of vision away from the
main point and observation point close to the plane of
the work.
}
5
2 0 1 4-0 6-2 7
Definition and examples
Anamorphosis -kind
of intentional
picture deformation
}
6
Definition and examples
Hans Holbein,
The Ambassadors, 1533,
London, National Gallery
Erh a rd Sch ön (1535)
The idea of anamorphosis appeared as a byproduct on the
investigation of oblique images and wide-angle views by
Piero della Francesca and Leonardo da Vinci.
Leonardo da Vinci, CODEX ATLANTICUS
7
2 0 1 4-0 6-2 7
8
www.exp loramu seu m.d e
20 1 4 -0 6 -2 7
2
2014-06-27
Definition and examples
Profet Jonah
9
www.exploramuseum. de
Erha rd Schön
2 0 1 4-0 6-2 7
Definition and examples -R. Paprocki- Krosno
11
2 0 1 4-0 6-2 7
10
20 1 4 -0 6 -2 7
Definition and examples -R. Paprocki- The Salt World
12
20 1 4 -0 6 -2 7
3
2014-06-27
Definition and examples -R. Paprocki- Niedzica
13
Definition and examples -R. Paprocki- Niedzica
2 0 1 4-0 6-2 7
14
Construction of perspectival anamorphosis with
deformation grids
}
In the R3 space we have a plane a (called wall) and a ver tical
plane ß, per pendicular to a . Let us assume that we ar e given an
eye point O, and also a squar es gr id in a plane ß. This gr id will be
pr ojected fr om O upon a wall a . At the beginning we consider the
images of the hor izontal lines of the gr id. The or thogonal
pr ojection of O upon the wall gives us the point O’ which is the
vanishing point of this set of par allel lines. Thus the images of the
hor izontal lines ar e lines r adiating out fr om O’.
20 1 4 -0 6 -2 7
Construction of perspectival anamorphosis with
deformation grids
}
To project the vertical lines upon the wall we use the diagonal method.
So we intend to construct the images of the diagonal AB of the grid. This
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
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
that PO’=O’O. Due to one of the most fundamental theorems of the
perspective theory PB is an image of AB. The images of vertical lines of
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
15
a
O
ß
16
a
A
20 1 4 -0 6 -2 7
4
2014-06-27
Descriptive geometry
}
}
}
}
}
A cone and a cylinder in Monge projection
We wish to represent three-dimensional figures in a plane.
We choose two mutually perpendicular planes: one is called the ground (or
horizontal) plane a while the other is called the vertical plane ß.
Their intersection is the ground line t.
We project a point A in a space orthogonally on those two planes.
This gives Av (in ß) and Ah (in a )..
This figures represent a cone and a cylinder
Sv
ß
Bv
t
A
Bv
Av
Av
Av
Ah
Bh Ah
Bh
Av
Ah
Ah
Sh
a
( Av,Ah) represents A
t
17
2 0 1 4-0 6-2 7
Definition and examples – conical anamorphosis
18
Conical anamorphosis
}
}
}
19
2 0 1 4-0 6-2 7
20 1 4 -0 6 -2 7
Imagine a right circular cone mirror
standing on the ground plane and an
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
the mirror – seen from O – it appears
to be P.
In other words, P’ satisfies the
following condition:
Let OP intersect the cone at U. A ray
from P’ striking the cone at U is being
reflected along UO.
20
O
U
P
P’
20 1 4 -0 6 -2 7
5
2014-06-27
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
trivial. We are going to
make use of this special
case to solve the general
problem.
Rotate the ground plane
around the (vertical) axis of
the cone till P coincides
with some point Q in p and
solve the problem Q. This
gives R. Apply the inverse
rotation to R to find P’.
Uv
Qv
Av
Oh
Qh
Rv
Rh
Ah
Ph
P’ h
21
2 0 1 4-0 6-2 7
22
Ov
Ov
Cylindrical anamorphosis
}
}
}
An eye point O and a
cylindrical mirror standing
on the horizontal (ground)
plane are given.
Let p be the polar plane of
O with respect to the
cylinder (i. e. the plane
determined by the lines of
contact of the tangent
planes through O).
Because of the eye does not
distinguish between points
on the same visual ray, we
assume that the light seen
from O is coming from
points in p.
23
20 1 4 -0 6 -2 7
Cylindrical anamorphosis
}
Pv
Qv
p
Oh
Ch
Ph
Qh
P’h
So we need to solve the
pr oblem: Let P be a point in p
and let C be the point wher e
the segment OP cuts the
cylinder. Deter mine the point
P’ in the gr ound plane so that
the r ay P’C r eflects along CO.
} We let Q be the point of
inter section of OP and the
gr ound plane. The laws of
r eflection and some
elementar y geometr y show
that P’h Ch and Ch Oh make the Oh
same angle with the tangent
line at Ch to the cir cular base
and that Ch P’h =Ch Qh .
} This solves the pr oblem.
24
Pv
Qv
p
Ch
Ph
Qh
P’h
6
2014-06-27
Application
}
Application
}
The most popular form of
anamorphosis are plane
anamorphosis. We can
include the entire group of
horizontal road signs to this
group. Disproportionately
stretched signs painted on
the street, from the view of
the road users take the
natural proportions.
25
}
2 0 1 4-0 6-2 7
Emmanuel Maignan (1642)
„St. Francis of Paola”,
Proper flat and reflective anamorphosis are used in interior
design, for example as a form of drawings on the walls
which, depending on the point of the observation, show
another image.
They also can be seen in the architecture, where the mirror
columns are not only the support of architecture, but also
an additional element of the design, in which different
figures made on the floor specifically for this purpose are
reflected.
26
20 1 4 -0 6 -2 7
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
descriptive geometry, 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
2 0 1 4-0 6-2 7
28
20 1 4 -0 6 -2 7
7
2014-06-27
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., Gabriel-Randour Ch. Descriptive geometry
and anamorphosis,
(users.skynet.be/ m athema/ 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.
29
2 0 1 4-0 6-2 7
8