Orthogonal and parallel Curves

[Edward Solomo]

The definition of parallel curve is well defined, such that given two curves, they must remain equidistant to each other.

For instance y = (x^2) + 4 and y = (x^2) - 8 are parallel curves in a function the maps x to y. These form parabolas whose vertical distance to one another remains constant.

In polar coordinates r = 1 and r = 2 form two circles whose radial difference remains constant at any given angle.

In parametric coordinates [ x = 2cos^2(t), y = sin^2(t)] and [ x = 8cos^2(t), y = 4sin^2(t)] form two ellipses whose radial differences remain constant at any given angle.

In general two curves f(x) and g(x) are parallel curves if f'(x) = g'(x) and f(x) is not equal to g(x). Which is the same as saying:

F[f'(x)] = g(x) - C, such that C is not equal to zero. This only applies to Cartesian graphing (mapping x to an orthogonal y-axis).

In the case of polar coordinates you get:

In general two curves f(THETA) and g(THETA) are parallel curves if f'(THETA) = g'(THETA) and f(THETA) is not equal to g(THETA). Which is the same as saying:

F[f'(THETA)]dTHETA = g(THETA)/C, such that C is not equal to one. This only applies to polar graphing (mapping some angle theta to a ray starting at the origin whose length is the resulting function on theta).

Overall two curves are parallel if the derivative of the function of the argument is the same for both of them at all times.

On orthogonal curves.

Two curves are said to be orthogonal if the derivative of the function of the argument is the negative reciprocal of the other at all times (this is my definition since I seem unable to find any math literature of the subject, however I'm sure this is what it what it would have said if I found it).

For instance the curves y = (x^2)/2 and -ln(|x|) are orthogonal curves.

Go to this graphing calculator website

http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

and type (x^2)/2 in the first line and -ln(abs(x)) in the second line to see what they look like (it's quite pretty).

In the case of Cartesian graphing given two curves f(x) and g(x), they are said to be orthogonal if f'(x) = -1/g'(x), or equivalently if f'(x)g'(x) = -1.

So given f(x), g(x) must equal -(F(1/f'(x))dx + C). So take the example y = (x^2)/2 then -F(1/x)dx = -ln(|x|) - C

Now my question turns towards polar graphing. Although the definition "Two polar curves are said to be orthogonal if the derivatives f'(theta) and g'(theta) are negative reciprocals of each other" OR "F[f'(theta)] = g(theta)/C, such that C is not equal to one" meets the requirements of orthogonal parallel curves, I am unable to think of a simple example of such curves in polar coordinates to see it visually.

Also, seeing that there is not much information on MY concept of orthogonal curves (when google searched), do these types of curves go by another name? Is there any practical use for these curves?

 

[alexfloo]

Your concepts of "parallel" and "orthogonal" here aren't really that analogous (at least in the same sense that they are in the case of lines). In the parallel case, you mean that the derivative at each point in one function is equal to the derivative at some (corresponding?) point on the other one. How these points actually correspond depends on the coordinate system.

In the orthogonal case, you say that the linearizations of those functions at a point (not all of them) are orthogonal in the sense of vectors, and intersect at that point.

Now, what "orthogonal in the sense of vectors" means doesn't depend on the coordinate system, it depends on how the inner product is defined. You've left the problem underspecified, because we do not know whether the "vectors" used to check whether the functions are orthogonal come from the values of the function, or the appearance of the function in the coordinate system. In other words, there are a few answers, and it's not necessarily clear which is best for the purpose. What is the goal?

 

[Edward Solomo]

The definition is as follows:

Given v= f(u) and v = g(u), the curves v = f(u) and v = g(u) is said to be orthogonal if and only if f'(u)*g'(u) = -1

 

[HallsofIvy]

On orthogonal curves.

Two curves are said to be orthogonal if the derivative of the function of the argument is the negative reciprocal of the other at all times (this is my definition since I seem unable to find any math literature of the subject, however I'm sure this is what it what it would have said if I found it).

Two curves are orthogonal if and only if their tangent lines at a point of intersection are perpendicular so that the derivative of one function is the negative of the derivative of the other at a point of intersection.

You may be confusing this with orthogonal families of sets such that, at any point, lines from each family intersect. For example, the family of all circles of the form $x^2+ y^2= c^2$ (circles, of any radius, with center at (0, 0)) is orthogonal to the family of all straight lines, y= cx, passing through the origin.

 

[Edward Solomo]

Two curves are orthogonal if and only if their tangent lines at a point of intersection are perpendicular so that the derivative of one function is the negative of the derivative of the other at a point of intersection.

You may be confusing this with orthogonal families of sets such that, at any point, lines from each family intersect. For example, the family of all circles of the form $x^2+ y^2= c^2$ (circles, of any radius, with center at (0, 0)) is orthogonal to the family of all straight lines, y= cx, passing through the origin.

Ignore the current definition of orthogonal curves, I am fully aware of what orthogonal curves means in the conventional sense. I am curious to know if there is a name in mathematical literature for the following definition:


The curves v = f(u) and v = g(u) are said to be orthogonal if and only if f'(u)*g'(u) = -1

This implies that the derivatives of the two functions are always perpendicular. Examples of such functions are y = (x^2)/2 and y = -ln(x) + C OR y = (x^3)/3 and y = 1/x OR (x^n)/n and y = (2-n)(x^(2-n)) for n greater than or equal to 3.

 

[alexfloo]

Yes, the definition you've selected is actually independent of the coordinate system, since it's defined in terms of derivatives. The functions x and -x are therefore orthogonal at x=0, even though in polar coordinates (where the curves would be "stretched" somewhat) it won't quite look that way.