Cartesian Co-ordinates

Values for determination of the location of points (and thus also of sets of points, e.g. curves, areas) within the plane or within the three-dimensional space (in the n-dimensional space in general). Therefore a frame of reference (co-ordinate system) must be defined. Most frequently is the cartesian (othogonal) system of co-ordinates.

The Cartesian System of Co-ordinates of the plane

It is made up of two orthographic straight lines of numbers (co-ordinate axes) which intersect in their Null points. They form an axial cross. The point of intersection is called origin, null point or co-ordinate starting point. The first axis (mostly illustrated horizontal) is normally named x-axis or axis of abscissae, the second axis y-axis or axis of ordinates. The orientation of the axes is defined in the following way: the positive x-axis fades into the positive y-axis by a 90-degrees-rotation with positive rotational direction (anti-clockwise).

Unless P is a point of the plane. If the parallel of the y-axis through P intersects the x-axis in x(p) and the parallel of the x-axis through P intersects the y-axis in y(p), we can assign the pair of numbers ( x(p) ; y(p) ) uniquely to the point P and vice versa. Then x(p) is named x-co-ordinate or abscissa,  y(p) y-co-ordinate or ordinate of P. If  ( x(p) ; y(p) )  is the pair of co-ordinates of the point P therefore we can denote  P = ( x(p) ; y(p) )  or in short  P( x(p) ; y(p) ).

The Cartesian System of Co-ordinates of the 3-dimensional Space

 cartesian co-ordinates

It is built up similar to the cartesian co-ordinate system of the plane. However there are three co-ordinate axes which are pairwise orthographic to each other. As a rule the third axis is called z-axis or axis of applicates. The orientation of the three axes based upon the so-called right-hand-rule. Therewith the three axes comply with a right-hand system.

Analogical to the cartesian co-ordinate system of the plane the triple of numbers  ( x(p) ; y(p) ; z(p) )  is assigned uniquely to a point P in space, i.e.  P = ( x(p) ; y(p) ; z(p) )  or in short  P( x(p) ; y(p) ; z(p) ).

  
See also
  
 Gauss' trihedral Gauss' trihedral
 spherical co-ordinates spherical co-ordinates