Parameter Method

To discribe parameter representations we take the following assumptions:

fx, fy, fz  are continuous functions in each considered interval  I  of real numbers resp. real pairs of numbers.  x, y, z  are real numbers too.

In the following, 3 parameter representations will be illustrated which are very important for mathematical cartography.

1. parameter representation of a curve in a plane

The set of all points  P = ( x ; y )  with

x = fx(t)     and
y = fy(t)     for  t from I          (1)

is a plane curve. (1) is named parameter representation of a plane curve.

Remarks:

The real number  t  is called parameter of the plane curve. You can interpret t as time; thus  P = ( x ; y ) = ( fx(t) ; fy(t) )  will be the location onto the plane at which a wanderer is positioned onto the (plane) curve at the time t.

Example:

If choosing the continuous functions

fx(t)  =  a1  +  b1 * t
fy(t)  =  a2  +  b2 * t

for (1) with real coefficients  a1, a2  and  b1, b2,  you get the parameter representation of a straight line in the plane.

2. parameter representation of a curve in the three-dimensional space

The set of all points  P = ( x ; y ; z )  with

x = fx(t)     ,
y = fy(t)     and
z = fz(t)     for  t from I          (2)

is a space curve. (2) is named parameter representation of a space curve.

Remarks:

The real number  t  is called parameter of the space curve. Analogical to section 1 you can interpret  t  as time too.

Example:

If choosing the continuous functions

fx(t)  =  a1  +  b1 * t
fy(t)  =  a2  +  b2 * t
fz(t)  =  a3  +  b3 * t

for (2) with real coefficients  a1, a2, a3  and  b1, b2, b3,  you get the parameter representation of a straight line in space.

3. parameter representation of an area in the three-dimensional space

The set of all points  P = ( x ; y ; z )  with

x = fx(u,v)     ,
y = fy(u,v)     and
z = fz(u,v)     for  (u,v) from I x I          (3)

is a area in space. (3) is named parameter representation of an area.

Remarks:

The real numbers  and v  are named parameter of the area in space.

Examples:

If choosing the continuous functions

fx(u,v)  =  a1  +  b1 * u  +  c1 * v
fy(u,v)  =  a2  +  b2 * u  +  c2 * v
fz(u,v)  =  a3  +  b3 * u  +  c3 * v

for (3) with real coefficients  a1, a2, a3  and  b1, b2, b3  as well as  c1, c2, c3,  you get the parameter representation of a plane in space.

If choosing the continuous functions

fx(u,v)  =  R  *  cos(u)  *  cos(v)
fy(u,v)  =  R  *  sin(u)  *  cos(v)
fz(u,v)  =  R  *  sin(v)

for (3) with the real coefficients  R,  -  / 2 ≤ ≤  / 2  and  -  ≤ ≤ ,  you get the parameter representation of the surface of a sphere in space.

Thereby R is the sphere radius. The Parameter v can be interpreted as geographic latitude and the Parameter u as geographic longitude.

  
See also
  
 parameter lines parameter lines