Primitive Relation for Elliptic Geometry

​
collinearity
show planes
move point(s)
1
This Demonstration shows that the binary relation
AB=π/2
(the distance of
A
from
B
equals
π/2
) is a suitable primitive notion for elliptic geometry. M. Pieri showed that the ternary relation of a point being equally distant from two other points (in symbols,
AB=BC
) can be used as the only primitive notion of Euclidean geometry of two or more dimensions[1]. Pieri's relation can also be used as a primitive defining relation for non-Euclidean geometries.
In the usual model of the elliptic plane, a point is a line through the origin in a three-dimensional Euclidean space. Such a line is determined by a point on a unit sphere. Two antipodal points on the sphere represent the same point of the elliptic geometry. The distance of two points is measured by the angle between their radius vectors and is always less than or equal to
π/2
. Points at a distance
π/2
from a given point lie on a great circle. Lines of the model are planes through two points and the center of the sphere; they are shown as great circles on the unit sphere.
Ignoring anything outside the sphere, an alternative view of the model is that points in elliptic 2D geometry are antipodal pairs of points of a unit sphere and lines are of great circles.
So the collinearity of three points means that they are on the same great circle. Perpendicularity of two segments means perpendicularity of the great circles that contain the segments. Distance
AR=π/4
is obtained by the existence of points
B
and
C
such that distances
AB
,
BC
and
AC
are
π/2
, and by the existence of points
Q
and
P
that form a special equilateral triangle
RPQ
​
so that
R
is the "midpoint" of
AB
(but the definition of midpoint is not yet available). Thus we have two definitions of midpoint, depending on which antipodal point is used. Points
A
and
C
are symmetric relative to
B
if
B
is the midpoint of
AC
or
A=C.
Finally,
A
and
C
are equidistant from
B
if there exists a point
X
such that
A
and
C
are symmetric relative to
X
and
AC
is perpendicular to
BX
.

Details

col(A,B,P)⇔(∃C)(AC=BC=PC=π/2)
(collinearity)
AB⊥DP⇔A≠B∧D≠P∧(∃C)(AC=BC=π/2∧col(D,P,C)
(perpendicularity)
AR=π/4⇔(∃BCPQ)(BC=AB=AC=π/2∧col(B,P,C)∧col(A,Q,C)∧col(A,R,B)∧P≠C∧Q≠C∧AP⊥BQ∧PR⊥AR)
(distance
π/4
)
mid(A,B,C)⇔col(A,B,C)∧(∃X)(AX=CX=π/4∧AC⊥BX)
(midpoint)
mex(A,B,C)⇔col(A,B,C)∧(∃X)(mid(A,X,C)∧BX=π/2)
(external midpoint)
sym(A,B,C)⇔mid(A,B,C)∨mex(A,B,C)∨A=B=C∨(A=C∧AB=π/2)∨(AC=π/2∧AB=BC=π/4)
(symmetric)
AB=BC⇔(∃X)(sym(A,X,C))∧BX⊥AB)
(Robinson's definition of Pieri's relation[2, pp. 72–73])

References

[1] M. Pieri, "La Geometria Elementare istituita sulle nozioni di punto e sfera," Memorie di matematica e di fisica della Società italiana delle Scienze, ser. 3(15), 1908 pp. 345–450.
[2] R. M. Robinson, "Binary Relations as Primitive Notions in Elementary Geometry: The Axiomatic Method with Special Reference to Geometry and Physics," in Proceedings of an International Symposium Held at the University of California, Berkeley, December 26, 1957–January 4, 1958, Amsterdam: North-Holland Publishing Company, 1959. doi:10.1017/S0022481200092690.

External Links

Elliptic Geometry (Wolfram MathWorld)
Spherical Geometry (Wolfram MathWorld)
Spherical Triangle (Wolfram MathWorld)
Between (Wolfram MathWorld)
Collinear (Wolfram MathWorld)
Midpoint (Wolfram MathWorld)
Euclidean Geometry (Wolfram MathWorld)
Geometry (Wolfram MathWorld)
Perpendicular Bisector (Wolfram MathWorld)
Equidistance and Betweenness in Euclidean Plane Geometry
Pasch's Axiom in Euclidean Geometry
Pieri's Ternary Relation and Euclidean Geometry
Orthogonality as well as Equidistance Can Be Used as the Sole Primitive Notion for Euclidean Geometry

Permanent Citation

Izidor Hafner
​
​"Primitive Relation for Elliptic Geometry"​
​http://demonstrations.wolfram.com/PrimitiveRelationForEllipticGeometry/​
​Wolfram Demonstrations Project​
​Published: April 24, 2018