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Coordinate Geometry.
Coordinate Geometry.  
QUESTION: Dear Prof Randy

http://en.wikipedia.org/wiki/Analytic_geometry
http://www.mathopenref.com/coordintro.html
http://www.math.com/school/subject3/lessons/S3U1L2GL.html
http://c-schools.net/PPT/Coordinate%20Geometry.ppt
http://www.youtube.com/watch?v=7h74ad7IWiI
http://www.math.wisc.edu/~robbin/461dir/coordinateGeometry.pdf
http://www.wolframalpha.com/examples/CoordinateGeometry.html

What could be the Impact to different disciplines and applications
of mathematics if Point (x,y) is replaced with Point (y,x).

i.e. x coordinate = Horizontal axis
    y coordinate = Vertical axis


Now instead of this standard convention, implement the following
given below

i.e. x coordinate = Vertical axis
    y coordinate = Horizontal axis


For examples :  

1st Quadrant Point A(3,2) will become Point A(2,3)
2nd Quadrant Point B(-2,3) will become Point B(3,-2)
3rd Quadrant Point C(-3,-4) will become Point C(-4,-3)
4th Quadrant Point D(2,-3) will become Point D(-3,2)

i.e. because the Points are now represented as y,x instead of x,y,
the Plotting of Points A,B,C,D on graph paper will be now different.

Can there be any impact while plotting Graphs viz Sine,Cosine, Tangent waves etc, vector algebra and other applications of mathematics similar to Coordinate Geometry if Point F (x,y) is represented as Point F(y,x) ?

Awaiting your reply,

Thanks & Regards,
Prashant S Akerkar

ANSWER: The impact of switching (x,y) to (y,x) can be viewed in a couple of ways.

First, it doesn't really matter what letters you assign to the axes as long as you are consistent.

Second, flipping the coordinates can be seen as just flipping a point in the plane about the line y = x, that is, creating a mirror image of the point. You can see this easily by plotting the points you listed in your question. This transformation is also given by the simple matrix manipulation

( 0  1 ) ( x )     ( y )
( 1  0 ) ( y ) = ( x ).

Third, switching axes will flip a curve in the x-y plane. For example, an ellipse will have the major and minor axes interchanged. Less symmetric curves will also be flipped across the y = x line.

Fourth, you can, in a sense, "flip" the functional relation y = f(x) to be x = f^(-1)(y), which is to take the inverse. However, I don't think this is what you have in mind.

In short, flipping the coordinates doesn't necessarily do much of anything. Any consequences can be accomodated by applying standard mathematical techniques.

Randy

---------- FOLLOW-UP ----------

Sine Wave
Sine Wave  
QUESTION: Dear Prof Randy

Thank you.

http://en.wikipedia.org/wiki/Sine_wave

Can the interchange of x - horizontal axes to y - horizontal axis and y - vertical axis to x - horizontal axes will effect the Sine, Cosine, Tangent and other trigonometric curves plotting ?

i.e. Will the Sine Wave plotted now will be seen in a different wave ?
This could be because the coordinates (0, 30 degree, 45 degree, 60 degree, 90 degree, 120 degree, 150 degree, 180 degree or respective conversion to radiant etc) will be plotted now on the Y axis ?

Awaiting your reply,

Thanks & Regards,
Prashant S Akerkar

ANSWER: Instead of propagating horizontally, the sine wave will propagate vertically. So, take your image and rotate it 90 degrees and you'll see what it looks like with sin(y) = x.

---------- FOLLOW-UP ----------

Sine Inverse Curve
Sine Inverse Curve  
QUESTION: Dear Prof Randy

Thank you.

http://www.intmath.com/analytic-trigonometry/7-inverse-trigo-functions.php

Can the interchange of x - horizontal axes to y - horizontal axis and y - vertical axis to x - horizontal axes will effect the Sine Inverse, Cosine Inverse, Tangent Inverse and other trigonometric curves plotting ?

i.e. Will the Sine Inverse Wave plotted now will be seen in a different wave ?

Awaiting your reply,

Thanks & Regards,
Prashant S Akerkar

Answer
Your picture really tells it all. If you switch x and y, the curve reflects about the y = x line. This is actually a very good representation of the "effect".

You should think about what the curves represent. The sine of a number can be interpreted as a ratio, as in trigonometry. The arcsin (sine inverse) of a number x (which, by the way, has to be -1<x<1) can be interpreted an an angle. Two different animals. So calling arcsin a "wave" is not really accurate.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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