Four yuan--combined with "real time rendering" in the four-dollar part of the

The four-dollar number, which was generated in 1843, is an extension of the plural, so it contains several complex operations. It was not used in graphic science until 1985.

The advantage of the four-dollar number is that the four-dollar number is more intuitive and convenient relative to the matrix and Euler angles. Four of dollars can also be used as interpolation in some directions, and Euler angles may not be very well done.

The four-dollar number is represented by four digits. In general, the first three and rotated axes are closely related, and the last one relates to the angle of rotation. Here are some mathematical backgrounds, which are important for the change of the four-dollar later.

Note that the four-tuple is an extension of the complex number, then it can be represented as: û= (Â, v), where V is a real number, and  is a virtual part, and â= i*qx + j*qy + k*qz wherein, I j k are imaginary parts, they are calculated as

I2 = j2= K2 =-1, JK =-kj=i, ik =-ki = j, ij =-ji = k, because â contains three parts, so the operation of some vectors can also be used on the imaginary part of the four-dollar number.

Multiply for two four-dollar numbers, such as:

The other calculations are as follows:

wherein the conjugate is conjugated meaning. We can introduce the division operation according to the norm operation:

And some other operational rules:

For a unit of four yuan, we can also use the following method in the expression, which is also a more intuitive way to express rotation changes

The calculation process is as follows, where UQ is required to be a unit vector:

Another way to express this is to solve the log and square operations:

A visual understanding of the unit's four-dollar number is as follows:

The above is the basic knowledge background of four yuan. The following is a description of the change in the four-dollar representation.

We use only one subset of the four-dollar number, which is very simple and powerful, with a number of four-dollar numbers for various rotational changes. The equation of change is:

Its q is a unit of four, in the form of Equation 4.36, and P is a four times the second point or vector, each part of P directly replaces the various parts of the four-dollar number, forming a new four-dollar number. Note that Q is a unit of four yuan, so q-1 = q*. Moreover, the unit four-dollar number and real numbers do not change the effect of four-dollar number, so the effect of the four-Q and-Q effects. In the process of converting from a matrix to a four-dollar number, it may be possible to get Q as well. The visual comprehension of Equation 4.40 is the diagram above. The following formula represents a change of two four-dollar number to one P.

Four-dollar and matrix conversions. Because using a matrix on hardware is faster than using a four-tuple, you can use a four-dollar number at the application level, but use a matrix at the computational level. The conversion between them is:

However, at the application level to use the four-dollar number, but also from the matrix to convert to four of dollars. By observing, you can see the following equation

The other three can be calculated until QW, QX, qy, or QZ. The following equation can be known through observation:

where TR (Mq) means the trace of the matrix, which is the same as the diagonal of the matrix. In the process of calculation, we know that we should try to avoid dividing by a smaller number. So 4.46 is not necessarily the best algorithm. Through the following comparison

It is known that the size of the QX qy QZ QW can be understood by comparing several elements of the matrix and the size of U. If the QW is the largest, then directly use 4.46 can be. If not, you can use the following method to figure out the maximum element value, and then decide to use the 4.44 element as the denominator.

Four yuan--combined with "real time rendering" in the four-dollar part of the