While Greek geometry was essentially confined to the study of lines and conics, the new analytic geometry opened the way to the study of an incredibly wider class of curves. This is the main reason behind its success. However, before reaching its full strength, analytic geometry had to overcome a few hurdles.
We shall not dwell on the trivial way to extend the idea of Cartesian coordinates from the plane to three dimensional space: simply choose three axis of coordinates, not in the same plane. The three dimensional case had been considered by Fermat and Descartes from the very beginning of analytic geometry, and later by La Hire (1640–1718) and many others.
Let us instead focus on the evolution of ideas concerning these systems of coor- dinates.
An important step was the recognition of negative numbers as possible coor- dinates: a coordinate is not a distance (thus a positive number), it is a distance equipped with a sign—that is, after all—an arbitrary number. It seems that the British mathematician Wallis (1616–1703) was the first to use negative coordinates.
But it is essentially the work of the famous British mathematician and physicist Isaac Newton (1642–1734), in particular a work on curves of higher degree pub- lished in 1676, which popularized the use of negative coordinates. In this work, Newton had classified 72 types of cubics, forgetting half a dozen of them.
The idea of “separating the variables” of an equation is due to the Swiss math- ematician Leonhard Euler (1707–1783). Instead of considering one equation with two variablesx,y, he writes these two variables separately in terms of some variable parameter.
1.3 More on Cartesian Systems of Coordinates 7
Fig. 1.5
For example, consider the following system of equations:
x=Rcosθ y=Rsinθ We can view it in two different ways (see Fig.1.5).
Ifθ is a varying parameter whileR >0 is fixed, all the pairs(x, y), for all the possible values ofθ, describe a circle of radiusR. The classical equation
x2+y2=R2
is recaptured by eliminating the parameterθbetween the two parametric equations:
simply square both equations and add the results.
However, ifθ is constant and R is the variable parameter, the same system of equations describes all the points(x, y)of a line making an angleθ with the hori- zontal axis! The elimination of the parameterR between the two equations is now straightforward:
xsinθ=ycosθ.
So before presenting a system of parametric equations, it is important to clearly identify the parameter.
Another useful technique was to clarify the rules by which coordinates transform when passing to another system of coordinates. As we have seen, Fermat had used methods of this type to study the conics empirically. But again it was Euler who developed the general theory of “coordinate changes”. In particular, he observed the very special form that these formulổ take in the case of rectangular systems of coordinates:
Definition 1.3.1 By a rectangular system of coordinates in the plane or in space is meant a Cartesian system of coordinates in which any two of the axes are always perpendicular.
Fig. 1.6
Notice that in those days the possible choice of different unit lengths on both axis was not considered, since the coordinate along an axis was defined as the distance from the origin to the projection of the point on this axis, with the adequate sign.
Let us follow Euler’s argument, in the case of the plane. Euler had developed an analogous theory in dimension 3; we omit it here since the general theory, in arbitrary finite dimensionn, will be studied in subsequent chapters, using the full strength of modern algebra. Let us also mention that the results of Euler were discov- ered independently, a little bit later, by the French mathematician Lagrange (1736–
1813).
Consider two rectangular systems of coordinates in the plane (see Fig.1.6). Let (a, b)be the coordinates ofOin the system of coordinates with originOand(x, y), (x, y), respectively, the coordinates of a pointP in the systems of coordinates with originO,O. We use the notationA,B,X,Y,X,Yto indicate the corresponding points on the axes. Writeθfor the angle between the directions of thexandxaxes.
Draw the various parallels to the axis and consider the points so obtained on the figure. We observe first that the two right angled trianglesOMY andP N X are isometric, proving that
KL=N X=MY=OYãsinθ=ysinθ.
Therefore
x=OX=OA+AX=OA+OK=OA+
OL−KL
=a+xcosθ−ysinθ.
An analogous argument withy yields the formulổ x=a+xcosθ−ysinθ
y=b+xsinθ+ycosθ.