The definition of parallel curve is well defined, such that given two curves, they must remain equidistant to each other.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Orthogonal and parallel Curves

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