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Test 2 study cards Proj. Geom.
Projective Geometry Test 2
20
Other
Graduate
10/26/2015

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Term
Fundamental Theorem of Projective Geometry
Definition
A projectivity is determined when three collinear points and the corresponding three collinear points are given.
Term
Pappus's Theorem
Definition
If the six vertices of a hexagon lie alternately on two lines, the three pairs of opposite sides meet in collinear points
Term
The axis
Definition
Every projectivity relating ranges on two distinct lines determines another special line, the "axis", which contains the intersection of the cross-joins of any two pairs of corresponding points.
Term
the dual of the axis
Definition
center
Term
any two harmonic sets are related...
Definition
by a unique projectivity
Term
A projectivity relating ranges on two distinct lines is a perspectivity iff...
Definition
the common point of the two lines is invariant.
Term
Graves' triangles
Definition
a cycle of three triangles, each inscribed in the next
Term
if two triangles are doubly perspective...
Definition
they are triply perspective
Term
Axiom 3.11: any two lines are incident...
Definition
with at least one point
Term
Axiom 3.12: there exists four points...
Definition
of which no three are collinear
Term
Principle of duality
Definition
every definition and every theorem remain true when we interchange the words point for line (and join and meet, collinear and concurrent, vertex and side, etc.)
Term
three dimensional duality
Definition
points, lines and planes are interchanged with planes, lines and points
Term
Desargues configuration
Definition
(m_c, n_d) where mc=nd and c of the n lines pass through each of the m points while d of the poiunts lie on each of the n lines.
(m points have c lines pass through and n lines contain d points)
Term
self-dual
Definition
a configuration where (n_d,n_d) like 3_2 which is a triangle, another example is a 10_3
Term
harmonic conjugate of c with respect to a and b
Definition
any three concurrent lines a, b, c determine a fourth line f, concurrent with them (see page 28 for construction help)
Term
3.31: a harmonic set of points is projected from any point outside the line by...
Definition
a harmonic set of lines
Term
3.32: Any section of a harmonic set of lines, by a line not passing through the point of concurrence ...
Definition
is a harmonic set of points
Term
3.33 If ABCD is perspective (or projective) to A'B'C'F' and H(AB,CF), then...
Definition
H(A'B',C'F')
Term
3.35 If H(AB,CF), then...
Definition
H(FC,AB)
Term
cevians
Definition
lines passing through vertices of a triangle (examples include the "median" and "altitude"
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