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Polar coordinates
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Polar coordinates of a point P.
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From polar to cartesian coordinates.
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From cartesian to polar coordinates.
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Polar equation of a curve.
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Examples
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Direction of a curve in polar coordinates
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Isogonal Curves
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Tangent line parallel to the polaraxis
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Tangent line orthogonal to the polaraxis
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Investigation of a curve ; an example
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From a cartesian equation t
o
a polar equation
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From a polar equation to a cartesian equation
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More examples of curves with polar equation
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Dot product in polar coordinates
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A line and its polar equation
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A line through the pole.
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A line d not containing the pole.
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A circle and its polar equation
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A circle with the pole as center.
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An arbitrary circle
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A special circle with
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A conic section and its polar equations
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Common points of two curves
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A problem with common points of two curves
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The cause of this problem
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A solution of this problem. A special property (P).
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Equations with property (P)
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The pole is a special point
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An extensive example.
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Rotating and the polar equation.
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Asymptotes in polar coordinates
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Visualization of dr/dt
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Asymptotes
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Area with polar coordinates
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General formula
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Area calculations
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Arc length in polar coordinates
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Length of a cardioid
Polar coordinates of a point P.
In the plane we choose a fixed point O, and we call it the pole.
Additionally we choose an axis x through the pole and call it the polar axis.
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View Full DocumentOn that xaxis, there is just 1 vector
E
such that abs(
E
)=1.
The pole and the polar axis constitute the basis of the polar coordinate system.
Now, we take a point P.
On the line OP we choose an axis u.
The number t is a value of the angle from the xaxis to the uaxis.
The number r is such that
P
= r.
U
The numbers r and t define unambiguous the point P.
We say that (r,t) is a pair of polar coordinates of P.
One point P has many pairs of polar coordinates. If (r,t) is a pair of polar coordinates, (r, t + 2.k.pi) is also
a pair of polar coordinates and additionally ( r, t + (2.k+1).pi ) is a pair of polar coordinates too.
Of course, k is an integer.
Examples
Figure 1:
polar coordinates of P are (2 , 0.3) or (2 , 3.44) or (2 , 5.98) .
..
Figure 2:
polar coordinates of P are (1.4 , 3.6) or (1.4 , 0.46 ) or (1.4 , 5.8) .
..
Figure 3:
polar coordinates of P are (3 , 5.7) or (3 , 0.58) or (3 , 2.56 ) .
..
The polar coordinates of the pole O are by definition (0,t) with t perfectly arbitrary.
From polar to cartesian coordinates.
A polar coordinate system is given and point P has polar coordinates (r,t). We choose a yaxis through the
pole O and perpendicular to the xaxis. So, we have cartesian axes x and y. Call (x,y) the cartesian
coordinates of P.
According to the previous definition, the cartesian coordinates of U are (cos t, sin t).
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 Spring '11
 MarkJohnson
 Regression Analysis

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