Entry

Math: Transformation: 2D: Rotation: Rotation: Formula: Can you derive the 2D rotation formula?

Feb 1st, 2006 15:13
Knud van Eeden,

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--- Knud van Eeden --- 01 February 2006 - 10:23 pm -------------------

Math: Transformation: 2D: Rotation: Rotation: Formula: Can you derive 
the 2D rotation formula?

===

Method: follow the old legs of the triangle

    Given

    old:
                                X
                                |
                                |
                xOld            | yOld
          O---------------------+


    new:
                X
                 \  yOld
                   \
                     \
                     /
                   /
                 /
               /  xOld
             /
          O/  angle h

By turning the old triangle as a whole, like a rigid shop hook, over 
an angle a,
the old X arrives in a new position.

---

Calculating the new x and the new y can
be done as follows:

---

You find the new x by adding the horizontal length1 together with 
horizontal length2:


          |     X     |
          |     |\  yOld
          |     |h \  |
          |     |    \|
          |     |    /|
          |     |  /  |
          |      /    |
          |    /| xOld|
          |  /  |     |
          O/ h  |     |
          <---------->
length1 = xOld . COS(h)
                <----->
       length2 = yOld.SIN(h)

from the figure it shows that:

xNew = length1 - length2

or thus

+----------------------------------------+
|xNew = xOld . COS( h ) - yOld . SIN( h )|
+----------------------------------------+

Further,

You find the new y by adding the vertical height1 together with 
vertical height2:

                X------------
                |\  yOld
                |h \          height2 = yOld . COS( h )
                |    \
                +------------
                |  /
                |/            height1 = xOld . SIN( h )
               /| xOld
             /  |
          O/h   +------------


from the figure it shows that:

yNew = height1 + height2

or thus

+----------------------------------------+
|yNew = xOld . SIN( h ) + yOld . COS( h )|
+----------------------------------------+

===

Internet: see also:

---

Math: Transformation: Rotation: Link: Can you give an overview of 
links?
http://www.faqts.com/knowledge_base/view.phtml/aid/39299/fid/1856

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