CONTENTS

• 2D Rotation about the origin - Derivation
• Note on 2D rotation
• 2D Rotation about an arbitrary axis
• 3D Rotation about the origin - Derivation
• 3D Rotation about an arbitrary axis
• Related topics

Rotation is the transformation along circular paths. Rotation angle determines the amount of rotation for each vertex.

Rotation about the origin - Derivation

Let be any point on the

Then the rotation of the point P about an angle about the origin maps it onto a point

From the figure, it is obvious that

Equating () and (), we get the equation of rotation about an angle .

Equation () can be expressed in matrix form.

is the rotation of the matrix about the origin through an angle

  Example If is rotated about an angle , determine the image point.

Rotation matrix is given by

Substituting the above values in equation ()

Therefore, the image point is given by

Note on 2D rotation

1. The origin of the plane is not affected by rotation.
2. If , every point of the plane remains unaffected.
3. Rotation through an angle yields matrix of the form

   

4. The product of the rotation matrix through an angle and that through is an identity matrix

   

2D Rotation about an arbitrary axis

To rotate a point by an angle about an arbitrary axis v that passes through the origin

1. Translate all the points through so that v coincides with the origin.
2. Rotate about origin through an angle

   

3. Translate all points back to

3D Rotation about the origin

3-Dimensional rotation provide flexible views of 3D objects. Rotation in 3D is quite complex than that in 2D. In 2D, the centre of rotation is fixed. However in 3D, the axis and the angle of rotation has to be defined. When one of the positive x, y, z coordinate is chosen as the axis of rotation, then the rotation is similar to that of 2D. The general convention is that the positive rotation is counter clockwise.

Rotation about Z axis is rotating every point from X to Y. The rotation matrix is given by

Rotation about X axis is rotating every point from Y to Z. The rotation matrix is given by

Rotation about Y axis is rotating every point from Z to X. The rotation matrix is given by

3D Rotation about an arbitrary axis

Let P be any point in the YZ plane. Rotation of the point about an arbitrary axis is done as follows

1. Rotate so that it is aligned with Z axis. This is done in two steps
   1. Rotate through about Z axis so that it is in XZ plane.

      

   2. Rotate through about Y axis, so that it is aligned with Z axis.

      

2. Rotate about Z axis through the desired angle A.

   

3. Rotate to its original direction. This is done by rotating in the reverse direction through an angle of and . The rotation matrix about an arbitrary axis u through an angle A is obtained as one matrix by multiplying all the transformation matrices mentioned above.

   

Related topics

Translate points along a straight line

Scale the distance of points from an origin

Skew points relative to an axis.

Project points from 3D to 2D

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