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I have a quadrilateral with coordinates $A(2 , 2), B(4 , 4), C(4 ,3), D(2 ,3)$. The coordinates have been converted to local coordinates with the new value as $A(0, 0), B(2 , 0), C(2 ,1), D(0 , 4)$. Can someone please how local coordinates work. Thanks

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There are many different ways to think about local coordinates, but in most cases you can imagine a function $f$ which either (1) takes every point and moves it to a new location or (2) renames each point without moving. In both cases, you can write $f(2,2)=(0,0)$, $f(4,4)=(2,0)$, $f(4,3)=(2,1)$, and $f(2,4)=(0,4)$.

In this particular case, the function in question is most likely nonlinear, so it might take some time to find an explicit expression.

Whether or not you actually need to know the function depends on the application. Is this in the context of computer science, linear algebra, differential geometry, or something else?

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Plane coordinates are usually translated and/or scaled. A translation usually adds the same value to all the x-coordinates and adds another same value to all the y-coordinates. A scaling usually multiplies all the coordinates by the same value but could scale x and y coordinates separately. Translating just moves the location of the figure while scaling changes the size of the figure.

Coordinates can be rotated.

Spherical coordinates or ellipsoidal coordinates are often projected to a plane and that's a more difficult transformation than translating or scaling.

Now a figure on a coordinate system could be morphed to meet certain requirements or conditions.

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