# Orthogonal trajectories in polar coordinates

• patric44
In summary, the book asks to find the orthogonal trajectories to the curves described by the equation r^{2} = a^{2}\cos(\theta). The attempt of a solution is to eliminate "a" from the equations and get \frac{dr}{d\theta} = -\frac{1}{2}tan(\theta). However, this step is not explained and it does not seem to be equal.
patric44
Homework Statement
in the polar coordinates find the set of curves that intersects with a right angle with the set of curves describes by r^{2} = a^{2}cos(theta)
Relevant Equations
r^{2} = a^{2}cos(θ)
there is a problem in a book that asks to find the orthogonal trajectories to the curves described by the equation :
$$r^{2} = a^{2}\cos(\theta)$$
the attempt of a solution is as following :
1- i defferntiate with respect to ##\theta## :
$$2r \frac{dr}{d\theta} = -a^{2}\;\sin(\theta)$$
2- i eliminated "a" from the two equations and get :
$$\frac{dr}{d\theta} = -\frac{1}{2}tan(\theta)$$
then the book said to set ##\frac{dr}{d\theta} = -r^{2}\frac{d\theta}{dr} ## ! , this step i don't get ? why would i do that , it doesn't seem to be equal ?!

patric44 said:
Homework Statement:: in the polar coordinates find the set of curves that intersects with a right angle with the set of curves describes by r^{2} = a^{2}cos(theta)
Relevant Equations:: r^{2} = a^{2}cos(θ)

there is a problem in a book that asks to find the orthogonal trajectories to the curves described by the equation :
$$r^{2} = a^{2}\cos(\theta)$$
the attempt of a solution is as following :
1- i defferntiate with respect to ##\theta## :
$$2r \frac{dr}{d\theta} = -a^{2}\;\sin(\theta)$$
2- i eliminated "a" from the two equations

$$\frac{dr}{d\theta} = -\frac12 \tan\theta$$

I wouldn't descrive that as "eliminating $a$", so much as "eliminating $r$".

then the book said to set ##\frac{dr}{d\theta} = -r^{2}\frac{d\theta}{dr} ## ! , this step i don't get ? why would i do that , it doesn't seem to be equal ?!

There is an abuse of notation here.

A curve $r = f(\theta)$ in plane polar coordinates is given by $\mathbf{r}(\theta) = f(\theta)\mathbf{e}_r(\theta)$. The tangent vector is, by the product rule, given by $$\frac{d\mathbf{r}}{d\theta} = \frac{df}{d\theta}\mathbf{e}_r + f(\theta) \mathbf{e}_\theta$$. If another curve $r = g(\theta)$ is orthogonal to this, then we must have $$\frac{df}{d\theta} \frac{dg}{d\theta} + fg = 0,$$ but because the curves intersect we have $f(\theta) = g(\theta)$ at this point, and so $$\frac{df}{d\theta} = - g(\theta)^2 \frac{d\theta}{dg}.$$ By an abuse of notation which suppress the fact that$r$ is given by a different function of $\theta$ on each side, this could be written $$\frac{dr}{d\theta} = -r^2 \frac{d\theta}{dr}.$$

Last edited:
patric44 and etotheipi
thank you so much, it make sense now

## 1. What are orthogonal trajectories in polar coordinates?

Orthogonal trajectories in polar coordinates refer to a set of curves that intersect at right angles to each other. These curves are formed by the intersection of two families of curves, where each family is defined by a single parameter.

## 2. How are orthogonal trajectories related to polar coordinates?

In polar coordinates, the curves are defined by a radius and an angle, rather than the typical x and y coordinates. Orthogonal trajectories are formed by the curves that intersect at right angles to these polar curves.

## 3. What is the significance of orthogonal trajectories in polar coordinates?

Orthogonal trajectories in polar coordinates have many practical applications in fields such as physics, engineering, and mathematics. They can be used to model physical phenomena, such as electric and magnetic fields, and also have applications in curve fitting and optimization problems.

## 4. How do you find orthogonal trajectories in polar coordinates?

To find orthogonal trajectories in polar coordinates, you first need to determine the polar equation of the given curve. Then, using the equation for orthogonal trajectories, you can solve for the parameter that defines the family of curves. This will give you the equation for the orthogonal trajectory.

## 5. Can orthogonal trajectories in polar coordinates intersect more than once?

Yes, it is possible for orthogonal trajectories in polar coordinates to intersect more than once. This occurs when the two families of curves have multiple points of intersection. In some cases, the curves may even form a closed loop, with each curve intersecting itself at multiple points.

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