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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: 
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 . 

The orthopolar of a circle centred on O is a quadrifolium. 

The orthopolar with respect to the axes of the circle passing by O is the curve .
For , we get the torpedo... 

... and for we get the regular bifolium. 

An example with a circle that does not pass through O. 

The orthopolar of the crosscurve is the circle . 

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© Robert FERRÉOL 2017