Page 406 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 31: Joachimsthal's Notations

Such connections are also symmetric, which means that Notes
D Y – D X = [X, Y],
Y
X
where [X, Y ] is the Lie bracket of vector fields.
31.3 Joachimsthal Theorem and Notation

If the curve of intersection of two surfaces is a line of curvature on both, the surfaces cut at a
constant angle. Conversely, if two surfaces cut at a constant angle, and the curve of intersection
is a line of curvature on one of them, it is a line of curvature on the other also
Ferdinand Joachimsthal (1818-1861) was a German mathematician and educator famous for the
high quality of his lectures and the books he wrote. The notations named after him and discussed
below serve one of the examples where the language of mathematics is especially auspicious for
derivation and memorization of properties of mathematical objects. Joachimsthal’s notations
have had extended influence beyond the study of second order equations and conic sections,
compare for example the work of F. Morley.
A general second degree equation
Ax + 2Bxy + Cy + 2Fx + 2Gy + H = 0 ...(1)
2
2
represents a plane conic, or a conic section, i.e., the intersection of a circular two-sided cone with
a plane. The equations for ellipses, parabolas, and hyperbolas all can be written in this form.
These curves are said to be non-degenerate conics. Non-degenerate conics are obtained when
the plane cutting a cone does not pass through its vertex. If the plane does go through the cone’s
vertex, the intersection may be either two crossing straight lines, a single straight line and even
a point. These point sets are said to be degenerate conics. In the following, we shall be only
concerned with a non-degenerate case.

The left-hand side in (1) will be conveniently denoted as s:
s = Ax + 2Bxy + Cy + 2Fx + 2Gy + H ...(2)
2
2
so that the second degree equation (1) acquires a very short form:
s = 0. ...(3)

A point P(x , y ) may or may not lie on the conic defined by (1) or (3). If it does, we get an identity
1
1
by substituting x = x and y = y into (1):
1
1
Ax + 2Bx y + Cy + 2Fx + 2Gy + H = 0, ...(4)
2
2
1
1 1
1
1
1
which has a convenient Joachimsthal’s equivalent
s = 0. ...(5)
11
For another point P(x , y ) we similarly define s and, in general, for points P(x , y ) or P(x, y) we
j
2
i
2
i
22
j
define s and s , where, for example,
ii
jj
s = Ax + 2Bx y + Cy + 2Fx + 2Gy + H. ...(6)
2
2
i
i
ii
i
i
i i
Thus, s = 0 means that P(x , y ) lies on the conic (3), s  0 that it does not.
i
i
ii
ii
There is also a mixed notation. For two points P(x , y ) and P(x, y), we define
j
j
i
i
s = Ax x + B(x y + xy ) + Cy y + F(x + x) + G(y + y) + H. ...(7)
i
j
j
i j
i j
ij
i j
i
j i
LOVELY PROFESSIONAL UNIVERSITY 399