Parametric Equations
      The parametric equations of a circle
         The parametric equations of a circle centered at the origin with radius r
         The parametric equations of a translated circle with center (x0, y0) and radius r
      The parametric equations of an ellipse
         The parametric equations of an ellipse centered at the origin
         The parametric equations of a translated ellipse with center at (x0, y0)
The parametric equations of a circle
The parametric equations of a circle centered at the origin with radius r
The parametric equations of a circle centered at the origin
with radius r,
x = r cos t
y = r sin t
where,   0 < t < 2p.
To convert the above equations into Cartesian coordinates,
square and add both equations, so we get
x2 + y2 = r2
as  sin2 t + cos2 t = 1.
The parametric equations of a translated circle with center (x0, y0) and radius r
The parametric equations of a circle with center 
(x0, y0) and radius r,
x = x0 + r cos t
y = y0 + r sin t
where,   0 < t < 2p.
If we write the above equations, 
  x - x0  = r cos t
 y - y0  = r sin t
then square and add them we get the equation
of the translated circle in Cartesian coordinates,
(x - x0)2 + (y - y0)2 = r2.
 
The parametric equations of an ellipse
The parametric equations of an ellipse centered at the origin
Recall the construction of a point of an ellipse using two concentric circles of radii equal to lengths of the
semi-axes a and b, with the center at the origin as shows
the figure, then
x = a cos t
y = b sin t
where,   0 < t < 2p.
To convert the above parametric equations into Cartesian
coordinates, divide the first equation by a and the second
by b, then square and add them,
thus, obtained is the standard equation of the ellipse.
 
The parametric equations of a translated ellipse with center at (x0, y0)
The parametric equations of a translated ellipse with center (x0, y0) and semi-axes a and b,
 
x = x0 + a cos t
y = y0 + b sin t
   where,   0 < t < 2p.
To convert the above parametric equations into Cartesian coordinates, we write them as
 
x - x0 = a cos t
y - y0b sin t
 
and divide the first equation by a and the second by b, then square and add both equations, so we get
Pre-calculus contents C
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.