I have a polar equation defined as:
$r = ae^{θ \tan m}$
where, $a$ and $m$ are constants, $θ$ is the angle between the horizontal axis from the origin $(x_c,y_c)$ to the coordinate. $e$ refers to exponent.
The equation is illustrated below:
How can I express the above polar equation in Cartesian form? (i.e. I want to get the Cartesian equation that describes the line which passes through the coordinates as indicated by the broken red lines in the above illustration)
The motive behind this is to find the gradient of the broken red line given specified coordinates which lie along the broken red line.
$\endgroup$1 Answer
$\begingroup$Your curve is a logarithmic spiral $r= ae^{b\theta} $ with $b=\tan m$. You can express such curve in a parametric form as: $$ x(t)=ae^{bt}\cos t $$ $$ y(t)=ae^{bt}\sin t $$
This is the simpler way to write out the cartesian coordinates.
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