b · (a – xb) = 0
Solving for x gives the formula for the projected vector:
Projection of a onto b =
b
See also MERCATOR’S PROJECTION; STEREOGRAPHIC
PROJECTION.
projective geometry The branch of GEOMETRY
that examines those properties of geometric figures
that remain unchanged by a central PROJECTION is
called projective geometry. Informally, the shadows
cast by a point source of light, such as the flame of a
candle or the small bulb of a flashlight, are central
projections, and so projective geometry examines
those properties that are the same for the shadow of
a geometric object and the object itself. For instance,
the “shadow” of a straight line is another straight
line, and so the notion of “straight” is a valid concept
in projective geometry.
French mathematician and engineer GIRARD
DESARGUES (1591–1661), through his studies of PERSPECTIVE in art, was struck by the fact that the image of
a CONIC SECTION under projection is another conic section. (For instance, the outline of the circular rim of a
flashlight, a conic section, aimed at the ground at an
appropriate angle produces the image of either an
ELLIPSE, a PARABOLA, or a branch of a HYPERBOLA.) His
study of this phenomenon founded the field of projective geometry. Unfortunately, his work was misunderstood at the time and was neglected. The subject was
later revived in 1822 by French mathematician Jean
Victor Poncelet (1788–1867) with his publication of
Traité des proprietés des figures (Treatise on the projective property of figures).
DESARGUES’S THEOREM is a fundamental result
about perspective. In order to unify special cases of the
theorem and to allow for the possibility of two lines
being parallel in its statement, Desargues developed the
notion of “points at infinity” so that one could appropriately speak of the point of any intersection of any
two lines in the plane. (Parallel lines are said to intersect at these points at infinity.) In this framework, any
two points in the plane determine a line, and any two
lines determine a point. Thus Desargues had created a
notion of geometry in which the words point and line
play dual roles, and can be interchanged in any true
statement about this geometry to obtain new (true)
statements in this geometry. This notion of duality was
first properly outlined by Poncelet.
Today, any system of objects, usually called points
and lines, satisfying the following five axioms is called
a projective geometry:
1. For any two distinct points, there is exactly one line
containing them both.
2. For any two distinct lines, there is exactly one point
common to both.
3. There is at least one line.
4. Each line contains at least three points.
5. Not all points lie on the same line.
(The last three axioms ensure that the geometry is not
trivial.)
A finite projective geometry is any arrangement of
a finite number of points and lines that obey these five
axioms. For instance, the diagram below, called the
Fano plane, is a projective geometry with seven points
and seven lines with exactly three points on each line
and three lines through each point. (The central triangle is considered a single line.) This is called the symmetric configuration of type 7 3 .
There is only one symmetric configuration of type
8 3 , three of type 9 3 , 10 of type 10 3 , and 31 of type 11 3 .
All the symmetric configurations up to type 18 3 have
been determined.
See also HISTORY OF GEOMETRY (essay).
a · b
––
|b|
2
projective geometry 419
The symmetric configuration of type 7 3
Solving for x gives the formula for the projected vector:
Projection of a onto b =
b
See also MERCATOR’S PROJECTION; STEREOGRAPHIC
PROJECTION.
projective geometry The branch of GEOMETRY
that examines those properties of geometric figures
that remain unchanged by a central PROJECTION is
called projective geometry. Informally, the shadows
cast by a point source of light, such as the flame of a
candle or the small bulb of a flashlight, are central
projections, and so projective geometry examines
those properties that are the same for the shadow of
a geometric object and the object itself. For instance,
the “shadow” of a straight line is another straight
line, and so the notion of “straight” is a valid concept
in projective geometry.
French mathematician and engineer GIRARD
DESARGUES (1591–1661), through his studies of PERSPECTIVE in art, was struck by the fact that the image of
a CONIC SECTION under projection is another conic section. (For instance, the outline of the circular rim of a
flashlight, a conic section, aimed at the ground at an
appropriate angle produces the image of either an
ELLIPSE, a PARABOLA, or a branch of a HYPERBOLA.) His
study of this phenomenon founded the field of projective geometry. Unfortunately, his work was misunderstood at the time and was neglected. The subject was
later revived in 1822 by French mathematician Jean
Victor Poncelet (1788–1867) with his publication of
Traité des proprietés des figures (Treatise on the projective property of figures).
DESARGUES’S THEOREM is a fundamental result
about perspective. In order to unify special cases of the
theorem and to allow for the possibility of two lines
being parallel in its statement, Desargues developed the
notion of “points at infinity” so that one could appropriately speak of the point of any intersection of any
two lines in the plane. (Parallel lines are said to intersect at these points at infinity.) In this framework, any
two points in the plane determine a line, and any two
lines determine a point. Thus Desargues had created a
notion of geometry in which the words point and line
play dual roles, and can be interchanged in any true
statement about this geometry to obtain new (true)
statements in this geometry. This notion of duality was
first properly outlined by Poncelet.
Today, any system of objects, usually called points
and lines, satisfying the following five axioms is called
a projective geometry:
1. For any two distinct points, there is exactly one line
containing them both.
2. For any two distinct lines, there is exactly one point
common to both.
3. There is at least one line.
4. Each line contains at least three points.
5. Not all points lie on the same line.
(The last three axioms ensure that the geometry is not
trivial.)
A finite projective geometry is any arrangement of
a finite number of points and lines that obey these five
axioms. For instance, the diagram below, called the
Fano plane, is a projective geometry with seven points
and seven lines with exactly three points on each line
and three lines through each point. (The central triangle is considered a single line.) This is called the symmetric configuration of type 7 3 .
There is only one symmetric configuration of type
8 3 , three of type 9 3 , 10 of type 10 3 , and 31 of type 11 3 .
All the symmetric configurations up to type 18 3 have
been determined.
See also HISTORY OF GEOMETRY (essay).
a · b
––
|b|
2
projective geometry 419
The symmetric configuration of type 7 3