Different Coordinate Systems on the Same Plane :: Translation and Rotation of Axis, Graph Sheets

Learning Analytical/Coordinate/Cartesian Geometry
...from page 6

Different Coordinate Systems on the same Plane

The Origin and the Coordinate Lines define the nature of the coordinate system formed by the coordinate plane. These two decide the definition for the location of other points on the line. These decide the physical length between the origin and a point in the plane.

The distance between any two points in the plane would not be affected by these changes (until the unit length relating to the coordinate lines is changed).

Different coordinate systems can be defined on the same plane by varying the position of the "Origin" (O) and the Coordinate Lines x’x and y’y’.

Varying the Coordinate System

This can be done by

  • Translation of Axis
    Changing the position of the "Origin" only without changing the "Unit Length" and without tilting the coordinate lines.

    This is done by ensuring that the coordinate lines stay parallel to their original locations while the origin is shifted to the new location.

    The darker lines indicate the new position. To indicate the change the coordinate lines are represented using capital letters.

    Bringing about a change in the coordinate system in this manner is called "Translation of Axis".

    Notice that the coordinates of the points "P", "A", "C" and "Q" have changed with the change in the location of the origin.

    Where "(h,k)" represents the coordinates of the new "Origin" in the old coordinate system.

    Coordinates of a point in the new coordinate system
          = (Coordinates of the point in the old system)
                − (Respective Coordinates of the new origin)

    ⇒ X = x − (h) and Y = y − (k)
    ⇒ (X,Y) = [{x − (h)}, {y − (k)}]
    [Where "(x,y)" and "(X,Y)" represent the coordinates of the same point in the old and the new coordinate systems respectively.]

    The origin has been shifted to (2, 1) ⇒ h = 2 and k = 1

    ⇒ Coordinates (in the new system) of
    Point "A" ⇒ (XA, YA) = [{(xA) − h}, {(yA) − k}]
    = [{(2) − (2)}, {(− 2) − (1)}]
    = [{2 − 2}, {− 2 − 1}]
    = (0, − 3)
    Point "C" ⇒ (XC, YC) = [{(xC) − h}, {(yC) − k}]
    = [{(− 4) − (2)}, {(3) − (1)}]
    = [{− 4 − 2}, {3 − 1}]
    = (− 6, 2)

  • Rotation of Axis
    Changing the position of the "axes" only without changing the "Unit Length" and without changing the location of the origin.

    This is done by rotating the axis only about the origin.

    The darker lines indicate the new position. To indicate the change the coordinate lines are represented using capital letters.

    Bringing about a change in the coordinate system in this manner is called "Rotation of Axis".

    Notice that the coordinates of the points "P", "A", "C" and "Q" have changed with the change in the location of the origin.

    Where "θ" represents the angle by which the axes are rotated in an anti clockwise direction.

    The Coordinates of a point in the new coordinate system is given by the following relations:

    • X-Coordinate
      X = x cos θ + y sin θ
    • Y-Coordinate
      Y = − x sinθ + y cosθ

    ⇒ (X,Y) = [{x cos θ + y sin θ}, {x sin θ + y cos θ}]
    [Where "(x,y)" and "(X,Y)" represent the coordinates of the same point in the old and the new coordinate systems respectively.]

    The axes have been rotated by 45o ⇒ θ = 45o

    ⇒ Coordinates (in the new system) of
    Point "A" ⇒ (Xa, Ya) = [{(xa cos θ) + (ya sin θ)}, {(− xa sin θ) + (ya cos θ)}]
    = [{(2 cos 45o) + (− 2 sin 45o)},
          {(− 2 sin 45o) + ( (− 2) sin 45o)}]
    =
    [{(2 ×
    1
    √2
    		 ) + (− 2 ×
    1
    √2
    		 )}, {(− 2 ×
    1
    √2
    		 ) + (− 2 ×
    1
    √2
    		 )}]
    =
    [{
    2
    √2
    		 −
    2
    √2
    		 }, {−
    2
    √2
    		 −
    2
    √2
    		 }]
    =
    (0, −
    4
    √2
    		 )
    = (0, − 2√2)
    Point "C" ⇒ (Xc, Yc) = [{(xc cos θ) + (yc sin θ)}, {(− xc sin θ) + (yc cos θ)}]
    = [{((− 4) cos 45o) + (3 sin 45o)},
          {(− ((−4) sin 45o) + ( (3 sin 45o)}]
    =
    [{(− 4 ×
    1
    √2
    		 )+(3 ×
    1
    √2
    		 )}, {(− (− 4) ×
    1
    √2
    		 )+(3 ×
    1
    √2
    		 )}]
    =
    [{−
    4
    √2
    		 +
    3
    √2
    		 }, {−
    4
    √2
    		 +
    3
    √2
    		 }]
    =
    [{− 2√2 +
    3
    √2
    		 }, {− 2√2 +
    3
    √2
    		 }]

  • Translation and Rotation of Axis
    Changing the position of both the "axes" and the origin without changing the "Unit Length".

    This is done by rotating the axis and shifting the origin.

    Translation First Rotation First Final Resultant

    This can be done by rotating first and then translating (Or) by translating first and then rotating. In both the cases the result would be the same.

    The darker lines indicate the new position. To indicate the change the coordinate lines are represented using capital letters.

    Bringing about a change in the coordinate system in this manner is called "Translation and Rotation of Axis".

    The coordinates of the points "P", "A", "C" and "Q" change with the change in the coordinate system.

    Where "θ" represents the angle by which the axes are rotated in an anti clockwise direction and (h,k) the coordinates of the New Origin with respect to the old coordinate system.

    The Coordinates of a point in the new coordinate system is given by the following relations:

    • X-Coordinate
      X = (x − h) cos θ + (y − k) sin θ
    • Y-Coordinate
      Y = − (x − h) sin θ + (y − k) cos θ

    ⇒ (X,Y) = [{(x − h) cos θ + (y − k) sin θ}, {− (x − h) sin θ + (y − k) cos θ}]
    [Where "(x,y)" and "(X,Y)" represent the coordinates of the same point in the old and the new coordinate systems respectively.]

    The axes have been rotated by 45o ⇒ θ = 45o

    The origin has been shifted to (− 2, 2) ⇒ h = − 2 and k = 2

    ⇒ Coordinates (in the new system) of
    Point "A" ⇒ (Xa, Ya) = [{((xa − h) cos θ) + ((ya − k) sin θ)},
          {((− (xa − h) sin θ) + ((ya − K) cos θ)}]
    = { [ (2√2) + (− 2√2)], [ {− (2√2)} + {(− 2√2)} ]}
    = [2√2 − 2√2], [ − 2√2 − 2√2]
    = (0, − 4√2)

    A (2, − 2) ⇒ xa = 2 and ya = − 2.

    The origin has been shifted to (− 2, 2) ⇒ h = − 2 and k = 2

    ((xa − h) cos θ) =
    [(2) − (− 2) ×
    1
    √2
    		 )]
    =
    [(2 + 2) ×
    1
    √2
    		 ]
    =
    [
    4
    √2
    		 ]
    = 2 √2
    ((ya − k) sin θ) =
    [(− 2) − (2) ×
    1
    √2
    		 )]
    =
    [(− 2 − 2) ×
    1
    √2
    		 ]
    =
    [−
    4
    √2
    		 ]
    = − 2 √2

    ⇒ Coordinates (in the new system) of
    Point "C" ⇒ (Xc, Yc) = [{((xc − h) cos θ) + ((yc − k) sin θ)},
          {((− (xc − h) sin θ) + ((yc − K) cos θ)}]
    =
    [ (− √2) + (
    1
    √2
    		 ) ], [ − (− √2) + (
    1
    √2
    		 )]
    =
    (
    −2 + 1
    √2
    		 ,
    2 + 1
    √2
    		 )
    =
    (−
    1
    √2
    		 ,
    3
    √2
    		 )

    C (− 4, 3) ⇒ xc = − 4 and yc = − 3.

    The origin has been shifted to (− 2, 2) ⇒ h = − 2 and k = 2

    ((xc − h) cos θ) =
    [{ (− 4) − (− 2) } ×
    1
    √2
    		 )]
    =
    [(− 4 + 2) ×
    1
    √2
    		 ]
    =
    [−
    2
    √2
    		 ]
    = − √2
    ((yc − k) sin θ) =
    [{(3) − (2)} ×
    1
    √2
    		 )]
    =
    [(1) ×
    1
    √2
    		 ]
    =
    [
    1
    √2
    		 ]
     

Graph Sheet » Representing the coordinate Plane on it

By coordinate plane we mean, a plane with a defined coordinate system. It is made up of two perpendicular coordinate lines with a common origin. We generally come across such a presentation on a white paper or on a graph sheet.

Graph sheets enable us to define the coordinate system on a plane and then locate the position of an object (generally considered as a point) on a plane within that coordinate system.

Graph Sheets are plain white papers with a number of horizontal and vertical lines evenly spaced forming a grid. It represents a plane.

One of the horizontal lines is chosen as the horizontal axis and another vertical line passing through it chosen as the vertical axis. The even space between two consecutive horizontal lines or vertical lines is the "Unit Length".

Based on horizontal line and the vertical line chosen to represent the axes, the graph sheet may be used to represent either all the four quadrants or only one of the quadrants.

First Quadrant Second Quadrant Third Quadrant Fourth Quadrant


Author Credit: The Edifier

...continued page 8




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