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ORTHOPOLAR OF A CURVE WITH RESPECT TO TWO LINES
| Notion studied by Henri Lazennec in 2015.
Homemade name. |
Given two secant lines  and  , consider the projection M of a point  on the polar line of
with respect to the lines
and .
When the point describes a curve  ,
the point M describes the orthopolar of the curve with respect to the lines
and .
In the following, we take the orthogonal lines
and , equal to the axes. In this case, if  , then  .
Therefore, we have the simple result in polar coordinates: |
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The orthopolar with respect to the axes of the curve  is the curve  . |
Examples (initial curve in blue, orthopolar in red):
| The orthopolars of lines that do not pass by O are the strophoids.
More precisely, the orthopolar with respect to the axes of the line  is the strophoid  .
It is the right strophoid for  . |
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| The orthopolar of a circle centred on O is a quadrifolium. |
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The orthopolar with respect to the axes of the circle passing by O  is the curve  .
For |
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... and for  we get the regular bifolium. |
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| An example with a circle that does not pass through O. |
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The orthopolar of the cross-curve  is the circle  . |
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© Robert FERRÉOL 2017
, we get the