 The blue dot is the
point (x,y) on the ellipse
being traced out in the complex way illustrated by the
blue
line
which has length equal to the
semimajor axis "a"
of the ellipse
and is directly tracing out a
circle.
The x component of the
vector
is clearly
x(t)=a*cos(E)
where E is the angle
measured counterclockwise
from the positive xaxis to the
blue
line
and in astro jargon
is called the
eccentric anomaly.
From the ellipse equation
(with "b" being the semiminor axis),
we find:
(x/a)**2+(y/b)**2 = 1 .
(y/b)**2=1(x/a)**2 = 1cos(E)**2 = sin(E)**2
(y/b) = ± sin(E) where the valid case is clearly +sin(E)
y = b*sin(E)
The ordered pair [x(E)=a*cos(E),y(E)=b*sin(E)]
constitute the
parametric equation
of an ellipse.
The radius R to the blue dot is
R=sqrt[(a*cos(E))**2+(b*sin(E))**2] .
Only in the case of a circle
where a=b is R=a
and the
parametric equation
reduces to [x(E)=a*cos(E),y(t)=a*sin(E)].

One of the great ironies
of the
history of astronomy
is that ancient authority on ellipses
Apollonius of Perga (c.262c.190 BCE)
is believed to have invented
epicycle models
for planetary orbits instead of
using ellipses which are the true
shape of planetary orbits to 1st order
(see Wikipedia: Deferent and epicycle: History).
Note Apollonius did NOT use
formulae in a modern sense.
He worked purely in words
and diagrams.
His understanding of ellipses
was based on his study of
conic sections which
are illustrated in Image 2 and described in the next list item.
 Image 2 Caption: "The conic sections
(circle,
ellipse,
parabola,
and hyperbola)
are 2dimensional figures formed by the
intersection
of a plane
with a cone at different
angles.
The theory of these figures
was developed extensively by the
ancient Greek mathematicians, surviving
especially in works such as those of
Apollonius of Perga (c.262c.190 BCE).
The conic sections pervade
modern mathematics."
(Slightly edited.)
 Apollonius'
"goof" in NOT using
ellipses for
planetary orbits
is explained by
noting that the planetary data in his day did NOT compel
elliptical orbits and
epicycle models seemed adequate and
were simpler to understand.
 Note in particular that
ancient Greek astronomers
had difficulty dealing with nonuniform motions which
planets have on their
elliptical orbits.
The ancient Greek astronomers
did NOT have
calculus
which is needed for the mathematical description of nonuniform motions
in general.
The approach they took to deal with nonuniform motions in
Heavens
was decompose them into
uniform circular motions
that compounded reproduced to some
accuracy/precision
the motions of the
planets.
In Aristotelian cosmology, the
uniform circular motions
were of compounded
celestial spheres that
carried the planets around.
In Ptolemaic system, the
uniform circular motions
were those on
the deferent and epicycle
of epicycle models.
 Compounded
uniform circular motions
are illustrated in the
animation below.
 Image 3 Caption: An animation
where "The red
curve
is an epicycloid
traced as the small circle
(radius
r = 1) rolls around the outside of the large
circle
(radius
R = 3)."
This animation
does NOT depict an actual
planet
epicycle model since
there are 3
apparent retrograde motions
and actual
planet
epicycle models
had only 1
apparent retrograde motions
when in
inferior conjunction
(for inferior planets)
or in opposition
(for superior planets).
 Before
1600
astronomers
working in the western Eurasian tradition of
astronomy
continued to rely on compounded
uniform circular motions
in their
Solar System models.
 The hold of compounded
uniform circular motions
was only broken by
Johannes Kepler (15711630)
who was one of the leading mathematical innovators of the
Scientific Revolution
of the 16th and
17th centuries
and also a person who was willing to do long tedious
calculations to deal with
nonuniform motions.
 In modern times, epicycles
have been used as mathematical tool in analyzing
the orbits of
astrobodies
in galaxies
(see Wikipedia:
Lindblad resonance).
However, these modern
epicycles have
been criticized too and maybe are on their way out of modern use
(Francis, C. 2011, arXiv:0911.1594v1,
"Lindblad's epicycles  valid method or bad science?").