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Parametric Equations |
The parametric equations of a
circle
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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) |
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The parametric equations of a
circle
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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, |
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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. |
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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 |
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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. |
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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 |
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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, |
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thus,
obtained is the standard equation of the ellipse. |
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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, |
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x
= x0
+ a cos t |
y
= y0
+ b sin t |
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To
convert the above parametric equations into Cartesian
coordinates, we write them as |
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x
- x0
= a cos t |
y
- y0
= b sin t |
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and divide the first
equation by
a
and the second by b, then square and add
both equations, so we get |
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Functions
contents B
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© 2004 - 2020, Nabla Ltd. All rights reserved. |