Return to Answer

$$ r= 2C \cos \theta,~~ r'= - 2C \sin \theta~d \theta $$

Dividing

$$\frac{r}{r'}=- \cot \theta~$$

A differential right triangle of sides $( r,r',\sqrt{r^2+r^{'2}})$ can be drawn and it has trig relation $$\tan \psi=\frac{r}{r'}$$

Replacing $\psi$ by $ \pi/2-\psi$ for orthogonal trajectory with CCW sign convention... and with luck in separating variables in polar coordinates,

$$\frac{r'}{r}= \cot \theta,~~ \frac{dr}{r}=\cot \theta ~ d \theta ~;$$ Integrate with arbitrary constant $2A$

$$\log r = \log \sin \theta + \log 2A, ~ r= 2 A \sin \theta; $$

The given set are all circles through the origin centered on x-axis and the O.T. are all circles through the origin centered on y-axis.

It is recommended using another letter symbol for the O.T. set.

$$ r= 2C \cos \theta,~~ r'= - 2C \sin \theta~d \theta $$

Dividing

$$\frac{r}{r'}=- \cot \theta~$$

A differential right triangle of sides $( r,r',\sqrt{r^2+r^{'2}})$ can be drawn and it has trig relation $$\tan \psi=\frac{r}{r'}$$

Replacing $\psi$ by $ \pi/2-\psi$ for orthogonal trajectory with CCW sign convention... and with luck in separating variables in polar coordinates,

$$\frac{r'}{r}= \cot \theta,~~ \frac{dr}{r}=\cot \theta ~ d \theta ~;$$ Integrate with arbitrary constant $2A$

$$\log r = \log \sin \theta + \log 2A, ~ r= 2 A \sin \theta; $$

The given set are all circles through the origin centered on x-axis diameter $2C$ and the O.T. are all circles through the origin diameter $2A$ centered on y-axis.

It is recommended using another letter symbol for the O.T. set to visualize them as arbitrary, but different.

$$ r= 2C \cos \theta,~~ r'= - 2C \sin \theta~d \theta $$

Dividing

$$\frac{r}{r'}=- \cot \theta~$$

A differential right triangle of sides $( r,r',\sqrt{r^2+r^{'2}})$ can be drawn and it has trig relation $$\tan \psi=\frac{r}{r'}$$

Replacing $\psi$ by $ \pi/2-\psi$ for orthogonal trajectory with CCW sign convention... and luck with separating variables,

$$\frac{r'}{r}= \cot \theta,~~ \frac{dr}{r}=\cot \theta ~ d \theta ~;$$ Integrate with arbitrary constant $2A$

$$\log r = \log \sin \theta + \log 2A, ~ r= 2 A \sin \theta; $$

The given set are all circles through the origin centered on x-axis and the O.T. are all circles through the origin centered on y-axis.

It is recommended using another letter symbol for the O.T. set.

$$ r= 2C \cos \theta,~~ r'= - 2C \sin \theta~d \theta $$

Dividing

$$\frac{r}{r'}=- \cot \theta~$$

A differential right triangle of sides $( r,r',\sqrt{r^2+r^{'2}})$ can be drawn and it has trig relation $$\tan \psi=\frac{r}{r'}$$

Replacing $\psi$ by $ \pi/2-\psi$ for orthogonal trajectory with CCW sign convention... and with luck in separating variables in polar coordinates,

$$\frac{r'}{r}= \cot \theta,~~ \frac{dr}{r}=\cot \theta ~ d \theta ~;$$ Integrate with arbitrary constant $2A$

$$\log r = \log \sin \theta + \log 2A, ~ r= 2 A \sin \theta; $$

The given set are all circles through the origin centered on x-axis and the O.T. are all circles through the origin centered on y-axis.

It is recommended using another letter symbol for the O.T. set.

$$ r= 2C \cos \theta,~~ r'= - 2C \sin \theta~d \theta $$ Dividing $$\frac{r}{r'}=- \cot \theta~$$ A differential right triangle of sides $( r,r',\sqrt{r^2+r^{'2}})$ has $$\tan \psi=\frac{r}{r'}$$

Replacing $\psi$ by $ \pi/2-\psi$ for orthogonal trajectory with CCW sign convention

$$\frac{r'}{r}= \cot \theta,~~ \frac{dr}{r}=\cot \theta ~ d \theta ~;$$ Integrate with arbitrary constant $2A$

$$\log r = \log \sin \theta + \log 2A, ~ r= 2 A \sin \theta; $$

The given set are all circles through the origin centered on x-axis and the O.T. are all circles through the origin centered on y-axis.

It is recommended using another letter symbol for the O.T. set.

$$ r= 2C \cos \theta,~~ r'= - 2C \sin \theta~d \theta $$

Dividing

$$\frac{r}{r'}=- \cot \theta~$$

A differential right triangle of sides $( r,r',\sqrt{r^2+r^{'2}})$ can be drawn and it has trig relation $$\tan \psi=\frac{r}{r'}$$

Replacing $\psi$ by $ \pi/2-\psi$ for orthogonal trajectory with CCW sign convention... and luck with separating variables,

$$\frac{r'}{r}= \cot \theta,~~ \frac{dr}{r}=\cot \theta ~ d \theta ~;$$ Integrate with arbitrary constant $2A$

$$\log r = \log \sin \theta + \log 2A, ~ r= 2 A \sin \theta; $$

The given set are all circles through the origin centered on x-axis and the O.T. are all circles through the origin centered on y-axis.

It is recommended using another letter symbol for the O.T. set.