Parametric Equations
Parametric
equations are a pair of equations that expresses a group of quantities as
explicit functions of a number of independent variables. The independent variables
are called ‘parameters’ and denoted by ‘t’.
Parametric
Equations of a Circle
The equation
of a circle with radius a is written as x2 + y2 = a2
in Cartesian coordinates.
The equation
of circle can be written in the parametric form as follows:
x = a cos t
y = a sin t
Note that
the parametric equations are non-unique, so, the same quantities can be written
as a number of different parameters.
A single
parameter is usually represented by the parameter t.
Parametric
Equations of a Parabola
The equation
of a parabola in Cartesian form is y = x2 or x = y2.
The equation
of parabola in parametric form can be written as follows:
x = t
y = t2
This
conversion of equation from Cartesian form to parametric form is called
parameterization.
It provides
great efficiency when we integrate or differentiate the curves.
Parametric
Equations of an Ellipse
The Cartesian
equation of an ellipse is x2/a2 + y2/b2
= 1.
The equation
of ellipse in parametric form can be written as follows:
x = a cos t
y = b sin t
where x and
y are the coordinates of any point on the ellipse, and a and b are the radius
on the x-axis and y-axis, respectively.
Parametric
Equations of a Hyperbola
The Cartesian
equation of a hyperbola is x2/a2 - y2/b2
= 1.
The equation
of hyperbola in parametric form can be written as follows:
x = a sec t
y = b tan t
where x and
y are the coordinates of any point on the hyperbola, and t is the parameter.
Parametric
Equations of a Line
The Cartesian
equation of a line is ax + by = c.
The equation
of a line is typically written as y = mx + c
Where m is
the slope of the line and c is the y-intercept of the line.
The equation
of the line in parametric form can be written as follows:
x = x0
+ at
y = y0
+ bt
where (x0,
y0) are the coordinates of any point on the line, and t is the
parameter.