The order that I do these operations is not important, We must be careful when combining rotations which are in a different plane, in this case we cannot combine them by adding the bivectors, we need other notations to do this.įor instance, in the case of linear translations, I could move an object upīy 3 units (represented by ) and then move the object to the right byĤ units (represented by ). It just happens that in 3 dimensional vector space that bivectors also have three dimensions and therefore 3D rotations have 3 degrees of freedom. For more information about rotations in higher dimensions see this page. The dimension of the bivector also gives the degree of freedom of rotations in the given dimension. A bivector is the result of multiplying two vectors, the following table shows the dimension of the bivector produced by the multiplication of vectors of a given dimensions. The plane can be defined by a bivector (see directed area box on right of this page). Any rotation can be represented by projecting the object onto a 2-dimentional plane and then rotating it through an angle. Rotations in a higher number of dimensions get more complicated. Rotations in two dimensions are relatively easy, we can represent the rotation angle by a single scalar quantity, rotations can be combined by adding and subtracting the angles. You can specify the position of a point using polarĬoordinates, this is covered separately from the topic of rotations. Order of rotation operations is important.There are also things to take into account when implementing rotations such We need to do both, and we don't want to keep transferring between notations.įor more information about these different notation methods, click on the links Linear and angular motion) so they can be used for things like kinematics where Matrices have the advantage of defining both rotations and translations (i.e. The instabilities associated with the previous notations. So quaternionĪrithmetic can be used to interpolate rotations in keyframe animation, without That complex number arithmetic represents operations in 2D space. Quaternion arithmetic can represent operations in 3D space in a similar way Therefore Axis and Angle is not a very good notation to use when combining Valid to say that the total rotation is the sum of the individual rotations, When applying one rotation and then applying another rotation, it is not.The space where the normal rules don't apply. The 'gimbal lock' problem, there are singularities at certain points in.When an object is rotating it suddenly jumps from 360 degrees back to zero.Notations like euler angles and Axis and Angle are intuitive easy to understand, In keyframing, we may want to generate in-between frames so we need to interpolate We need toĭo things like, working out the effect of 2 or more subsequent rotations, also, These means of specifying rotations have different pros and cons. Representing Rotation with Translation (isometry).Rotation about origin (orthogonal transformation).There are different ways to specify and perform this rotation, these methods One method of holding this information is not suitable for all needs, therefore Rotational quantities are more difficult to represent than linear quantities, We have a reference orientation we can always define orientation as a rotation However both rotation and orientation can be defined in the same way, provided Takes a starting orientation and turns it into a possibly different orientation. I think of orientationĪs the current angular position of an object and rotation as an operation which The orientation and subsequent rotations of the object. When simulating solid 3D objects we need a way to specify, store and calculate
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