Elementary Coordinate Systems

At a time when GPS location is in full swing, it appears opportune to recall the most elementary methods permitting us to locate the position of a point in the plane or in space. It is no accident that the plane and the sphere play a fundamental role in the initial processes of location

The term "coordinates" is attested from at least 1754, as being the expression from geometry which designates "some elements permitting to place something or someone precisely ". By analogy, they came to be employed very quickly in astronomy, in geography... Today the term has passed into everyday language as indicating information about where to find someone : telephone, address, email...

In the mathematical sense, the coordinates of a point M situated in a space E (or in a part of E) are scalars which permit us to locate, without ambiguity, this point in the system to which it belongs.

In the case where M belongs to a plane or a surface, the principle is to determine two families of curves plotted on the surface such that all curves of one family have a unique point of intersection with those of the other family and such that for all points M of the surface there exist one curve of each family which passes through M. In the case of 3-dimensional space, this time you define three families of surfaces. Each point will be located by the intersection of three surfaces, one for each of the families.

Cartesian Coordinates

The simplest method of locating a point is the system of cartesian coordinates. As its name indicates, it was René Descartes who made known this useful method of location on the affine plane or the affine space of dimension 3, a method which generalizes to a space of dimension n.

In such a space, one defines a reference consisting of an origin O and of a basis of the associated vector space $\Re^n$. This basis is composed of two unit vectors for a plane, of three vectors for a space of dimension 3, and n vectors for a space of dimension n. Along with the origin, this basis permits us to define coordinates axes in the space, these axes being orientated straight lines passing through the origin. In a similar manner, a point M of a 3-dimensional space will be defined in a unique manner by the vectorial equation \[ \vec{OM} = x \bf i\rm + y \bf j\rm + z \bf k\rm \] where \[ \bf i\rm, \bf j\rm, \bf k\rm \] forms the basis of $\Re^3$ and $(x,y,z)$ are called the coordinates of M: $x$ is the abscissa, $y$ is the ordinate, and $z$ is the height.

The plane or the space of dimension n could be supplied with a metric, permitting us to define, in addition, the distance between two points. Among the possible metrics, the one most used in elementary maths are the "Euclidean" distances. With such distances are associated the notion of the scalar product of two vectors, the distance between two points M and N then being the square root of the scalar product of the vector $\vec{MN}$.

One can therefore choose an "orthonormal" reference frame $(\bf i\rm, \bf j\rm, \bf k\rm)$ of vectors, each pair of different vectors being orthogonal (their scalar product is zero) and with their norm being 1 (their scalar square equals 1).

In such a reference frame, the scalar product of two vectors is very easy to calculate \[ (x \bf i\rm, y \bf j\rm, z \bf k\rm) \bullet (x' \bf i\rm, y' \bf j\rm, z' \bf k\rm) = xx' + yy' + zz' \] and therefore also the distance between two points.

Once all points are identified with the aid of these cartesian coordinates, other problems arise. One of the most classic consists in finding the "equation" of a collection of points, a confirmed relation between the coordinates of its points.

Therefore, on the plane supplied with a cartesian coordinate frame you can have, for example

  • straight lines obeying equations of degree 1, of the form $ax + by + c=0$
  • conics with equations of degree 2, of the form $ax^2 +bxy + cy^2 +dx +ey +f=0$
  • cubics with equations of degree 3
  • ...and so on

One can also represent the functions $f$ by their graphs, the collection of points M of the plane supplied with a cartesian reference of coordinates $(x,y)$ such that $y=f(x)$.

In the space of dimension 3, a unique equation will in general be confirmed by the collection of points of a surface. This is the case for example for a plane, which verifies an equation of first degree. A curve will be defined, as regards to it, by two equations to the extent where it corresponds to the intersection of two surfaces.

Polar Coordinates, and Cylindrical and Spherical Coordinates

It's in the Euclidean plane that one commonly uses "polar" coordinates. To start off, they still consist of an origin O and of a "unit" vector. This specifies, similar to the abscissa in cartesian coordinates, a half-line $[Ox)$, called the polar axis. A point M of the plane is absolutely defined by the distance OM, denoted $r$ (or sometimes $\rho$ [place cursor here for translator's note]), and by the angle $\theta$ which expresses the measure, in a trigonometrical sense, of an angle "orientated" between $[Ox)$ and the half-line $[OM)$.

One passes easily from cartesian coordinates to polar coordinates via the relations \[ x=r \cos\theta; \ y=r\sin\theta;\ r = \sqrt{x^2+y^2} \]

A curve of the plane could in this way be defined by a polar equation, a relation between $r$ and $\theta$.

A circle of center O will have a simple equation ($r = R$) but a straight line or a circle with a different center than O will be represented by more complicated equations.

Thus \[ r=h\cos (\theta - \theta_0)\] is the equation of the straight line situated at a distance h from the origin in the direction $\theta_0 + \frac{\pi}{2}$ and \[r= \frac{h}{\cos(\theta - \theta_0)}\] is the equation of a circle whose center is situated on the straight line making an angle $\theta_0$ with the axis $Ox$.

By extension from polar coordinates in the 2-dimensional plane, one can define cylindrical coordinates in the space $\Re^3$ very simply. It is sufficient just to add a third dimension to these polar coordinates, often denoted z just like the height in 'ordinary' cartesian coordinates, which measures the height of a point with respect to the 2-dimensional plane represented by the polar coordinates $(r, \theta)$.

From the cartesian coordinates $(x,y,z)$ one can obtain the cylindrical coordinates $(r, \theta, z)$ from the following formulae. \[ r = x^2 + y^2; \ \ \theta = \tan^{-1}\frac{y}{x}; \ \ z=z\]

Inversely, you can convert the cylindrical coordinates $(r, \theta, z)$ to cartesian coordinates $(x,y,z)$ by using the following formulae \[ x= r\cos\theta; \ \ y=r\sin\theta; \ \ z=z\]

Another generalization of the polar coordinates of the 2-dimensional plane to the 3-dimensional space is given by spherical coordinates.

A point of space y is represented by the distances $\rho$ from the origin O (the pole) and by two angles.

In an orthonormal representation $(O, \bf i \rm, \bf j \rm, \bf k \rm)$ of $\Re^3$, let the orthogonal projection of M onto the plane (at $z=0$) be m . The spherical coordinates of M are the triplet $(\rho, \theta, \phi)$ where $\rho$ is the distance OM, where $\theta$ (between 0 and $2\pi$ inclusive) is the measure in radians of the angle $(\bf \bf i \rm , \vec{Om})$ and $\phi$ (between $0$ and $\pi$ inclusive) is the measure in radians of the angle $(\bf k \rm, \vec{OM})$ (where $\bf k \rm$ is the basis vector in the direction of z).

Spherical Coordinates

For a point on the terrestial sphere, $\rho$ would be the radius of the Earth augmented by any extra altitude, $\theta$ corresponds to the longitude and $\phi$ to the latitude (although the latitude would need to be 'tweaked' to correspond exactly with the values $0-90^\circ$ North/South used on a map).

This paragraph diverges from the original article, indicating the difference in conventions in different countries. I have used $(r, \theta)$ as the convention for representing the 2-dimensional polar coordinates. This is so that when representing the spherical coordinates as ($\rho, \theta, \phi$) it is clear that $\theta$ is analogous to the $\theta$ in the previous sentence but the other coordinates are not. Other conventions can produce confusion (in my view) by using the same symbol for different things in the different coordinate systems. The original article is translated as follows : For reasons connected to their norms, physicists invert $\theta$ and $\phi$, and for certain of them $\phi$ is designated as a measure in radians of the angle $(\bf k \rm, \vec{OM})$. $\phi$ is then called co-latitude, such a system is very common in astronomy. The first part of this statement is true in the sense that physicists and mathematicians, even in the same country, have a tendency to invert the names for these coordinates; whereas the second part just describes the convention I have used here and which is actually used by mathematicians in my part of the world. In passing, I could say that the term co-latitude is indeed a term used in astronomy : for example, for a person standing on the Earth at a latitude of $50^\circ$ north would find that stars on the celestial sphere at a co-latitude of $50^\circ$ or greater would be circumpolar (i.e. they would be in the night sky all year long if they occupy a position on the celestial sphere at, or above, a celestial latitude of $+40^\circ$ ).

For the geographical representation, we have just seen that we the coordinates are the altitude, the latitude and the longitude. Nevertheless, according to the precision hoping to be achieved, this would need to be adapted taking into account the fact that the Earth is not a sphere. Therefore, it would be replaced with a geodesic system of ellipsoid type which models the form of the Earth.

It is these elements used by the GPS system which obtain signals from whichever objects enter into the field of their satellites allowing them to calculate the location of these objects by "trilaterisation" .

Alain Zalmanski, Tangente, Nr 157

  • Conformal Transformations

    When two orientated curves cut each other, we can define an angle at the point of incidence on considering the angle formed by the tangents at this point.

    If these two curves undergo a geometrical transformation, it is interesting to look at how this angle evolves. When the angle is unchanged, we say that the transformation is conformal. This is evidently the case if you consider a rotation or an enlargement, but conformal tranformations are not limited to these applications alone - on the contrary, they are relatively numerous! In a plane referenced by straightfoward orthonormal coordinates axes we can use mapping to make a correspondence between the points of the plane and the complex numbers. The majority of simple complex functions like $z\rightarrow z^2, z \rightarrow e^z, z \rightarrow \frac{1}{z}$ are associated with conformal transformations. They correspond to applications which which are dscribed as holomorphic. The three preceding functions transform a square net of the plane into the following figures respectively, where all the curves intersect at right angles.

    Another well-known conformal transformation is the Mercator Projection. It is a cylindrical projection used for geographical maps, in which the meridians and the parallels form orthogonal lines. Because they rely on a conformal transformation, these maps are used in navigation because measured angles can be transferred directly from them into the real world. The downside of this is a consequent deformation of proportions on a grand scale.