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.
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.
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.
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 u 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 ≤ v ≤
/ 2 and -
≤ u ≤
, 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 | |
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parameter lines |